Kinetic Theory of the Alfvén Wave Acceleration of Auroral Electrons
Robert L. Lysak and Yan Song
University of Minnesota
Abstract
Recent observations have indicated that in addition to the classical “inverted-V” type electron
acceleration, auroral electrons often have a field-aligned distribution that is broad in energy and
sometimes shows time dispersion indicating acceleration at various altitudes up the field line.
Such acceleration is not consistent with a purely electrostatic potential drop, and suggests a wave
heating of auroral electrons. Alfvén waves have been observed on auroral field lines carrying
sufficient Poynting flux to provide energy for such acceleration. Calculations based on the linear
kinetic theory of Alfvén waves indicate that Landau damping of these waves can efficiently
convert this Poynting flux into field-aligned acceleration of electrons. At high altitudes along
auroral field lines that map into the plasma sheet boundary layer, the plasma gradients are
relatively weak and the local kinetic theory can describe this wave-particle interaction. At lower
altitudes, the gradient in the Alfvén speed becomes significant, and a non-local description must
be used. A non-local theory based on a simplified model of the ionospheric Alfvén resonator is
presented. For a given field-aligned current, the efficiency of the wave-particle interaction
increases with the ratio of the thermal velocity of the electrons to the Alfvén speed at high
altitudes. These calculations indicate that wave acceleration of electrons should occur at and
above the altitude where the quasi-static potential drops form.
Introduction
The auroral zone is one of the most intriguing regions in the Earth’s magnetosphere. A
variety of plasma physics processes occur on auroral field lines, from large-scale MHD
phenomena to the microphysics of plasma instabilities, solitary waves, and radio emissions. It is
well known that the aurora occurs on magnetic field lines along which field-aligned currents
(FAC’s) flow (e.g., Birkeland, 1908; Iijima and Potemra, 1976). It has long been believed that
field-aligned potential drops associated with the FAC’s cause the field-aligned acceleration of
electrons related to the formation of the aurora (e.g., Winckler et al., 1958; Evans, 1968, 1974;
Mozer and Fahleson, 1970; Frank and Ackerson, 1971; Block, 1972; Gurnett and Frank, 1973;
Mozer et al., 1977, 1980). Theories to explain the formation of parallel electric fields and
auroral acceleration have fallen into two categories: kinetic models that follow the adiabatic
motions of auroral electron populations and fluid models that describe the time development of
the parallel electric fields. However, most of the kinetic theories for the parallel electric field are
quasi-static in nature. While such models show that parallel electric fields can self-consistently
exist, they do not describe the time-dependent evolution of the parallel electric field. Fluid
models can describe this evolution, but do not directly model the particle acceleration that
produces the aurora. Clearly, a synthesis of the two approaches is necessary for a complete
understanding of auroral acceleration.
The main features of the classic “inverted-V” particle acceleration can be well described
by a quasi-static potential drop; however, some aspects of auroral acceleration cannot be
understood in the context of these quasi-steady models. In particular, a number of rocket and
satellite observations have identified nearly field-aligned electron beams precipitating into the
auroral zone (Johnstone and Winningham, 1982; Arnoldy et al., 1985; Robinson et al., 1989;
McFadden et al., 1986, 1987, 1990, 1998; Clemmons et al., 1994; Lynch et al., 1994, 1999;
Knudsen et al., 1998; Chaston et al., 1999, 2000). Upgoing and counterstreaming electrons have
also been observed (Sharp et al., 1980; Klumpar and Heikkila, 1982; Burch et al., 1983; Block
and Fälthammar, 1990; Boehm et al., 1995), especially in the downward auroral current region
(Marklund et al., 1994, 1995, 1997, 2001; Carlson et al., 1998; Ergun et al., 1998). Features in
the non-field-aligned part of the distribution function, such as electron conics (Menietti and
Burch, 1985; Temerin and Cravens, 1990; André and Eliasson, 1992; Burch, 1995; Eliasson et
al., 1996) are also evidence that quasi-static potential drops do not explain all facets of auroral
electron acceleration. Thus, it appears that auroral electron acceleration operates in at least two
different modes: acceleration in a quasi-static potential drop and time-dependent acceleration
associated with wave activity.
Both steady-state and time-dependent auroral acceleration require a source of energy.
The Poynting flux associated with the auroral current system transfers energy from the outer
magnetosphere (i.e., the magnetopause and/or the plasma sheet) to the auroral acceleration
region (e.g., Mozer et al., 1980). Correlations between electric and magnetic field measurements
on low-altitude satellites and rockets have indicated that the Poynting flux is mostly downward,
although upward Poynting fluxes are sometimes observed (Sugiura et al., 1982; Sugiura, 1984;
Primdahl et al., 1987). Measurements of the quasi-static Poynting flux have indicated that
fluxes of 1-10 mW/m2 are common (Kelley et al., 1991; Gary et al., 1994, 1995; Thayer et al.,
1995; Kletzing et al., 1996) and that fluxes of over 40 mW/m2 are sometimes observed (Gary et
al., 1994). Small Poynting fluxes (< 1 mW/m2) associated with kinetic Alfvén waves have been
observed on Freja (Volwerk et al., 1996). Recent observations from Polar (Wygant et al., 2000;
Keiling et al., 2000) have shown that the Poynting flux associated with Alfvénic fluctuations at
the plasma sheet boundary layer at 4-6 RE have Poynting fluxes of 1-2 mW/m2, which map to
values of up to 100 mW/m2 at ionospheric altitudes. Moreover, these Alfvénic fluctuations also
occur on small perpendicular scales, and are associated with field-aligned electron acceleration
(Wygant et al., 2002). Observations of large parallel electric fields from FAST also indicate that
such fields are found in regions of strong Alfvénic activity (Ergun et al., 2001). Although
energy input to the aurora from the kinetic energy of inflowing particles cannot be ignored, it is
clear that the electromagnetic energy input is a significant energy source for auroral acceleration,
especially during the dynamic processes that first establish the potential drop. The
electromagnetic energy input from the outer magnetosphere has recently been emphasized by
Haerendel (2000) and Song and Lysak (2001).
Models of the quasi-static process are most often based on the adiabatic motion of
charged particles in the magnetic mirror geometry, such as the Knight (1973) relation, which
describes the response of the electrons in such a geometry to the presence of a parallel potential
drop. Such a model must be combined with a model of quasi-neutrality (e.g., Chiu and Schulz,
1978; Stern, 1981) or a full solution of the Poisson equation (Chiu and Cornwall, 1980) in order
to establish a self-consistent model of the potential drop. Such models have had a resurgence
recently in light of results from the FAST satellite (Temerin and Carlson, 1998; Rönnmark,
1999; Schriver, 1999; Jasperse et al., 2000). A detailed model of this sort has been presented by
Ergun et al. (2000), who used 8 populations of ionospheric and magnetospheric particles in a
Vlasov model to construct a self-consistent potential profile. This model shows that one or two
narrow-scale transition layers exist with large, localized parallel electric fields.
On a more localized scale, the development of parallel potential drops has been studied
by the use of BGK solutions of plasma double layers and solitary waves (e.g., Block, 1972; Swift,
2
1975; Borovsky, 1984, 1993a; Koskinen and Mälkki, 1993; Berthomier et al., 1998). While these
steady-state solutions show that such structures can exist, they do not describe how they form.
The time evolution of these structures has been studied through particle-in-cell (PIC) simulations
(e.g., Goertz and Joyce, 1975; Sato and Okuda, 1981; Barnes et al., 1985). Since these PIC
simulations generally model a limited region of space, the results depend on which boundary
conditions are imposed on the simulation box. Such localized models, while essential for
understanding the detailed behavior within the region of parallel electric fields, avoid the
important question of how the energy that accelerates the auroral particles is provided. A global
scale electrostatic model of the auroral field line has been presented by Schriver (1999). This
model is driven by the injection of ion fluxes at the upper end of the system; however, since this
simulation is electrostatic, the electromagnetic energy input is not included.
