NASA-CR-1 95820
mt UNIVERSITY OF ALABAMA IN HUNTSVILLE
UAH RESEARCH PROPOSAL 94-429
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IONOSPHERIC PLASMA OUTFLOW IN RESPONSE TO
TRANSVERSE ION HEATING:
SELF-CONSISTENT MACROSCOPIC TREATMENT V
Third year funding request
and
annual report
for
NAGW-2903
Prepared by
Principal Investigator
Department of Electrical and Computer Engineering
The University of Alabama in Huntsville
Huntsville, AL 35899
205-895-6678
Submitted by
The University of Alabama in Huntsville
Sue B. Weir
Research Administrator
May 1994
Ionospheric Plasma Outflow in Response to Transverse Ion
Heating: Self-Consistent Macroscopic Treatment
Grant # NAGW-2903, NASA/Headquarters
P. I.: Dr. N. Singh
Brief Summary of W ork Performed Since July 1 . 1993 :
In the previous grant year we examined the effect of transverse ion heating on the polar wind
outflow using both a hydrodynamic model [Singh, 1992] and a semikinetic, small-scale simulation
model [Singh and Chan, 1993]. These studies demonstrated that the transverse ion heating
creates significant plasma perturbations in the polar wind; the perturbation consists of (1) plasma
cavity formation for extended heating, (2) formation of a density depletion and an enhancement
on its top in response to a localized heating, (3) generation of upward pointing electric fields like
m a double layer in response to a localized heating, and (4) generation of waves by the ion-ion
instability.
The direct observational evidence of the cavity formation in response to the transverse
heating came from recent rocket experiments called TOPAZ I and m. The experiments revealed
that filamentary cavities, with depletions as high as 80% and aligned with the Earth's magnetic
field, occur at altitudes ~ 10 3 km. These cavities were observed in conjunction with intense lower
hybrid waves and transversely heated ions [Kinter et al, 1992], As the rocket cut across the
filamentary plasma cavities with width ~50 m, the waves appear as spikes with a duration of 50
ms. The original interpretation of the cavity formation was given in terms of lower hybrid wave
collapse [Vago et al, 1992],
We examined the various likely processes for creating the cavities and found that the mirror
force acting on the transversely heated ions is the most likely mechanism [Singh, 1994; Singh and
Chan, 1993], The pondermotive force causing the wave collapse was found to be a much weaker
force than the mirror force on the transversely heated ions observed inside the cavities along with
the lower hybrid waves.
Using a hydrodynamic model for the polar wind we modeled the cavity formation and found
that for the heating rate obtained from the observed waves, the mirror force does create cavities
with depletions as observed. Some initial results from this study were published in a recent
Geophysical Research Letters [Singh, 1994] and were reported in the Fall AGU meeting in San
IFrancisco. We have continued this investigation using a large-scale semikinetic model.
1
We have also continued our investigation on the microprocesses driven by the transverse ion
heating [Singh and Chan, 1993]. In the previous study we performed simulations using a small-
scale semilcinetic code. We have extended this code to be fully kinetic by treating both electrons
and ions kinetically. The goal of this study is to examine how the process of the transverse ion
heating energizes electrons parallel to the magnetic field as observed from satellites.
Tasks for the Grant Period Beginning July 1, 1994 :
We will continue to study the microprocesses responsible for the transfer of energy from the
transversely heated ions to the electrons. During the present grant period we developed the code.
We plan to perform a systematic set of simulation runs and analyze them theoretically.
We will also continue to study the process of the cavity formation by the lower hybrid waves,
which has been now observed from satellites as well [Holback et al, 1993], Using the semikinetic
modeling we plan to examine the issue of bulk versus trail heating of the ions. When the lower
hybrid waves are relatively slow they affect the entire velocity distribution function of the polar
wind ions. On the other hand, for relatively fast waves only a tail heating is likely. In the former
case a strong density depletion occurs, while in the latter case only weal depletions are possible.
A systematic theoretical study of this type comparing the results from the modeling and the
observations will greatly improve our understanding of the filamentary cavities observed in the
auroral region.
Publications Under Grant # NAGW-2903
1 . N. Singh, Pondermotive versus mirror force in creation of the filamentary cavities in auroral
plasma, Geophys. Res. Lett., 21, 257, 1994.
2. N. Singh and C. B. Chan, Numerical simulation of plasma processes driven by transverse ion
heating, J. Geophys. Res., 98, 11,677, 1993.
3. C. W. Ho, J. R. Horwitz, N. Singh and G. R. Wilson, Plasma expansion and evolution of
density perturbations in the polar wind: Comparison of semi-kinetic and transport models, J.
Geophys. Res., 98, 13,581, 1993.
4. N. Singh, Cavitons or simply plasma depletions by transverse ion heating, EOS, 74, 532,
1993. 1993 Fall Meeting, Dec. 7-11. 1993, San Francisco.
2
References
Holback, B. R. Bostrom, A. I. Ericksson, P. O. Dovner, and G. Holmgren, Characteristics of
spikelets seen from Freja, EOS, 74, 526, 1993.
Kinter, P. M., J. Vago, S. Chesney, R. L. Amoldy, K. A. Lynch, C. J. Pollock, and T. E. Moore,
Phys. Rev. Lett., 68, 2448, 1992.
Singh, N., Plasma perturbations created by transverse ion heating events in the magnetosphere, J.
Geophys. Res., 97, 4235, 1992.
Vago, J. L., P. M. Kinter, S. W. Chesney, R. L. Amoldy, K. A. Lynch, T. E. Moore, and C. J.
Pollock, Transverse ion acceleration by localized lower-hybrid waves in the topside auroral
ionosphere, J. Geophys. Res., 97, 16,935, 1992
3
Third year funding for NAGW-2903
THE UNIVERSITY OF ALABAMA IN HUNTSVILLE
UAH RESEARCH PROPOSAL 94 429
COST ESTIMATE FOR A ONE-YEAR PERIOD
(July 1, 1994 - June 30, 1995)
A. SALARIES AND WAGES
1. Dr. N. Singh, Principal Investigator •
16% x 7 weeks x $1 ,589.74/wk.
16% x 6 weeks x $1,675.68/wk.
16% x 31 weeks x $1 ,742.70/wk.
16% x 7 weeks x $1 ,742.70/wk.
2. Secretary
5% x 3/12 yr. x $18,889
5% x 9/12 yr. x $19,645
3. Undergraduate Research Assistant
10 hr. wk. @ $5. 20/hr.
TOTAL SALARIES AND WAGES
B. FRINGE BENEFITS (21% A.1. & A. 2.)
C. OPERATING COSTS
1. Supplies, reproduction
2. Page charges
TOTAL OPERATING COSTS
D. TRAVEL
1 . See below
TOTAL DIRECT COST
E. INDIRECT
1. 42.0% MTDC
2. 42.5% MTDC
TOTAL INDIRECT
TOTAL ESTIMATED COST
UAH changes from a quarter to semester academic year beginning 8/19/94. This change affects the way
an academic appointment s time is computed. UAH's annual merit increase occurs on October 1 The four
components of Dr. Singh's salary listed above are:
7 weeks of summer 1994 (7/1/94-8/18/94) figured on the weekly rate of an FY’94 academic year (39 weeks)
base of $62,000 ($62,000/39)
6 weeks of academic year 1995 (8/19/94-9/30/95) figured on the weekly rate of the new semester academic
year (37 weeks) from the $62,000 base. ($62,000/37)
31 weeks of the remainder of the 1995 academic year figured on the new base of $64 480 (4% increase)
($64,480/37)
7 weeks of summer 1995 (May 15- June 30) at same weekly rate as academic year 1995
D.1 . Travel to professional meeting to present paper/Washington, DC used for estimation purposes = $1 .280
air fare = $623 (travel agent quote), per diem = $ 1 44 x 3 days (GSA rate), registration = $125. misc. = $50
/I/ See paragraph 2. a. of financial data sheet
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.w -y- i- <t
GEOPHYSICAL RESEARCH LETTERS, VOL. 21, NO. 4, PAGES 257-260, FEBRUARY 15, 1994
Pondermotive versus mirror force in creation of the
filamentary cavities in auroral plasma
Nagendra Singh
Department of Electrical and Computer Engineering , The University of Alabama in Huntsville
Abstract . Recently rocket observations on spikelets of lower-
hybrid waves along with strong density cavities and
transversely heated ions were reported. The observed thin
filamentary cavities oriented along the magnetic field in the
auroral plasma have density depletions up to several tens of
percent. These observations have been interpreted in terms of
a theory for lower-hybrid wave condensation and collapse.
The modulational instability leading to the wave condensation
of the lower-hybrid waves yields only weak density
perturbations, which cannot explain the above strong density
depletions. The wave collapse theory is based on the
nonlinear pondermotive force in a homogeneous ambient
plasma and the density depletion is determined by the balance
between the wave pressure (pondermotive force) and the
plasma pressure. In the auroral plasma, the balance is
achieved in a time ^ 1ms. It is shown here that the
mirror force, acting on the transversely heated ions at a
relatively long time scale, is an effective mechanism for
creating the strong plasma cavities. We suggest that the
process of wave condensation, through the pondermotive
force causing generation of short wavelength waves from
relatively long wavelength waves, is a dominant process until
the former waves evolve and become effective in the
transverse heating of ions. As soon as this happens, mirror
force on ions becomes an important factor in the creation of
the density cavities, which may further trap and enhance the
waves. Results from a model of cavity formation by
transverse ion heating show that the observed depletions in
the density cavities can be produced by the heating rates
determined by the observed wave amplitudes near the lower-
hybrid frequency. It is found that the creation of a strong
density cavity takes a few minutes.
Introduction
In a recent paper, Vago et al [1992] reported interesting
results from rocket (TOPAZ HI) observations on lower-
hybrid waves and associated heating of ions transverse to the
magnetic field lines in the auroral plasma. The rocket
observations reveal that intense lower-hybrid waves occur in
thin plasma cavities oriented along the geomagnetic field
lines. In the cavities, plasma depletions up to 80% have been
reported. As the rocket crosses the cavity, the lower-hybrid
waves appear as spikelets of 50- to 100-ms duration giving
the cavity width 50-100 m across the magnetic field lines.
Lower-hybrid wave amplitudes up to 300 mV/m have been
reported. The characteristic energy of the transversely heated
Copyright 1994 by the American Geophysical Union.
Paper number 93GL03387
0094-8534/94/93GL-03387S03 . 00
ions is reported to be 6 eV. However, the energy
spectrograms for the reported spikelet events show
acceleration up to ~30 eV [Vago et al, 1992].
Vago et al [1992] have interpreted their observations in
terms of the theory for the collapse of lower-hybrid waves
[Morales and Lee, 1975; Sotnikov et al, 1978]. According to
this theory, the nonlinear pondermotive force associated with
the wave expels plasma forming density cavities. In view of
the dispersion property of the lower-hybrid wave, the wave
number is enhanced in the depletion region. The consequent
refraction of the waves into the cavity leads to wave trapping
and its intensification, which in turn, intensifies the process of
cavity formation. Eventually the wave collapses into a
filamentary structure like the observed spikelets. This process
can create density depletions of a few percent; the theory of
modulational instability operative in this process yields a
density perturbation 8n/n = (o^ e /Q c co ni )W/nT, where
(D^, and <D; h are the electron plasma, cyclotron, and
lower-hybrid frequencies, respectively; W is the wave
electrostatic energy density and nT is the thermal energy
density of the plasma. For the parameters of the observations
5n/n~4xl0" 2 . Therefore there is a difficulty in explaining
the observations in terms of the wave collapse involving the
pondermotive force alone.
Recently Singh [1992] and Singh and Chan [1993] reported
that a natural consequence of transverse ion heating is the
formation of a density cavity. For spatially extended bulk
heating of ions, the density depletion can be deep and it
extends along the magnetic field lines without a significant
density enhancement on top of the cavity. On the other hand,
for a localized heating the density cavity and enhancement go
hand-in-hand. The cavity formation is caused by the plasma
expulsion by the upward mirror force acting on the
transversely heated ions. Since transversely heated ions are
an integral part of the observations, we examine here the
relative roles of the mirror and pondermotive forces in the
cavity formation.