In contrast to these kinetic models of auroral acceleration, the time development of
auroral currents and fields has often been considered in terms of Alfvén wave propagation
(Mallinckrodt and Carlson, 1978; Goertz and Boswell, 1979; Lysak and Dum, 1983; Seyler,
1988). While the Alfvén wave does not carry a parallel electric field that can accelerate auroral
particles according to ideal MHD, a parallel electric field does develop when the spatial scale
size perpendicular to the background magnetic field becomes comparable to the ion acoustic
gyroradius (Hasegawa, 1976) or the electron inertial length (Goertz and Boswell, 1979). At
altitudes of less than about 2-3 RE where much of the auroral acceleration is thought to occur, the
electron inertial effect is most important (Lysak and Carlson, 1981). This has led to a variety of
models that have considered the development of parallel electric fields due to electron inertia
(e.g., Seyler, 1988; Lysak, 1993, 1998; Lysak and Song, 2000; Rönnmark and Hamrin, 2000;
Seyler and Wu, 2001). In particular, some models have invoked Alfvén field line resonances that
collapse to the electron inertial scale to explain auroral arc formation (Rankin et al., 1993, 1994,
1999a,b; Streltsov and Lotko, 1995, 1996, 1999; Streltsov et al., 1998; Rankin and Tikhonchuk,
1998; Bhattacharjee et al., 1999). In addition to these global resonances, a resonant cavity can
be formed by the rapid decrease of the plasma density above the ionosphere, which causes an
increase in the Alfvén speed from less than 1000 km/s in the ionosphere to greater than 100,000
km/s in the acceleration region at a few thousand kilometers altitude. The resonant frequency of
this cavity, which has been called the ionospheric Alfvén resonator (IAR) by Polyakov and
Rapoport (1981; see also Trakhtengertz and Feldstein, 1981, 1984, 1987, 1991; Lysak, 1986,
1988, 1991, 1993), is in the 0.1-1.0 Hz range. This density gradient also implies that the region
where the drift velocity corresponding to a given field-aligned current maximizes is also
localized, as is the effect of the electron inertial term. It should be noted that waves in this
frequency range have been observed in the auroral zone by Viking (Block and Fälthammar,
1990; Marklund et al., 1990), Freja (Louarn et al., 1994; Grzesiak, 2000), and FAST (Sigsbee et
al., 1998; Chaston et al., 1999), and at higher altitudes by Geotail (Sigsbee et al., 1998).
Although Alfvén wave models have been very useful in describing the evolution of
auroral fields and currents, fluid models of Alfvén waves are inadequate to self-consistently
describe particle acceleration due to the fields of these waves. One approach to model the
particle acceleration due to Alfvén waves has involved PIC simulations that take Alfvén wave
dynamics into account (e.g., Otani et al., 1991; Goertz et al., 1991; Hui and Seyler, 1992;
Silberstein and Otani, 1994; Clark and Seyler, 1999; Génot et al., 1999, 2000, 2001a,b). These
works have had some success at modeling features of auroral particle acceleration, but are still
limited to small scales, so that they do not reflect the large-scale resonant structures described
above. Large scale modeling of electron acceleration by Alfvén waves has been considered by
3
test particle models (Kletzing, 1994; Thompson and Lysak, 1996; Chaston et al., 1999, 2000,
2002; Kletzing and Hu, 2001). This work, in particular that of Chaston et al. (2000, 2002) and
Kletzing and Hu (2001), has shown that the dispersed electron signatures frequently observed in
field-aligned electron bursts can be reproduced by the Alfvén wave model. At higher altitude,
electron acceleration has been associated with observations of small-scale Alfvén waves at the
plasma sheet boundary layer (Wygant et al., 2002) and has been shown to be consistent with the
kinetic Alfvén wave dispersion relation in the regime where electron pressure rather than
electron inertia supports the parallel electric field (Lysak and Lotko, 1996; Lysak, 1998).
These models based on the electron inertial term treat the entire electron population as
cold, neglecting the fact that hot magnetospheric plasma exists on auroral field lines, and indeed,
can be the dominant component of the plasma in the auroral acceleration region (e.g.,
Strangeway et al., 1998). In an effort to include the kinetic effects of hot electrons, Rankin et al.
(1999b) and Tikhonchuk and Rankin (2000) have developed a model in the context of field line
resonances that includes a kinetic description of electrons bouncing between mirror points in the
two hemispheres. While this model is a first step toward including electron kinetic effects into
an Alfvén wave model, it is restricted to field line resonance cases and only considers one of the
electron populations that exist on auroral field lines.
The purpose of this paper is to apply this kinetic approach to Alfvén waves that propagate
along the plasma sheet boundary layer, as observed by Polar (Wygant et al., 2000, 2002; Keiling
et al., 2000), and their extension into the ionosphere, where they are influenced by the
ionospheric Alfvén resonator. At high altitudes, the plasma parameters vary slowly, and a local
approach can be used to investigate the kinetic effects, as in previous work (Lysak and Lotko,
1996; Lysak, 1998). Near the ionosphere, however, the plasma parameters vary rapidly, and the
electron kinetics must be treated non-locally. While Rankin et al. (1999b) and Tikhonchuk and
Rankin (2000) concentrated on the effects of mirroring particles, the present paper will
concentrate on the kinetics of passing electrons that propagate from the magnetosphere to the
ionosphere, as well as ionospheric particles that propagate into the magnetosphere. In the
remainder of this paper, we will first review the theory of kinetic Alfvén waves in a uniform
plasma, concentrating on the effects on the electron distribution. Then the kinetic theory of
Alfvén waves in the ionospheric Alfvén resonator will be presented, based on a simplified model
that neglects the dipole magnetic geometry as in Lysak (1991). Although this model does not
include all aspects that are important in this problem, it does give a simplified framework in
which the kinetic theory can be understood. The paper will conclude with a discussion of the
results and an outline of future work to extend the results presented here.
Local Kinetic Theory of Alfvén Waves
Recently, Polar observations of strong Alfvénic turbulence in the plasma sheet boundary
layer (PSBL) have drawn renewed attention to Alfvénic acceleration of auroral electrons.
Wygant et al. (2000) noted that large amplitude (~ 100 mV/m) Alfvén waves are seen by Polar in
the PSBL at 4-6 RE, on field lines that are conjugate to the aurora as seen by the UVI instrument.
The Poynting flux estimated in these waves is the order of 1 mW/m2 at the spacecraft, which
maps to over 100 mW/m2 at the ionosphere. The waves are seen most often when Polar crosses
the PSBL during the expansion phase of substorms (Keiling et al., 2000, 2001), and expand into
the central plasma sheet during large storms (J. R. Wygant, personal communication).
In addition to these larger-scale waves, there is also a considerable amount of fine
4
structure in the Alfvén waves, and these smaller-scale waves have a larger ratio of the wave
electric to magnetic fields. This enhanced ratio could indicate that the waves have been modified
by kinetic effects (Lysak, 1998), or that waves reflected from an insulating ionosphere or from
the auroral acceleration region (Vogt and Haerendel, 1998) are present. The absence of any
significant phase shift between the electric and magnetic fields and the lack of reflected Poynting
flux suggest that kinetic effects are the most important influence on this ratio. Wygant et al.
(2002) used this ratio and the measured plasma parameters together with the theoretical results of
Lysak (1998) to find that the perpendicular wavelength of these waves was comparable to the
electron inertial length. The theory was then used to estimate the ratio of the parallel to
perpendicular potentials in the wave, and the estimated parallel potential was consistent with the
energy of field-aligned electrons measured by the Hydra instrument on Polar. Angelopoulos et
al. (2002) compared the Polar observations with those at Geotail at 18 RE, and found that the
Poynting flux was much larger at Geotail, indicating that dissipation of the wave energy had
taken place between the two satellites.
These observations suggest that an evaluation of energy flow and dissipation of kinetic
Alfvén waves is in order. Such an evaluation should start with an analysis of the local dispersion
relation, which can be written in matrix form as (Lysak and Lotko, 1996):
 ε xx − nz2
nz n x   E x 
(1)