The question arises here as to which force, the mirror or the
pondermotive force or their combined effect, is driving the
process of plasma depletion in the observed cavities. The
purpose of this letter is to compare these forces for the
parameters of the plasma during the observed wave spikelet
events [Vago et al, 1992]. We find that in the initial stage of
the cavity formation with weaker fields of about 25 mV/m
[Vago et al, 1992], even a slight transverse ion heating causes
a mirror force exceeding the pondermotive force. Only when
the wave field intensifies to values greater than 200 mV/m,
the two forces become comparable. In view of this, we
suggest that the evolution of the density cavity and the lower-
hybrid waves occurs in two stages; in the early stage the long
wavelength lower-hybrid waves, probably generated by the
auroral electron beams [Vago et al, 1992], undergo a
puscsxnc page blank not rlmeo
257
258
Singh: Pondermotivc vs. Mirror Force
modulations! instability [Sotinikov et al, 1978; Shapiro et al,
1993] generating waves with shorter and shorter wavelengths
until they become effective in transverse heating of ions. For
strong waves, such a heating is expected to occur when the
difference between the perpendicular wave phase velocity and
the wave trapping width becomes comparable to the velocity
spread of the thermal ions. As soon as the heating begins to
enhance the perpendicular temperature of the ions, the mirror
force becomes an important cause for the plasma depletion in
the density cavities; the depletions in the cavity can be up to
several tens of percent, which cannot be achieved by the
pondermotive force alone. Calculations show that the
observed amplitude levels of the lower-hybrid waves produce
sufficiently strong heating to create density cavities with
depletions as observed.
Comparison of Pondermotive and Mirror Forces
The effects of pondermotive force on the nonlinear
evolution of lower-hybrid waves have been studied by
including different types of nonlinearities. In the early work
of Morales and Lee [1975] the nonlinearity considered arose
from the motions of charged particles along the fields of the
wave. In later studies [Sotnikov et al, 1978; Shapiro et al,
1993] it was shown that a much stronger nonlinearity arises
due to the £ * B drift of the electrons, where £ is the wave
electric field and B is the ambient magnetic field. This latter
nonlinearity gives rise to a stronger pondermotive force than
that given in the early work of Morales and Lee [1975]. The
magnitude of the pondermotive force given by Shapiro et al
[1993] is
F ell = _ i
e p m pe 2 d
( 1 )
where F c || is the pondermotive force acting on electrons,
a> pc and Qe are the electron plasma and cyclotron
frequencies, respectively, e 0 is the permittivity of free space,
^ is the ambient plasma density, and Ej_ is the wave
electric field component perpendicular to the ambient
magnetic field.
In order to assess the relative importance of the above
pondermotive force and the mirror force acting on
transversely heated ions, we compare these two forces. The
latter force is given by
< 2)
where and Til arc * respectively, the perpendicular and
parallel ion temperatures in energy units, and B(z) is the
geomagnetic field. Along the auroral flux tubes,
B(z) = B 0 (Rc / z) 3 , where B Q is the magnetic field when the
geocentric distance z = Ro* the Earth’s radius. Using this
information, (2) can be written as
F m = 3(T x -TJ 1 )/z (3)
It is worthwhile to point out that for an isotropic ion
temperature (T^ =T,h), mirror force F m =0 When
> Tin, there is a upward force on the ions.
The plasma conditions for the reported spikelet events are as
follows [Vago et al, 1992]: plasma density n = 10 lo nr 3 ,
electron and ion temperatures ~ 5000° k , electron cyclotron
l* j" yru-: vj-jk;*
frequency f^ = 10® Hz and the plasma frequency
~ 1 0 6 Hz . The plasma predominantly consists of 0 + ions
at an altitude of about 10 3 km. In such a plasma, the lower-
hybrid frequency is given by o^, =o p j(l + o)| e /f^) -l/2 *
0.7a>pj .where Op is the ion-plasma frequency.
For evaluating the pondermotive force F e ||, we need to
estimate the parallel scale length in the variation of |E x (z)| .
We assume that it is determined by the intimate relationship
between the parallel (Lfl) and the perpendicular (L x ) scale
lengths of the lower-hybrid waves, namely,
L a = (m i /m t ) ,/2 L 1 (4)
L,| is the width of the observed filamentary cavity; for the
purpose of calculations we assume L x = 100m and find
L|| = 17 km. If the lower-hybrid wave is excited by an auroral
electron beam with energy of a few hundred eV, the above
values of L x and L|| are about ten times the perpendicular
and parallel wavelengths of the wave, respectively. The
differential 9/3z appearing in (1) can be approximated by
alEj.1 2 / 3z = |Ejf / L)| , which yields
F e „ = 6.3xl(r 24 |E 1 | 2 N (5)
Taking the nominal value of the electric field
|EjJ = 25 mV/m for the time just before the cavity forms
[Vago et al, 1992], we find F eli == 4 x 1(T 27 N This force is
transmitted to ions by an anbi polar electnc field if the
pondermotive force is the only force acting on the plasma.
At an altitude of K^km, the mirror force from (3) is
F m = 6.5 x 10' 26 AT, where AT = T^ - Tju, and it is expressed
in eV. Comparing F c || with F m , we find that the latter
becomes more effective than the former as soon as ions are
transversely heated giving
AT>0.16 eV (6)
that is, even for a slight perpendicular heating the parallel
mirror force begins to dominate the parallel pondermotive
force on ions.
It is important to point out that as the pondermotive and the
mirror forces act to create the plasma cavity, the electric field
is enhanced by wave trapping and the transverse ion
temperature is enhanced by the ion heating. Thus the
pondermotive and the mirror forces evolve simultaneously.
The exact nature of the evolution and their relative
importance have not been studied so far. However, we find
that even in the late stage of the evolution the mirror force is
an important factor. For example, if we assume that in the
late stage when the deep cavities have formed,
E x = 200mV/m and AT = 6eV, F eM =2.5x 10" 25 N and
F m = 4 x 10 -25 N.
A major difficulty with the wave collapse theory, based on
the nonlinear pondermotive force, is in explaining the
observed levels of plasma depletions in the density cavities.
This theory predicts that the quasineutral density perturbation
in the plasma is given by [Shapiro et al, 1993]
Sn/no^-^-eolEi^a+Ti) (7)
Assuming |E X | = 300 mV/m, the maximum value of the
Singh: Pondermotive vs. Mirror Force
259
electric fields reported by Vago et al [1992], = Tj = 0.5 eV
and n 0 = 10 lo m -3 , we find 5 n / n 0 = 4 x 1 0 -2 , which is at
least an order of magnitude smaller than the strong plasma
depletions in the observed cavities. The main reason for the
weak plasma depletion is that the density perturbations are
determined by the balance between the wave pressure
(pondermotive force) and the plasma pressure in a
homogeneous ambient plasma. The theory of Shapiro et al
[1993] shows that the balance is achieved at a time scale
T wc = 0- 1 cd jjj 1 , which is less than a millisecond in the auroral
plasma. The subscript H wc" on r refers to the time scale of
wave collapse in a homogeneous ambient plasma. However,
if the wave collapse has created strong waves effective in
heating ions, the continued heating of ions over a relatively
long time can create strong cavities. We demonstrate this by
a model calculation.