  = 0
ε zz − nx2   Ez 
 n z nx
where the elements of the dielectric tensor are given by:
c 2 1 − Γ 0 ( µi )
ε xx = 1 + 2
µi
VA
Γ (µ )
ε zz = 1 + 02 2 e (1 + χZ ( χ ) )
k z λ De
(2)
(3)
where µi ,e = k x2ρi2,e , χ = ω/k||ae, ρi (ρe) is the thermal ion (electron) gyroradius, ae = (2Te / me )1/ 2 is
the electron thermal speed, λ De = (ε 0Te / ne 2 )1/ 2 is the Debye length, Z(χ) is the plasma
dispersion function and Γ 0 (µ) = e −µ I 0 (µ) is the modified Bessel function. As usual, the
dispersion relation is found by taking the determinant of the matrix in equation (1). For the
parameters of the Polar observations, we have χ 1 and VA c ; thus, expanding the Z function
and using the Padé approximation for the Bessel function (e.g., Johnson and Cheng, 1997), the
dispersion relation can be written as:
(4)
ω2 ≈ k z2VA2 1 + k x2ρi2 + k x2ρ2s 1 − i πχ 


where ρ2s = Te mi / e 2 B 2 is the ion acoustic gyroradius. Note the imaginary term that gives the
Landau damping of the wave. Expanding the Z function in the cold electron limit, χ 1 , gives
the inertial limit of the Alfvén wave dispersion relation, including the effects of hot ions
1 + k x2ρi2
ω2 ≈ k z2VA2
(5)
1 + k x2 λ e2
(
)
where λe is the electron inertial length. It should be noted that there is essentially no Landau
damping in the cold plasma limit since the wave phase velocity is on the tail of the electron
distribution.
5
The polarization relations between the components of E and B can be found by solving
the full equation (1) and by using Faraday’s law (Lysak, 1998). In the warm electron limit, these
relations become:
k k ρ2
Ez
≈ − z x2 s2 1 − i πχ
(6)
Ex
1 + k x ρi
(
)
Ex
1 + k x2ρi2
≈ VA
By
1 + k x2ρi2 + k x2ρ2s
while for cold electrons, the wave components are related by
(7)
k k λ2
Ez
≈ z x2 e2
Ex 1 + k x λ e
Ex
≈ VA
By
(8)
(1 + k ρ )(1 + k λ )
2 2
x i
2 2
x e
(9)
Note that the parallel electric field Ez is produced by the electron pressure term in the warm
electron limit, through the ion acoustic gyroradius ρs, or by the effect of electron inertia for cold
electrons, and not, as is sometimes stated, by ion gyroradius effects. Indeed, the ion gyroradius
effect appears in the denominator of equation (6), indicating that hotter ions tend to reduce the
parallel electric field in the warm plasma case. Finite ion gyroradius does not affect the parallel
electric field in the cold electron limit, as is seen in equation (8). It should also be noted that the
sign of the parallel electric field reverses between the warm electron limit, equation (6), and the
cold electrom limit, equation (8). Equations (7) and (9) demonstrate that the ion gyroradius
effect increases the Ex/By ratio, as was indicated by the Polar observations. This ratio also has a
small imaginary part in the warm plasma case; however, the full numerical solution indicates that
it is negligible except when the wave is strongly damped. It should be noted that equations (4)(9) give the limiting cases of these quantities in the warm and cold plasma limits; these ratios are
plotted without these approximations in Lysak and Lotko (1996) and Lysak (1998).
We wish to consider the energy flow and dissipation produced by these waves. The
starting point for such a calculation is Poynting’s theorem for the wave energy averaged over a
wave period, which we can write in the form
∂W
(10)
+ ∇ ⋅ S = −P
∂t
where the wave energy, Poynting flux, and power dissipation are defined by
I
2
Re E* ⋅ ε ⋅ E
B
1
1
S=
W=
+
P = Re j* ⋅ E
(11)
Re E* × B
4
4µ0
2µ0
2
We consider a situation in which we assume that a wave with frequency ω and perpendicular
wave number kx is emitted from a source at z = 0. Then we consider the propagation of the wave
away from this source. Since this process is steady-state after averaging over the wave period,
the time derivative term in equation (10) goes to zero. Thus, in this scenario, the wave Poynting
flux decays away from the source due to the Landau dissipation P. To calculate the Poynting
flux and the dissipation, the dispersion relation given by equation (1) is solved for the complex
parallel wave number kz, and equation (1) together with Maxwell’s equations can be used to
(
)
(
6
)
(
)
calculate the Poynting flux and power dissipation associated with the wave. It should be noted
that the non-vanishing wave field components in this case are Ex, By, and Ez.
Although the Poynting flux of the ideal MHD Alfvén wave is strictly along the magnetic
field, the kinetic effects cause the wave energy to move across field lines as well. The parallel
component of the Poynting flux is
2
VA B y
Ex
1
*
Sz =
Re Ex By =
Re
(12)
2µ0
2µ 0VA
Ex
while the perpendicular Poynting flux is given by:
2
 k V   VA By * k E 
Ex
1
ω
*
x z

Re Ez By = −
Re  z A  
(13)
Sx = −

2µ0
2µ 0VA k xVA
ω   Ex  k z Ex 



Comparing equations (12) and (13), we can relate the two components of the Poynting flux as
 Re  k V / ω V B / E * k E / k E  
 ( z A ) ( A y x ) ( x z z x )   ω
(14)
Sx = −  
Sz

k
V
Re
/
V
B
E
(
)
x
A
A y
x




As noted above, the imaginary part of By/Ex is very small, so the perpendicular Poynting flux is
closely related to the real part of the scaled parallel electric field. Calculation of the
dimensionless quantity in braces in equation (14) shows that this quantity is less than 0.3. The
quantity ω / k xVA = λ x / VA τ , where λx is the perpendicular wavelength and τ is the wave period.
The Polar observations (Wygant et al., 2002) indicate that the perpendicular wavelength is the
order of 10 km, while a typical value of the Alfvén speed in the plasma sheet boundary layer is
1000 km/s and wave periods may be estimated to be the order of 1 minute. This suggests that
ω / k xVA < 10−3 ; therefore, the deviation of the Poynting flux from the parallel direction is quite
small for these parameters.
The energy dissipation can be determined by looking at the work done on the electrons
by the wave:
2
 k V   VA By *  k E  
Ex
1
1
ω
*
*
x z
Re −ik x By Ez =
Im  z A  
(15)
P = Re jz Ez =
 