Modeling of Cavity Formation
In order to demonstrate the effectiveness of the mirror force
in creating the cavity with the observed levels of wave
amplitude, we adopt a model based on hydrodynamic
transport equations for the O + ions in the polar wind [Singh,
1992]. Electrons are assumed to obey the Boltzmann law
with a temperature of 0.5 eV. The cavity formation and the
wave must evolve simultaneously. In this paper we cannot
study this simultaneous evolution. However, we can develop
a feel for the depth and time constants of the plasma cavity
formation by considering plasma depletions by the mirror
force for the representative values of the observed wave
amplitudes. From Vago et al [1992], we estimate that the
power spectral density \j/<10~* V 2 m' 2 Hz -1 . Thus, the
heating rate is limited to 3T x /3t<0.14 eV/s [Singh and
Schunk, 1984]. We calculate the response of the plasma to
such a heating in an auroral flux tube by considering different
transverse heating rates. We consider a portion of a flux tube
from an altitude of 1000 to 2800 km. First a polar wind type
of flow consisting of 0 + ions is established in it, with
boundary conditions at 1000 km altitude as follows: density
n o = 10 4 cm” 3 , equal electron and ion temperatures
To =0.5 eV, and flow velocity V 0 = V ti , the ion thermal
velocity. At the top end of the flux tube we assume the flow
is continuous. The perpendicular ion heating is switched on
at t = 0 for altitudes h > 1200 km. Figures la and lb show
the temporal evolution of the density and perpendicular
temperature, respectively, in the flux tube for a relatively low
heating rate of 0.014 eV / s. The temporal evolution up to 3
minutes are shown: At t = 1 min. , the cavity is quite weak.
By the time t = 3 min. , the cavity has grown to about
5n / n Q = 10% and it extends to an altitude of 2200 km. The
corresponding evolution of T i± shows a typical feature of
extended heating [Singh, 1992]; the temperature increases
with the altitude inside the cavity and then it saturates, with
saturation value increasing with time. The maximum
temperature inside the cavity at t = 3 min. is about
T] x = 7 T^j = 3. 5 eV.
Figures 2a and 2b show the evolution of n(r) and Tj/r)
for a stronger heating rate of 0.14 eV corresponding to
\\f = I0~* V 2 m -2 H~ ] . These figures show temporal and
spatial evolution of and T i± as in Figures la and lb.
Fig. 1. (a) Evolution of density depletion in response to
transverse ion heating above an altitude of 1200 km with a
heating rate of 0.014 eV corresponding to a lower-hybrid
wave level y = 10” 9 v^m^Hzr 1 . (b) Evolution of T x .
Note that for the low heating rate, the relative plasma
depletion Snln<\QPA.
respectively. The density depletions are generally much
stronger in Figure 2a than that in Figure la. For example at
t = 2 min., the maximum depletion is 28% at h = 1600 km
where = 36T 0 = 18 eV. At t = 3 min. , at the same
altitude, the density depletion is 36% with nearly the same
value of T x .
The heating rates considered above are within the range
given by the observed power spectral density. Therefore it
appears that within a few minutes after the onset of the
relatively strong lower-hybrid waves, the observed levels of
plasma depletions can be achieved. The results shown in
Figures 1 and 2 indicate that in order to create density cavities
with depletions of several tens of percent, the power spectral
density near the lower-hybrid waves must exceed
10“ 9 V 2 nr 2 Hz -1 , and the heating must last over a few
minutes. It is worth pointing out that at time scales t>t w ,
the pondermotive force may continue to participate in the
density depletion process because of the inhomogeneous
nature of the auroral plasma. This is especially true when the
wave amplitudes are sufficiently strong to yield comparable
pondermotive and mirror forces.
Finally we discuss the parallel and perpendicular sizes of the
density cavity. The axial size of the cavity depends on the
field-aligned extent of the heating region and the duration of
the heating. However, for a localized heating, the plasma
expulsion produces density enhancement on top of the cavity
[Singh, 1992; Singh and Chan, 1993]. On the other hand,
extended heating produces a continual expulsion of the
plasma into the steadily decreasing density of the polar wind
with increasing altitude, without producing hardly any density
1.6 2.2 2.8 1 1.6 2.2 2.8
ALTITUDE (1000 km)
Fig. 2. (a) Same as Figures la, but for a heating rate of 0.14
eV. (b) Same as Figure lb with the above heating rate. Note
that the relative depletion in Figure 2a is much stronger than
that in Figure la.
260
Singh: Pondermotive vs. Mirror Force
enhancement as seen from Figures 1 a and 2a for t — 1 and 2
minutes. Since the rocket observations do not seem to report
the existence of density enhancements, it appears that the
heating occurs over an extended region along the auroral field
lines. The size of the cavity along the field lines depends on
the heating time, as seen from Figures la and 2a. In view of
the observed level of waves and depletions in the density
cavity, the heating must be tasting at least over a few minutes,
and extending over hundreds or even thousands of kilometers.
The filamentary nature of the plasma cavities having
width 100 m probably follows from the extremely small
half-cone angle (0 C ) of the group-velocity resonance cone of
the lower-hybrid waves [Morales and Lee, 1975]. Near the
lower-hybrid frequency, 0 C = (m* / n^) 172 - 5 x 10" 3 rad.,
which is complementary to the phase-velocity resonance cone
angle. Since the electrostatic energy of the lower-hybrid
wave is confined within a cone with its axis along the
geomagnetic field and half cone angle 0 ^ 0 C , the observed
cavity width (w) of ^ 100 m reveals that the electrostatic
lower-hybrid waves are excited by the auroral electron beams
and absorbed by the plasma through the transverse heating of
ions within a distance < w /0 C = 20 km. The temporal and
spatial features of the excitation of such waves by auroral
electron beams and absorption by the thermal plasma are a
challenging problem and remain to be studied.