ω   Ex   k z Ex  
2
2µ0
2µ0VA VA



Note that only the parallel current and electric field enter into this expression since the
polarization current is out of phase with the perpendicular electric field. Again, it is useful to
relate this to the parallel Poynting flux, so we can write
 Im  k V / ω V B / E * k E / k E  
 ( z A ) A y x ( x z z x )   ω
P= 
(16)
 Sz
Re VA By / Ex

 VA


This power dissipation is related to the imaginary part of the scaled parallel electric field, which
is directly related to the Landau damping, as can be seen from equations (4) and (6). Note that
this scenario gives a characteristic parallel scale length of VA / ω = VA τ / 2π . For VA = 1000 km/s
and a wave period of 60 s, this scale length is 10,000 km.
Figure 1 plots the dimensionless power dissipation given by the factor in braces in
(
)
(
)
(
)
(
7
)
equation (16) in the same format as was used in Lysak and Lotko (1996) and Lysak (1998) as a
function of the parameters ve2 / VA2 = Te / meVA2 and k x c / ω pe . Since the plasma sheet ions are
generally somewhat hotter than the electrons, this plot shows results for temperature ratios Ti/Te
= 1, 2, 5, and 10. Since the dissipation rate is proportional to the Poynting flux and since the
perpendicular Poynting flux is negligible equation (10) in the steady state can be written as
∂S z
= − P = −2κS z
(17)
∂z
This equation indicates that the Poynting flux decays spatially as S z ( z ) = S z (0)e −2 κz . Thus, κ can
be interpreted as a spatial damping rate of the wave, with the factor of 2 being present since the
Poynting flux is a quadratic quantity. This decay of the Poynting flux is due to the conversion of
the electromagnetic energy to the energy of the Landau-resonant electrons.
According to the results of Wygant et al. (2002), the Polar observations occur in a regime
of this plot where ve2 / VA2 = Te / meVA2 is between 3 and 30. The observed ratio of the electric field
to the magnetic electric field is in the range 2-10, indicating that kxc/ωpe ranges from 0.5 to 3 if
Ti/Te = 1 and 0.1 to 1 for a temperature ratio of 10. For these parameters, it can be seen that the
normalized power dissipation is about 0.1 for equal temperatures and 0.01 for hotter ions. Thus,
the local heating of electrons is favored by cooler ion temperatures. For the canonical numbers
of 1000 km/s for the Alfvén speed and 1 minute for the wave period, this indicates that the decay
scale length L = 1/ 2 κ ~ 15 RE for the equal temperature case, and up to 10 times larger for Ti/Te
= 10. Over this distance, 64% of the wave Poynting flux is converted to electron energy. This
would indicate that 10% of the wave Poynting flux is converted to particle energy over a distance
of 1.5-15 RE. Since κ scales as 1/ VA τ , the efficiency of the energy conversion is higher for
lower Alfvén speeds and shorter wave periods.
A difficulty in these considerations is that it is difficult to determine the actual frequency
of the wave. The time variation of the fields measured by Polar is most likely due to the Doppler
shift due to the relative motion of the spacecraft and plasma sheet. Theoretical considerations
combined with the ratio of the perpendicular electric and magnetic perturbations can be used to
estimate the perpendicular wavelength, as in Wygant et al. (2002); however, only the parallel
phase velocity, i.e., ω/kz can be determined from the theory and available measurements, but not
ω or kz separately.
This ambiguity also leads to uncertainty in the applicability of the local model to the
Polar measurements. The local approximation requires that the wavelength of the wave be
smaller than the scale of variations in the Alfvén speed along the auroral field line. Thus, low
frequency global modes, such as those considered by Rankin et al. (1999b), must be treated with
a non-local model. However, higher frequency waves, such as the 0.9 Hz waves seen on Geotail
by Sigsbee et al. (1998), will be well described by the local model. Generally, in the tail where
the typical scales along field lines may be many Earth radii and the Alfvén speeds are less than
1000 km/s, waves with frequencies above 10 mHz (wave periods less than 100 s) can be
described by the local approximation. The Polar measurements are in the transition region where
the local approximation is beginning to break down. In this inner magnetospheric region, the
local theory must be replaced by a non-local theory that takes into account the mode structure
along field lines. A first step toward such a model will be described in the next section.
8
Non-local kinetic theory of the ionospheric Alfvén resonator
As noted above, when the gradients in the plasma parameters become comparable to a
wavelength, the local theory described above is not valid. Use of such a non-local theory is
particularly important when one considers the kinetic theory of fundamental field line resonances
(e.g., Rankin et al., 1999b; Tikhonchuk and Rankin, 2000), or when treating modes of the
ionospheric Alfvén resonator, as will be presented here. These modes intrinsically have
variations on the same scale as the inhomogeneities of the plasma. In such cases, one must
directly integrate the Vlasov equation, taking into account the inhomogeneous structure of the
plasma. Integration over velocities can then lead to an integral equation giving the field-aligned
current as an integral over the parallel electric field. If the field-aligned current is known, this
integral equation must be solved for the parallel electric field. This procedure is outlined in the
Appendix.
Carrying out this procedure is in general quite complex, and all of the different possible
particle orbits should be included. Tikhonchuk and Rankin (2000), for example, have included
trapped orbits in a dipolar magnetic field. For our studies of waves on field lines that connect to
the PSBL, this formulation may not be appropriate since the field lines may be open, or long
enough so that the particles may scatter before they mirror. In addition, the presence of an
equilibrium parallel electric field leads to the presence of multiple particle populations that can
be reflected by the parallel electric field or trapped between the magnetic mirror and the
electrostatic mirror, as in the calculations of Chiu and Schulz (1978), Chiu and Cornwall (1980),
and Ergun et al. (2000).
To avoid these complications while still obtaining new information about the kinetic
effects, we will adopt the simplified model of the ionospheric Alfvén resonator on straight
magnetic field lines, as in the work of Lysak (1991). We will calculate the wave fields and the
field-aligned current according to this model, and then solve the Vlasov equation to determine
the electron kinetic response. This background model assumes long perpendicular wavelengths
so that to lowest order the parallel electric field may be neglected. The effect of the kinetic
particles will then be treated as a perturbation on this background. In this approximation, the
wave equations can be determined from Faraday’s Law and Ampere’s Law including the
perpendicular polarization current. Assuming perturbations at a frequency ω, these equations
become
∂By
∂Ex
iω
(18)
= iω B y
= iωε xxµ 0 Ex ≈ 2
Ex
∂z
∂z
VA ( z )
where the perpendicular dielectric tensor component εxx is given by equation (2). Note that the
approximate form in the second equation holds in the long wavelength limit, where the Bessel
function factors in equation (2) reduce to unity, and for VA2 c 2 , which does not always hold on
auroral field lines. When this inequality is not valid, the Alfvén speed should be replaced by
VA2 /(1 + VA2 / c 2 ) in equation (18).
The plasma density in this model is given by
n ( z ) = nI e − z / h + ε 2
(19)
(
)
Note that the scale height h is typically a few hundred kilometers in the auroral zone. In this case
the Alfvén velocity and the electron inertial length λe can be written as
λ2
V2
(20)
VA2 ( z ) = − z / hAI 2
λ e2 = − z / heI 2
e
e
+ε
+ε
9
where VAI and λeI are the Alfvén velocity and inertial length, respectively, at the ionosphere. The
solutions to the wave equations (18) with the Alfvén speed profile (20) can be written in terms of
Bessel functions. There are two solutions, one corresponding to a wave with downward
Poynting flux, and the other with upward Poynting flux. Let us consider the downwardpropagating wave, which has the solution (Lysak, 1991)
δEx = −ik x Φ 0 J iξ ε ξ0 e − z / 2 h
(21)
0
(
)
where ξ0 = 2hω / VAI and the amplitude of the wave is given in terms of the perpendicular
potential drop Φ0. Now the magnetic perturbation can be written as
kΦ ξ
i ∂δEx
J i′ξ ε ( ξ )
(22)
δBy = −
=− x 0
VAI ξ0
ω ∂z
0
where the prime on the Bessel function indicates differentiation with respect to ξ = ξ0 e − z / 2 h .
Note that these expressions are valid in the limit k x2 λ 2M 1 ; for general values of this parameter,
a numerical solution of the eigenmode equations must be obtained as in Lysak (1993). Now
using Ampere’s Law, the field-aligned current becomes
ik x δBy ik x2 Φ 0 ξ
J i′ξ ε ( ξ )
(23)
δjz =
=
µ0
µ 0VAI ξ0
If that cold electrons carry the field-aligned current, the electron equation of motion gives the
parallel electric field
m
Φ
(24)
δEz ,cold = e2 ( −iωδjz ) = 0 k x2λ e2ξJ i′ξ ε ( ξ )
2h
ne
This cold plasma expression will be used for comparison with the kinetic expression below.
Now we can evaluate the perturbation in the distribution function. We will neglect any
background parallel electric fields, so that the unperturbed parallel velocity is constant, and so
we can write the travel times as τ( z1 , z2 ) = ( z2 − z1 ) / vz . First, consider the upgoing particles,
where the perturbed distribution function is given by equation (A8). These particles leave the
ionosphere before interacting with the wave, so that we can take δf+(0) = 0. Thus, the upgoing
distribution function can be written as
z
∂f
iω z − z ′ / v
(25)
δf + ( z ) = − q 0+ ∫ dz ′δEz ( z ′ ) e ( )
∂W 0
The solution for the downgoing particles is more complicated. Strictly speaking, there is no
upper boundary z = L, since the model extends to infinity. However, at high enough altitudes,
where e − z / h ε 2 , the solution becomes a plane wave solution (Lysak, 1991). Thus, at this
location we can adopt the uniform plasma solution as described in the previous section. The
perturbed distribution function in this case is given by
∂f
ivz
δf − ( L ) = −
q 0 δEz ( L )
(26)
ω + iη − k z vz ∂W
where k z = −ω / VM is the wave number of the wave in the uniform region as can be
approximated by equation (4). Note that this wave number is negative since the wave is
propagating in the –z direction. The small positive imaginary number η has been inserted to take
into account the Landau prescription for resolving the resonance at ω = kzvz. As usual the limit
η → 0+ is implicit. Inserting equation (26) into the solution given by equation (A9) gives the
0
0
z
10
distribution for negative velocities
L
∂f 0− 
ivz
− iω( L − z ) / v
− iω z ′ − z / v 
δf − ( z ) = q
e
δEz ( L ) + ∫ dz ′δEz ( z ′ ) e ( ) 
(27)
−
∂W  ω + iη − k z vz
z

In order to compare these results with those from the local theory, we need to calculate
the current carried by these hot particles, as described in the Appendix. Inserting equations (25)
and (27) into equation (A11), we find
∞
L
 ∂f z
∂f
iω z − z ′ / v
iω z ′ − z / v
δjz = − q 2 ∫ dv v  0+ ∫ dz ′e ( ) δEz ( z′ ) + 0− ∫ dz ′e ( ) δEz ( z′ )
∂W z
 ∂W 0
0
(28)

i ∂f 0−
vVM
iω L − z / v
−
e ( ) δEz ( L ) 
ω ∂W v − VM (1 + iη / ω)

Note that this expression simplifies if the upgoing and downgoing distribution functions are the
same
L
∞
∂f iω z − z′ / v
δjz ( z ) = − q 2 ∫ dz ′δEz ( z ′ ) ∫ dv v 0 e
∂W
0
0
(29)
∞
∂f 0
vVM
iq 2
iω L − z / v
+
dv v
e ( ) δEz ( L )
ω ∫0
∂W v − VM (1 + iη / ω)
z
z
Choosing a form for the distribution function, we can evaluate these integrals. We take a
Maxwellian for the electron distribution,
me − mv / 2T
me −W / T
(30)
= n0
f 0 = n0
e
e
2πTe
2πTe
2
z
e
e
It can be seen that ∂f 0 / ∂W = − f 0 / Te . Then the first of the two integrals in equation (29)
becomes
L
∞
n q2
me me v dv − mv / 2T iω z − z′ / v
(31)
δj1 ( z ) = 0 ∫ dz ′δE z ( z ′ )
e
e
me 0
2πTe ∫0 Te
Note that this integral can be written in the form of equation (A12)
2
e
L
δj1 ( z ) = ∫ dz ′σ ( z , z ′ ) δE& ( z ′ )
(32)
0
where the conductivity kernel is given by
 2iω ∞
iω z − z ′ / a ζ 
(33)
d ζ ζe −ζ e
−

∫
 πae 0

where the thermal velocity ae = (2Te / me )1/ 2 and the integral has been written in terms of the
dimensionless variable ζ = v/ae. Note that the factor in front of the brackets is simply the
conductivity for the cold plasma case. The dimensionless integral in equation (33) is plotted in
Figure 2 as a function of the parameter ω z − z ′ / ae . It can be seen that the main contribution to
σ ( z, z′) =
in0 q 2
me ω
2
e
this integral occurs when z − z ′ <
~ 2πae / ω , i.e., roughly the distance a thermal particle travels
in one wave period. It can also be verified that the bracketed term integrated over the spatial
coordinates is 1. Thus, in the limit of a cold plasma, the bracketed term becomes a delta-function
11
and the cold plasma conductivity is recovered.
Turning to the boundary value integral represented by the second line of equation (29),
similar manipulations can cast this in the form
∞