Conclusion and Discussion
The main conclusions of this paper are as follows: (1) A
comparison of the pondermotive and mirror forces show that
the latter force on the transversely heated ions is an important
factor in creating the strong plasma depletions in density
cavities as observed during the lower-hybrid wave spikelet
events [Vago et al, 1992]. (2) The mirror force becomes a
significant force from the very early stage when ions are even
slightly heated causing T x to exceed TJj by a fraction of an
eV. (3) A model of the polar wind type of flow including
transverse ion heating [Singh, 1992] shows that the heating
rates given by the observed levels of lower-hybrid waves can
produce density depletions consistent with the measured
densities in the filamentary density cavities. The
measurements indicate density depletions of a few tens of
percent to be a common occurrence, but some events
indicated depletions up to 80%, which were reported to occur
with strong lower-hybrid waves. Model shows that for the
strong depletions up to several tens of percent, the power
spectral density must exceed 10 _ ®V 2 m“ 2 Hz *. (4) The
model shows that for the observed wave levels and the
depletions in the cavities, the heating events should last over a
few minutes which is much longer than the time scale for the
wave collapse in a homogeneous plasma. For such heating,
the cavity extends to several hundred kilometers along the
geomagnetic field lines.
The effectiveness of the mirror force in density depletions is
contingent upon the transverse heating of ions. The lower-
hybrid waves, generated by auroral electron beams having
energies of several hundred eV, are generally too fast to
affect wave-particle interactions with the ions. For such an
interaction the perpendicular phase velocity of the wave
should be comparable to the thermal speed of the ions. In
order to achieve this, the long wavelength fast waves undergo
wave condensation and collapse through a modulation
instability, generating short wavelength slow waves [Sotnikov
et al, 1978; Shapiro et al, 1993]. During this stage, the
nonlinear pondermotive force drives the modulational
instability and creates plasma perturbations, which are weak
and not as strong as the observed density depletions. The
theory of Shapiro et al [1993] indicates that this occurs at a
time scale of » O.lco" 1 * 0.4 ms for a wave amplitude of
25 mV/m. As soon as the short wavelength waves become
effective in transverse heating of the ions, mirror force
becomes a significant mechanism by which density depletions
are created. The density perturbations created by the mirror
force may further facilitate the wave trapping and
enhancement.
Acknowledgments . This work was supported by a NASA
grant NAGW-2903.
References
Morales, G. J. and Y. C. Lee, Nonlinear filamentation of
lower-hybrid cones, Phys. Rev. Let., 35 , 930, 1975.
Singh, N., Plasma perturbations created by transverse ion
heating events in the magnetosphere, J. Geophys . Res., 97 ,
4235, 1992.
Singh, N. and C. B. Chan, Numerical simulation of plasma
processes driven by transverse ion heating, J. Geophys. Res.,
98, 11,677, 1993.
Singh, N. and R. W. Schunk, Energization of ions in the
auroral plasma by broadband waves: Generation of ion
conics, J. Geophys. Res., 89, 5538, 1984.
Sotnikov, V. I., V. D. Shapiro, and V. I. Shevchenko,
Macroscopic consequences of collapse of the lower- hybrid
resonance, Sov. J. Plasma Phys., Eng. TransL, 4, 252, 1978.
Vago, J. L., P. M. Kintner, S. W. Chesney, R. L. Amoldy, K
A. Lynch, T. E. Moore and C. J. Pollack, Transverse ion
acceleration by localized lower-hybrid waves in the topside
auroral ionosphere,/. Geophys. Res., 97, 16, 935, 1992.
Shapiro, V. D., V. I Shevchenko, G. I. Solov’ev, V. P.
Kalinin, R. Gingham, R. Z. Sagdeev, Ashaur-Abdalla, J.
Dawson and J. J. Su, Wave collapse at the lower-hybrid
resonance, Phys. Fluids, B5, 3148, 1993.
Nagendra Singh, Department of Electrical and Computer
Engineering, The University of Alabama in Huntsville,
Huntsville, AL 35899.
Received: September 27, 1993
Revised: November 1, 1993
Accepted; November 23, 1993
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98. NO. A7, PAGES 11 .677-1 1 .687. JULY 1. 199.1
Numerical Simulation of Plasma Processes Driven by Transverse Ion Heating
Nagendra Singh and C. B. Chan
Department of Electrical and Computer Engineering, University of Alabama, Huntsville
Numerical simulation is performed to study the plasma processes driven by transverse ion
heating in a diverging flux tube. It is found that the heating drives a host of plasma processes, in
addition to the well-known phenomenon of ion conics. The additional processes include formation
of a density cavity topped by a density enhancement, formation of a reverse and forward shock
pair with a "double-sawtooth 1 * structure in the flow velocity. The downward electric field near the
reverse shock generates a doublestreaming situation consisting of two upflowing ion populations
with different average flow velocities. A double streaming also occurs above the forward shock,
where the ions energized by the heating are overtaking the relatively slow ions in the ambient polar
wind. The energized ions appear as "elevated” ion conics with a low-energy cutoff depending on
the distance from the heating region. The parallel electric fields generated by the transverse
ion heating have the following noteworthy features; the electric field near the forward shock is
essentially unipolar, and it points upward, and for the heating localized in both space and time,
the field has the features of a weak double layer. The electric field in the reverse shock region is
modulated by the ion-ion instability driven by the multistreaming ions. The oscillating fields in
this region have the possibility of heating electrons. The results from the simulations are compared
with results from a previous study based on a hydrodynamic model. Effects of spatial resolutions
afforded by simulations on the evolution of the plasma are discussed, demonstrating how a crude
resolution can miss out plasma instabilities, affecting the plasma flow.
1. Introduction
Transversely heated ions are a common feature of the
Earth’s magnetosphere. Since the early observations of such
ions during the late seventies [e.g., Whalen et ai, 1978;
Klumpar , 1979], a great deal of work has gone into un-
derstanding the generation and transport of such ions [e.g.,
Chang , 1986; Klumpar, 1986]. However, most treatments on
the transport employ the test particle approach, in which a
perpendicularly heated ion is transported under the action of
the upward mirror force proportional to the gradients in the
magnetic field. Only recently, time-dependent models have
been employed to study the generation and transport pro-
cesses and their effects on the ambient plasma [Ganguli and
Palmadesso, 1987; Brown et ai t 1991; Singh , 1992]. The
aspect of the plasma perturbations created by the trans-
verse ion heating was emphasized by Singh [1992]. Among
the noteworthy features of the plasma perturbations are the
formation of density depletion and enhancement, and gen-
eration of parallel electric fields. For impulsive heating, an
interesting feature of the parallel field is that it occurs in
the form of a nearly unipolar upward pointing electric field
pulse, which moves upward with a velocity of several tens of
kilometers per second. However, a large-scale model dealing
with distances of thousands of kilometers is limited in its
temporal and spatial resolutions. On the other hand, elec-
tric fields seen in the auroral plasma [Temerin et ai, 1982]
have spAtial size of a few meters and corresponding time
scale of about a few milliseconds. Therefore, in the previ-
ous work of Singh [1992], it was not clear at all how such
an electric field pulse can be compared with weak double
layers.
The purpose of this paper is to study the perturbations
Copyright 1993 by the American Geophysical Union.
Taper number 92JA02789.
n 1 48-022 7/93/92.1 A-0 2 789$05. 00
created by the transverse ion heating, using a small-scale
particle-in-cell code having the capability of resolving dis-
tances of a few Debye lengths and time of a few millisec-
onds. The particle simulation reveals the same basic feature
of the plasma perturbations generated by the transverse ion
heating as seen from the large-scale hydrodynamic study,
namely, the formation of a density cavity topped by a den-
sity enhancement, and eventually, the evolution of the den-
sity perturbation into a reverse-forward shocks pair. The
unipolar upward pointing electric field occurs near the for-
ward shock. The maximum electric field in the pulse is a few
millivolts per meter, and its spatial dimension is a few tens
of meters. These features of the pulse, including its upward
velocity of about 50 km/s, have striking resemblance with
the weak double layer seen from satellite [Temerin et ai,
1982; Bostrom et ai, 1988].
Kinetic simulations show additional noteworthy features
involving multistreaming of ions. Above the perturbations
in the density, ions with relatively large energies stream up-
ward, setting up an ion conic type of flow on top of the am-
bient polar wind. In the midst of the density depletion and
the enhancement, two streams of up flowing ions appear,
which eventually couple together through ion-ion instabil-
ity. It is interesting to point out that if the grid size in the
simulation is increased beyond a certain limit, the ion-ion
instabilities are not seen. This implies an important limita-
tion of large-scale models, in which the usage of large grid
size eliminates the possibility of coupling the ion streams.
Futhermore, the large grid size and the corresponding large
time steps eliminate the process of steepening of a compres-
sive density perturbation forming a shock, like the forward
shock near the density enhancement.
The fast ions above the forward shock appear like “ele-
vated” ion conics with a low cutoff energy, which increases
with increasing distance from the heating region. Futher-
more, the density of such conics decreases monotonically
with the distance. This suggests that ion conics can be
found far from the regions of strong density perturbations
in the plasma, created by the heating process.
1 1 ,677
pKBCgXte PAGE W-ANK WOT FKMtB
11.678
Singh and chan: Plasma Processes Driven by Transverse Ion Heatino
The re«t of the paper is organised as follows. The simu-
lation technique is described in section 2. Numerical results
on the plasma perturbations are described in section 3. The
paper is concluded in section 4.
2. Simulation Model
We use a particle-in-cell code to solve for the dynam-
ics of ions flowing along a diverging flux tube (Figure 1).
The electrons are assumed to obey the Boltzmann distribu-
tion, which in conjunction with the quasineutrality condi-
tion yields the electric field parallel to the magnetic field.
As mentioned in the introduction, the flux tube simulated
is artificial in the sense that magnetic field is reduced by a
factor of 2 over a distance of about s„ ,<■ = 7.5 km. This is
done to hasten the transport of the transversely heated ions
by the mirror force. The ion heating occurs over a limited
region of space (Figure 1). Ions in this region are given a
random impulse Sw x in the perpendicular direction accord-
ing to a Maxwellian probability density function given by
[Brown et ai, 1991; Puri, 1966]
The energy of the ions is given by
Wxf = 4* Swx 4 2\rw± i\6w\cosj (2)
where w±i and wxf are the perpendicular energies of the
ions at the beginning and end of a time step, and is an
angle between 0 and 2 ir randomly chosen from a uniform
probability density function. The heating rate is related to
0 according to cr = 1.14 A«(*wa/*<)i wh « e “ 1 / 2mVj “ *
m is the ion mass and Vj. is the perpendicular velocity.
The ion motion is advanced by solving the equation of
motion
= M 97 ( )
where m and q are the mass and charge of an ion, E g is
the electric field, is the magnetic moment of the ion, and
dB/ds is the gradient in the magnetic field.
The parallel electric field, is calculated by assuming
that the plasma remains quasineutral, i.e. f n e ~ n^, where
n e and n, are the electron and ion densities, respectively.
Futhermore, electrons are assumed to be a massless isother-
mal fluid. The electron momentum equation gives
t
DENSITY N 0
TEMPERATURE T Q
Fig- 1. Geometry of the simulated flux tube. jVo( 20 cm )
and To(0.3 eV) are the boundary value* of plasma density and
ion temperature.
*»«-
kT< 1 Sn t
< n« ds
( 4 )
where k is the Boltsmann constant and is the electron
temperature.
As we will see later, the scale lengths in the plasma per.
turbations studied here are several tens of meters while the
plasma Debye length is a few meters. Therefore the space
charge effects are ignorable and quasineutrality is a good
approximation. The assumption of electrons being massless
eliminates the effects of velocity gradients in the flow. Since
in the present calculations we assume that there is no field-
aligned current and n, ~ n; , it is implied that V e — Vi ,
where V, and V, stand for electron and ion flow velocities,
respectively. In the calculations presented here, we assume
T e = 1 e V, for which electron thermal velocity, Vi, ~ 400
km/s. In the perturbations discussed in this paper, the flow
velocities V, and V < 15 V,<, where V ti is the ion thermal
velocity, which is about 5.5 km/s. Therefore we find that
V 3 ~ Vi* << V,*. This ensures that the assumption of elec-
trons being massless is justified. Futhermore, it also justifies
the assumption of electrons being isothermal.