in0 e 2 me
me vdv − m v / 2T 
v2
iω( L − z ) / v
δj2 ( z ) =
−
e
v

 δEz ( L ) e
∫
me ω 2πTe 0 Te
(34)
 v − VM (1 + iη / ω) 
2
e
e
≡ σ L ( z , L ) δEz ( L )
This expression, evaluated at z = L, would give the local, kinetic response for the plasma if the
integral were over all velocities; however, in this case, only the downgoing particles are
included. It can be verified that the first term in the square brackets gives the local, cold plasma
response. It can also be seen that when the resonant velocity is much greater than the particle
velocity, the second term is smaller by the ratio v/VM.
As an example of the response of the hot plasma to Alfvén waves in the ionospheric
resonator, consider a model in which ε = 0.1, i.e., the Alfvén speed is 10 times large in the
magnetosphere than in the ionosphere. The model Alfvén speed profile for this case is shown in
Figure 3. If we assume a typical value for the scale height of 600 km (0.1 RE), the range of this
calculation goes roughly to 3 RE geocentric distance. The cold ionospheric density will be given
by the exponential term in equation (19), while the kinetic particles will have a density
n0 = nI ε 2 . We will take the density and temperature of the upgoing and downgoing kinetic
particles to be the same, although in general they need not be. We will assume that the fieldaligned current is given by the Alfvén resonator model given by equation (23). We note that
while this model is certainly not an exact description of the auroral field line, it contains the
necessary features to illustrate the physics of the electron kinetic effects on a realistic field line.
A generalization of this model to a more realistic situation will be the subject of future work.
Note that the cold ionospheric particles have a response given by the local conductivity,
σcold = incold ( z )e 2 / me ω . Then the parallel electric field is given by the solution of the integral
equation
L
δjz ( z ) = σcold ( z ) δEz ( z ) + ∫ dz ′σ ( z , z ′ )δEz ( z ′ ) + σ L ( z , L ) δEz ( L )
(35)
0
At high altitudes, the solution to this equation must become equal to the kinetic solution of the
previous section, which can be derived from equation (3)
in0e 2 2
δjz = −2
χ (1 + χZ ( χ ) ) δEz ≡ σ kin δEz
(36)
me ω
where χ = ω / k& ae is found by solving the dispersion relation given by equation (1) using the
magnetospheric parameters. At low altitudes, the conductivity is given by the cold plasma
response in the ionosphere. Thus the first trial solution can be written as
δjz ( z )
(37)
δE0 ( z ) =
σcold ( z ) + σ kin
Note that the kinetic conductivity is smaller by a factor ε2 at low altitudes, but dominates at high
altitudes where the cold density goes to zero. Thus, this trial solution satisfies the boundary
conditions for this problem.
This trial solution itself is only an approximate solution to equation (35), and so it is
supplemented by a series of Bessel functions that constitute a complete set for functions that
12
vanish at z = 0 and as z → ∞ (e.g., Jackson, 1975, section 3.7). Thus the full trial solution is
N
(
δEz ( z ) = c0 δE0 ( z ) + ∑ ci J1 η1i e− z / 2 h
i =1
)
(38)
Here η1i denotes the ith zero of J1(x). A least-squares minimization procedure is used to calculate
the set of ci that gives the best solution to equation (35).
We will present the results of this calculation in terms of an effective conductivity given
by the current divided by the local value of the parallel electric field, normalized to the cold
plasma conductivity
δjz ( z ) in ( z ) e 2
(39)
σeff ( z ) =
=
σˆ eff ( z )
δEz ( z )
me ω
Figure 4 plots the value of this dimensionless conductivity σˆ eff for four values of the thermal
speed of the hot particles, ae = (2Te / me )1/ 2 , normalized by the Alfvén speed in the ionosphere.
The real part of the effective conductivity is plotted as a solid line, while the dashed line gives
the imaginary part. A value of k x2 λ 2M = 0.01 has been used in these plots. Note that this value
corresponds to the left boundary in Figure 1. Figure 4 shows results for ae/VAI = 1, 3, 10, and 30,
2
= 0.005 to 4.5 in Figure 1. (Recall that the ve in Figure 1 are defined
corresponding to ve2 / VAM
without a factor of 2 in the thermal speed.) It should be noted that a typical Alfvén speed in the
ionosphere is about 600 km/s, which is the thermal velocity for a 1 eV electron. Thus, these four
plots roughly correspond to energies of 1, 10, 100, and 1000 eV. It can be seen in panel (a) that
for ae/VAI = 1, the effective conductivity is nearly unity, indicating that the plasma response is
essentially that of a cold plasma. Panel (b) shows that the real part of the conductivity in
enhanced somewhat by thermal effects, while panels (c) and (d) show that the real part is
reduced when the thermal speed is comparable or greater than the magnetospheric Alfvén speed.
Note that the real part of the normalized conductivity refers to the inductive part of the parallel
electric field, as in the electron inertial effect; thus, values of the real part less than 1 indicate
regions where the kinetic effects will lead to a localized enhancement of the inductive parallel
electric field.
The dashed lines in Figure 4 give the imaginary part of the normalized conductivity,
which corresponds to a contribution to the field-aligned current that is in phase with the parallel
electric field. It is this component that contributes to the dissipation of the wave. The total
power dissipation averaged over a wave cycle can be given by
2
1
1
1 n(z)e
2
2
*
Im σˆ eff δEz
P = Re δjz δEz = Re σeff δEz = −
(40)
2
2
2 me ω
This dissipation accelerates the electrons along the field line, and provides damping to the wave.
Figure 5 plots this dissipation normalized to the incident Poynting flux as a function of the
thermal speed for the runs shown in Figure 4, with ae/VAI = 1, 3, 10, and 30 corresponding to the
solid, dotted, dashed, and dash-dot lines, respectively. For higher values of z, the dissipation
becomes uniform and is significant for the larger values of the thermal speed; thus, this damping
is the extension of the Landau damping discussed in the previous section. In addition to this
uniform Landau damping, there is also dissipation for z/2h between 3 and 5, in the transition
region where the Alfvén speed gradient is significant. Note that for the hottest case, there is a
region in which the dissipation is negative. In this region the local current and electric field are
in opposite directions locally and the electrons lose energy. The total integrated dissipation
(
)
13
relative to the input Poynting flux is 0, 0.04%, 3.0%, and 7.7% for the four cases. This increase
is due to the increase of the parallel electric field due to the thermal effects, since these
calculations are taken under conditions of equal current. It should also be noted that increasing
the perpendicular wave number leads to an increase in this damping; however, in this case the
simplifying assumptions made here that the kinetic effects do not modify the wave structure are
no longer justified.
Discussion
The model described above gives a preliminary calculation of hot plasma effects on the
ionospheric Alfvén resonator. It should be noted that the typical value of the Alfvén speed in the
ionosphere is comparable to the thermal speed of a 1 eV electron (600 km/s). Thus, a plasma
sheet temperature of 100 eV-1 keV corresponds to ae/VAI ~ 10-30 in Figures 4 and 5. The scale
height 2h is typically in the 1000 km range, so the transition region at z ~ 10h is at about 1 RE
altitude. Note that we have also made the assumption that the presence of the hot particles has
not modified the mode structure or the spatial profile of the parallel electric field. This is in the