3. Numerical Results
3.1. Summary of Results From the Fluid Model
The origin of this paper lies in a previous paper [Sinyh,
1992], in which plasma perturbations created by transverse
ion heating were studied, using a large-scale model based on
fluid equations for the plasma. Since our goal in this paper
is ,0 examine how the results from a kinetic treatment of
ions compare and contrast with the results from the fluid
treatment, it is useful to briefly review the latter results.
Figures 2a to 2 h show the basic nature of the perturbations
in density, flow velocity, parallel temperature, perpendicular
temperature, and the parallel electric field when the heat-
ing occurs over 5s over a heating region of 2 10- km length at
an altitude of 5500 km. The heating rate is 240 eV/s and
electron temperature is assumed to be 10 eV. Figures 2a to
2 h show the evolution of the perturbation up to t = t< =
2 min. Note that t„ = n x 30 s. We find that at an early
time ( t < t\) the basic feature of the perturbation is the
formation of a plasma cavity topped by a density enhance-
ment (Figure 2a). At later times, the density perturbation
evolves into a reverse-forward shock pair, as indicated by
“R” and “F”. The leading edge of the perturbation is the
forward shock (F) and the trailing edge of the density en-
hancement is the reverse shock (R). The entire perturbation
is seen moving upward. However, the trailing edge of the
perturbation moves much slower than the leading edge ( ),
resulting in the creation of an extended cavity which ex-
pands upward. Figure 2 b show, that the flow velocity is
perturbed over the entire region of the density perturbation
and it has the feature of a double sawtooth; the tooth near
the forward shock is sharp, while near the reverse shock it »
relatively shallow. When the heating continues for a longer
time, the reverse shock also evolves into sharp jumps [5myh,
1992]. „ .
The temperature profile for Tj shows a cooling in the
plasma cavity and an increase in the density enhancement
between the reverse and forward shocks. The transverse
heating yields a maximum perpendicular temperature of 100
e V at t = ti and the maximum temperature adiabatically
decreases later on. The enhancement in T x is limited to
altitudes below the forward shock. Later we show how this
Sinoh and Chan: Plasma Processes Driven by Transverse Ion Heatino
11,679
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t ft -
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1 • *
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Fig. 2. Plasma perturbations in response to an impulsive heating both in time and space st 5 s, As* =
60 km). Electron temperature T c = 10 eV. (a) to (d) The evolutions of n(j), u(j), T^(s) and Tx(i), respectively,
(e) to {h) The evolution of the electric fields distribution.
feature is appreciably modified when the ions are treated
kinetically.
The electric field perturbations for the transverse heating
are shown in Figures 2e to 2 h. The most noteworthy feature
of the electric field distribution shown in this figure is its
evolution to predominantly unipolar upward pointing elec-
tric field near the leading edge of the density bump when
t > 1 min. This dominant electric field pulse prop-
agates upward with a velocity of about 60 km/s, which is
about twice the ion acoustic speed with 10-eV electrons.
Such upward propagating electric field pulses appear quite
similar to the predominantly unipolar electric fields observed
in the auroral plasma [Temerin et al., 1982]. However, the
observed fields are generally interpreted as ion acoustic dou-
ble layers which have scale length of a few tens of Debye
lengths 100 m). In contrast, in the hydrodynamic calcu-
lations we have a spatial resolution of 80 km and temporal
resolution of Is, which are, respectively, the intergrid spacing
and the time step used in our calculations. In the follow-
ing discussion, we present results from a kinetic treatment
of ions with spatial and temporal resolutions capable of re-
solving ion dynamics at a time scale of ion-plasma period.
3.2. Results From a Small-Scale Kinetic Model
We first ran the simulation without any heating until a
polar wind type of flow is set up in the artificial flux tube.
For the parameters chosen here, this takes about 1200
where u is the ion plasma frequency at s = 0, where nor-
malized density is unity. When the flow is established, the
heating is switched on over the spatial region 50 < sj \,n <
250. The heating rate dW^jdt is about u^okTo, where k is
the Boltzmann constant and To is the ion temperature at the
boundary s = 0. The flux tube length is S, na z = 7500A<k.
For the parameters chosen here the plasma density at the
bottom of the flux tube is 20 cm -3 and temperature To =
0.3 eV, giving ion Debye length \.u ss 1 m, S mftr = 7.5 km
and heating rate (dW±/St) is 1800 eV/s. The heating is
kept on over a time period of A<* = A0ta^ o sr 5 ms.
11,680
Singh and Chan: Plasma Processes Driven by Transverse Ion Heating
3.3. Perturbation in Phase Space |
The evolution of the heated ions is shown in Figures 3a
and 3 b which give the temporal evolution of phase-space in
5 - Vj, and 5 - V L planes, respectively. At t = 1400, where
t = tufptoy heated ions are still relatively localised near the
heating region. At later times they flow upward under the
action of the mirror force leading to an increase in parallel
velocity(energy) at the expense of the perpendicular veloc-
ity (energy). As the ions flow upward, the phase-space plots
show that double streaming develops both near the top and
bottom of the perturbation. The two streams at the top
consist of transversely heated ions, which have gained con-
siderable parallel energies under the action of the mirror
force, and the ambient polar wind ions. The relative paral-
lel velocity between these two ion populations is sufficiently
high and therefore they do not show any sign of ion-ion in-
teraction causing instability [Gresillon and Doveil, 1975].
In the bottom most part of the perturbation ions appear
to be primarily accelerated in their parallel velocities, but
above a certain height depending on time, another stream
appears. The latter stream is relatively slower. In the re-
gions where these streams overlap, there are vortices in the
5 — Vjj plots. These vortices are the consequence of ion-ion
instability, which we shall discuss later on. By the time i as
2200, the major part of the perturbation in terms of trans-
versely heated ions has almost exited from the top of the
0 1875 3750 5625 7500
Distance S/A^
Fig 3a. Phase-space plots in 5 - V]j plane.
flux tube, but there are still perturbations persisting in V|
extending to much lower heights.
The distribution function of the ions in the perturbation
region (3750 < S/X,u 7500) is shown for l = 1800 in Fig.
ure 4a, which gives the scatter plots of ions in V x - V|
plane. Transverse acceleration of ions and associated par-
allel acceleration due to the mirror force is clearly seen.