spirit of plasma wave calculations that are done in the limit of weak damping or growth (e.g.,
Krall and Trivelpiece, 1973, section 8.6) in which the fluid equations are used to find the real
part of the dispersion relation while the full kinetic description is used to calculate growth or
damping. A more complete treatment, which will be left for future work, would be to consider
the modifications of the mode structure caused by the kinetic effects.
Another significant simplification in the present calculations is the use of a straight
magnetic field line model. The presence of the dipole magnetic geometry, and its extension into
the plasma sheet for PSBL field lines, modifies the zero-order particle orbits that should be
considered in equations (A3)-(A9). The magnetic mirror will lead to the reflection of hot
incoming particles, as was considered in Tikhonchuk and Rankin (2000). In addition, the
linearization of the Vlasov equation should be around a self-consistent equilibrium including a
zero-order parallel electric field, such as that calculated by Ergun et al. (2000). An important
feature that is introduced in such a model is the electrostatic reflection of ionospheric electrons
by an upward parallel electric field. It is worth noting that the bounce time of superthermal
electrons emitted from the ionosphere as backscattered primaries or secondary particles is
comparable to the 1 second wave period of waves in the Alfvén resonator. Thus, it is likely that
a bounce resonance feature would result. Such features are in fact seen in the test particle
calculations of Thompson and Lysak (1996). Indeed, as the evolution of an auroral arc
progresses, the density and temperature of such backscattered and secondary electrons might be
expected to increase and the evolution of Alfvén waves would be modified.
It is worth noting that the type of electron acceleration produced here is distinct from the
inverted-V monoenergetic electron beams that have been extensively studied. These calculations
indicate that a region of heated electrons will be produced at altitudes at and above about 1 RE in
altitude, and these heated electrons will propagate down the field lines. These electrons are
heated primarily parallel to the magnetic field, and so they should overcome magnetic mirroring
and precipitate into the ionosphere. It is worth noting that Polar Hydra data shows parallel
heating of electrons in the region of large Alfvénic turbulence (Wygant et al., 2002). This event
shows a bi-directional electron beam, which may be due to the non-resonant sloshing of
electrons in the wave, along with a uni-directional population in the direction of wave
propagation, suggestive of a Landau resonant tail. More coordinated wave-particle observations
14
of this type would be useful in determining the flow of energy from the plasma sheet into the
auroral zone.
A final question that remains is the source of the waves produced at the plasma sheet
boundary layer. If these waves are being dissipated by the acceleration of auroral electrons, they
will be damped if not regenerated in some way. There are a number of possible mechanisms for
producing these waves. First of all, since there are gradients perpendicular to the magnetic field
in the PSBL, linear phase mixing will produce small perpendicular wavelength waves if large
scale Alfvén waves are present in the boundary layer region. Such arguments have long been
invoked to explain the narrow scales of field line resonances (e.g., Mann et al., 1995) and in the
ionospheric Alfvén resonator (Lysak and Song, 2000). Shear Alfvén waves can be produced by
linear mode conversion at the plasma sheet boundary, as in the original suggestion by Hasegawa
(1976), who postulated that kinetic Alfvén waves can be produced from surface waves on the
boundary. Variations on this model invoke mode conversion from plasma sheet waveguide
modes (Allan and Wright, 1998) or from compressional waves in the plasma sheet (Lee et al.,
2001). Nonlinear mode conversion can also occur during bursty reconnection (Song and Lysak,
1989b, 1994). Linking these waves to reconnection is an appealing idea since it is thought that
the PSBL maps back to a distant neutral line. Such bursty reconnection in the presence of a By
component in the lobe field would naturally emit shear Alfvén waves due to the conservation of
magnetic helicity (Song and Lysak, 1989a; Wright and Berger, 1989).
Clearly, further study of the energy flows in the magnetotail and the dissipation of wave
energy into the acceleration of auroral particles is necessary. The observations of Wygant et al.
(2002) and Angelopoulos et al. (2002) have shown that a great deal of the energy flow in the tail
is mediated by wave propagation through this region. Observations from Polar, Geotail and now
Cluster should be coordinated to give a complete picture of these energy flows. Auroral
satellites such as FAST as well as sounding rocket and ground observations can view the results
of the dissipation of this wave energy into the acceleration of auroral particles. Thus, future
observations should give further motivation to theoretical discussions of energy flow and
dissipation in the tail.
In summary, recent observations as well as theoretical considerations presented here
suggest that Alfvén waves can be efficient accelerators of auroral electrons. Such acceleration
can be distinguished from the inverted-V electron acceleration in that it is confined in pitch angle
but has a broad spectrum of energy. Landau dissipation both in the relatively uniform plasma
sheet as well as its analog in the ionospheric Alfvén resonator can convert significant amounts of
wave energy into the field-aligned acceleration of auroral electrons. While the calculations
presented here only give a preliminary picture of these wave-particle interactions, they provide
the basis for more sophisticated further studies of this interaction.
Appendix: Evaluation of the kinetic conductivity
To evaluate the non-local kinetic conductivity, we must integrate the Vlasov equation
over unperturbed particle orbits. The first step in such a calculation is to calculate the
perturbations in the electron distribution function, assuming that the profile of the parallel
electric field perturbation is known. Then, this perturbation is integrated over all velocities to
calculate the current. The result from this procedure will be an integral equation given the fieldaligned current in terms of an integral over the parallel electric field, which can then be solved
for the parallel electric field when the field-aligned current is given. The following procedure
15
follows a similar procedure outlined by Rankin et al. (1999b) and Tikhonchuk and Rankin
(2000).
First, we must know the unperturbed particle orbits. For electrons interacting with lowfrequency Alfvén waves, the magnetic moment of the electron orbit is an adiabatic invariant, and
can be taken to be constant. A second constant of the motion can be taken to be the bounce
action for periodic orbits, or equivalently, the total energy of the electron, given by
1
W = mvz2 + µB ( z ) + qΦ ( z )
(A1)
2
where as usual µ = mv⊥2 / 2 B is the magnetic moment and Φ is the electrostatic potential. Note
that while the total energy is conserved for the unperturbed orbit, when Alfvén waves are present
it can be changed by the parallel electric field of the wave. Thus, the first-order Vlasov equation
in this case can be written as
∂ 
dW ∂f 0± ( µ, W )
∂
=0
(A2)
 + vz  δf ± ( z , µ, W , t ) +
∂z 
∂W
dt
 ∂t
Here z represents the path length along the field line, and the background and perturbation
distribution functions are separated into particles with positive parallel velocity (“+” subscript)
and those with negative parallel velocity (“–” subscript). Here we will take positive velocity to
be upward away from the ionosphere. Note that in general the unperturbed particle orbits may
have turning points where they are reflected either by the magnetic gradient or by the
background parallel electric field.
At such points zt the distributions must
satisfy δf + ( zt ) = δf − ( zt ) . The energy change is given by the work done by the parallel electric
field, dW / dt = qδEz vz .
Equation (A2) can be solved by integrating over the unperturbed trajectory of a particle
over past times
t
∂f
(A3)
δf ± ( z, t ) = δf ± ( z0 , t0 ) − q ∫ dt ′δEz ( z′, t ′ ) vz ( t ′ ) 0±
W
∂
t
0
where the dependencies on the constants µ and W are implicit. Here the orbit is defined by
t
z ( t ) = z0 + ∫ dt ′vz ( t ′ )
(A4)
t0
and we have adopted the notation that z = z(t) and z′ = z(t′). However, it can be seen that the
velocity can be defined as a function of position using equation (A1) and so it is more convenient
to write equations (A3) and (A4) as functions of position. Thus, equation (A4) can be inverted to
find
z
dz ′
≡ t 0 + τ ( z0 , z )
t = t0 + ∫
(A5)
′
v
z
(
)
z z
0
where the last form defines the travel time τ. Then equation (A3) can be written as
z
∂f 0±
δf ± ( z , t ) = δf ± ( z0 , t − τ ( z0 , z ) ) − q
dz ′δEz ( z ′, t − τ ( z ′, z ) )
∂W z∫
(A6)
0
If we now assume that the wave fields and the perturbed distribution function oscillate at a
frequency ω, we can cancel out the exp(–iωt) time dependence of each term and write
16
∂f
iωτ z , z
δf ± ( z ) = δf ± ( z0 ) e ( ) − q 0±
∂W
0
z
∫ dz′δEz ( z′) e
iωτ( z ′, z )
(A7)
z0
Considering a bounded system between z = 0 and z = L, we can write the two solutions as
z
∂f
iωτ 0, z
iωτ z ′, z
δf + ( z ) = δf + ( 0 ) e ( ) − q 0+ ∫ dz ′δEz ( z ′ ) e ( )
∂W 0
(A8)
L
∂f
iωτ z ′, z
(A9)
δf − ( z ) = δf − ( L ) e
+ q 0− ∫ dz ′δEz ( z ′ ) e ( )
∂W z
Note the change of sign in equation (A9) due to the switch in the limits of integration. It can be
seen that if we know the parallel electric field as a function of position, these integrals can be
evaluated.
Next we must evaluate the field-aligned current. This can be found be integrating the
perturbed distribution function over all velocities, which gives
iωτ( L , z )
∞
0
0
−∞
δjz ( z ) = q ∫ dvz vz δf + ( x, vz ) + q ∫ dvz vz δf − ( x, vz )
(A10)
This can be written in terms of a single integral by defining v = vz in the first term and v = –vz in
the second term
∞
δjz ( z ) = q ∫ dv v δf + ( z , v ) − δf − ( z , −v ) 
(A11)
0
Since the distribution function can be represented as an integral over z ′ of the parallel electric
field as in equations (A8) and (A9), the field-aligned current can be written in the form
δjz ( z ) = ∫ dz ′ σ ( z , z ′ ) δEz ( z ′ )
(A12)
Note that in general terms relating to the boundary terms in equations (A8) and (A9) must be
added to equation (A12). If the field-aligned current can be determined from the wave
equations, then the integral equation (A12) can then be solved for the parallel electric field.
Acknowledgements
This work has benefited from discussions with John Wygant, Chuck Carlson, and
Vassilis Angelopoulos. We thank the referees for helpful comments that have improved the
presentation of the paper. The work is supported by NASA grants NAG5-8082 and NAG5-9254.
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Figure Captions
Figure 1. Normalized power dissipation to the kinetic dispersion relation for a uniform plasma
with equal temperatures, as a function of k x c / ω pe and ve2 / VA2 = Te / meVA2 for temperature
ratios of (a) 1; (b) 2; (c) 5; (d) 10. Note that the power dissipation is favored by cooler
ions.
Figure 2. Plot of the dimensionless integral in equation (33) that gives the conductivity kernel as
a function of the parameter ω | z − z ′ | / ae . The solid curve gives the real part of the
integral, while the dotted line gives the imaginary part. Mixing between particles of
different velocity causes this kernel to go to zero when the distance between the points is
greater than the distance a thermal particle travels in one wave period.
Figure 3. The model Alfvén speed profile as given by equation (20) for a value of ε = 0.1.
Figure 4. The effective normalized conductivity as defined by equation (39) for a model with ε =
0.1, k x2 λ 2M = 0.01 and 2hω / VAI = 2.4 for various thermal speeds ae/VAI = (a) 1.0; (b) 3.0;
(c) 10.0; and (d) 30.0. Solid line gives the real part while the dotted line gives the
imaginary part. Note that negative imaginary parts in this presentation indicate regions of
dissipation.
Figure 5. Relative dissipation normalized to the input Poynting flux for the ionospheric Alfvén
resonator models shown in Figure 5. Here the solid line is for ae/VAI = 1.0, the dotted line
for 3.0, the dashed line for 10.0 and the dash-dot line for 30.0. Note that in addition to
the region of Landau damping in the uniform region on the right half of the figure, there
is additional dissipation in the region of the Alfvén speed gradient for z/2h < 5.
25
100.00
(a) (VA/ω)P/Sz, Ti/Te= 1.0
100.00
(b) (VA/ω)P/Sz, Ti/Te= 2.0
0.10
10.00
ve2/VA2
ve2/VA2
10.00
1.00
0.10
100.00
0.
50
1.0
kxc/ωp
0.0 0.10
1
0.01
0.1
10.0
(c) (VA/ω)P/Sz, Ti/Te= 5.0
100.00
0
0.5
0.01
0.1
1.00
0.10
1.00
0
0.0 .10
1
0.10
1.0
kxc/ωp
10.0
(d) (VA/ω)P/Sz, Ti/Te=10.0
0.01
1
0.0
10.00
ve2/VA2
ve2/VA2
10.00
1.00
0.10
0.01
0.1
1.00
0.10
0.0
1
1.0
kxc/ωp
0.1
0.0
0
0.01
0.1
10.0
1
1.0
kxc/ωp
10.0
Figure 1. Normalized power dissipation to the kinetic dispersion relation for a uniform plasma
with equal temperatures, as a function of k x c / ω pe and ve2 / VA2 = Te / meVA2 for temperature
ratios of (a) 1; (b) 2; (c) 5; (d) 10. Note that the power dissipation is favored by cooler
ions.
26
Conductivity kernel
0.6
σ(z,z’)
0.4
0.2
0.0
-0.2
0
5
10
ω|z-z’|/a
15
20
Figure 2. Plot of the dimensionless integral in equation (33) that gives the conductivity kernel as
a function of the parameter ω | z − z ′ | / ae . The solid curve gives the real part of the
integral, while the dotted line gives the imaginary part. Mixing between particles of
different velocity causes this kernel to go to zero when the distance between the points is
greater than the distance a thermal particle travels in one wave period.
27
Alfven Speed Profile, ε= 0.100
12
10
VA/VAI
8
6
4
2
0
0
5
10
z/h
15
Figure 3. The model Alfvén speed profile as given by equation (20) for a value of ε = 0.1.
28
(a) a/VAI= 1.00
1.5
(b) a/VAI= 3.00
2
1.0
1
σeff
σeff
0.5
0.0
0
-0.5
-1
-1.0
-1.5
-2
0
5
10
z/2h
15
20
0
(c) a/VAI= 10.00
4
10
z/2h
15
20
15
20
(d) a/VAI= 30.00
1.0
0.5
σeff
2
σeff
5
0
-2
0.0
-0.5
-4
-1.0
0
5
10
z/2h
15
20
0
5
10
z/2h
Figure 4. The effective normalized conductivity as defined by equation (39) for a model
with ε = 0.1, k x2 λ 2M = 0.01 and 2hω / VAI = 2.4 for various thermal speeds ae/VAI = (a) 1.0;
(b) 3.0; (c) 10.0; and (d) 30.0. Solid line gives the real part while the dotted line gives the
imaginary part. Note that negative imaginary parts in this presentation indicate regions of
dissipation.
29
Dissipation, a/VAI= 1.0, 3.0, 10.0, 30.0
0.015
0.010
Dissipation
0.005
0.000
-0.005
-0.010
-0.015
0
5
10
z/2h
15
20
Figure 5. Relative dissipation normalized to the input Poynting flux for the ionospheric Alfvén
resonator models shown in Figure 4. Here the solid line is for ae/VAI = 1.0, the dotted
line for 3.0, the dashed line for 10.0 and the dash-dot line for 30.0. Note that in addition
to the region of Landau damping in the uniform region on the right half of the figure,
there is additional dissipation in the region of the Alfvén speed gradient for z/2h < 5.
30