However, we also find some ions gaining only a parallel en-
ergy corresponding to the increase in parallel velocity up to
Vjj 15V (i . This parallel acceleration is the consequence of
the random nature of the ion heating; ions gain perpendic-
ular energies at some stage of the heating and then lose a
part of it at a later stage, after they have moved upward,
and converted a part of the earlier gained energy into their
O
r.
a
Uc
<u
a,
10
t=J200
0 1875 3750 5625 7500
Distance S/A &
Fig. 3b. Phase-space plots in 5 - Vj_ plane.
Sinok and Chan: Plasma Processes Driven by Transverse Ion H latino
11,681
0 6.25 12.5 18.75 25
Perpendicular Velocity Vj JVa
Fig. 4 . Distribution of heated Ions in V ± - V jj plane for (a) en-
tire perturbation (3700 < j/A^ > 7500) and (6) above the forking
point (s > OOOOAjj) at T= 1800.
parallel velocity component. However, the important fea-
ture of the random heating is the production of ion conics
elevated in parallel energy. The elevation in parallel energy
is more clearly seen if only the ions above the strong per-
turbation in the density, where double streaming occurs, are
examined. This is shown in Figure 4 b for ions with 5 > 6000
A,< at i = 1800.
3.4. Perturbation in Average Flow Properties
We now compare the basic features of the plasma per-
turbations produced by the heating in the hydrodynamic
(Figures 2a to 2h) and kinetic models. For the latter model,
the evolution of the perturbations in the bulk plasma pa-
rameters such as density, flow velocity, and effective parallel
and perpendicular temperatures are shown in Figures 5a to
5 d. It is important to point out that the comparison is not
quantitative, only the basic features of the perturbations are
compared here. As expected the localized heating creates a
density cavity topped by a density enhancement. The en-
tire perturbation rides on top of an upward expanding polar
wind into a plasma cavity created by the heating. Unlike in
the hydrodynamic model (see Figure 2a), the leading portion
of the density enhancement has a perturbation extending to
relatively large distances. The extended perturbation is the
consequence of the fast ions running ahead of the major per-
turbation in the density (see Figure 3). However, like in the
hydrodynamic model [Sin<?h, 1992], there is a sharp gradi-
ent near the leading edge of the density enhancement and
it occurs where the 5 - Vjj phase-space plot forks into two
distinct branches consisting of the ambient polar wind and
the transversely accelerated ions as indicated by downward
arrows in Figure 3a. Below the fork, the hydrodynamic pre-
dictions are expected to be true and above it, double streams
occur with large relative velocities, and the hydrodynamic
model fails. The sharp gradient in the density profile is the
forward shock found from a hydrodynamic model [Stngh,
1992], The shock separates the fast streaming ions above it
from the mixed, and relatively warm just below it.
The velocity profiles in Figure 5 b show that at t = 1400,
the perturbation is beginning to develop a double-sawtooth
structure and it is fully developed at t = 1600. The lower
sawtooth in the perturbation occurs near the trailing edge
of the density enhancements, where downward electric fields
occur and retard the upward flow of transversely heated ions.
This retardation of ions produces the doublestream (Figure
3a) feature in the reverse shock region. The hydrodynamic
model fails to handle such a double-streaming. The top saw-
tooth occurs near the leading edge of the density enhance-
ments, the forward shock. However, due to the fast ions
running ahead of the forward shock, the slope of the lead-
ing tooth is considerably reduced. For later times shown in
Figure 56, the upper sawtooth has exited from the top and
only the lower sawtooth can be seen.
It is worth noting that above the forking point in 5 — Vjj
space (Figure 3a), where double streams occur, the average
flow velocity does not give the true velocity of the trans-
versely heated ions because the relatively dense cold stream
(polar wind) weighs down the flow velocity. As mentioned
earlier, this region is not treated properly by a hydrody-
namic model.
In Figure 5c, we show the evolution of the effective parallel
temperature calculated from the equation
N
T'„(IAS) = - V)’/.V (5)
J = I
where N is the number of particles in a cell of length As =
75Arfi, and lAs is the distance from s = 0 with l as an integer.
The parallel temperature profiles show a cooling of ions in
the lowest part of the perturbation. Cooling occurs as the
polar wind expands into the plasma cavity created by the
transverse ion heating. Such a cooling is also predicted by
the hydrodynamic model (Figure 2c). However, the effective
temperature is seen to be elevated considerably beyond the
forking point in the phase-space plots in 5 — Vjj plane (Figure
3a). This is simply because above the forking point there
are double streams and the concept of a single temperature
for the entire ion population is not valid.
The evolution of the effective perpendicular temperature
is shown in Figure 5 d. In this case also, it is worth mention-
ing that above the forking point in 5 — Vjj phase-space, there
are two streams and the effective temperature does not give
the true picture of the heated ions because the relatively
dense cold ion stream (polar wind) weighs down the tem-
perature significantly. It is important to point out that the
heated ions above the the forking point in the 5 - Vy plots
(Figure 3a) are completely lost in a hydrodynamic model,
and these are the ions which appear as ion conics (Figure
4a and 46). There are heated ions even below the forking
point, but they represent an ion population having under-
gone a bulk heating, as a consequence of the merger of the
polar wind and transversely heated ions. The hydrodynamic
model can properly handle this portion of the perturbation.
11,682
SlNOH AND CHAN: PLASMA PROCESSES DRJVRN BY TRANSVER/ *N HEATINO
Fif . 5a. Perturbation in plasma velocity.
0 1875 3750 5625 7500
PARALLEL DISTANCE S (Debye length)
Fig. 5b. Perturbation in flow velocity.
3.5. Parallel Electric Field Generation
Figure 6 shows the evolution of the parallel electric field
generated by the transverse ion heating. The plot at t =
1200 shows essentially the noise in the simulation system just
before the heating. At t = 1400, we notice the development
of a triplet in the electric field perturbation, consisting of
upward (positive) fields in its bottom most part, downward
(negative) fields in the middle, and a relatively localired soli-
tary pulse with upward fields near its top. As the composite
perturbation evolves, the solitary electric field pulse moves
upward with a nearly constant speed; the propagation of
the pulse is indicated by the slant line giving the trajec-
tory of the peak of the pulse in J*f plane. The trajectory
is obtained by projecting the peak point on the horisontal
axis and joining the projection points in the panels for f =
1600, 1800 and 2000 in Figure 6. The slope of this line gives
the propagation speed to be about SAV tlf which is about
46 km/s for the parameters chosen for the run. The pulse
width of the electric field is about 200 m. The maximum
field strength is about 6 x \0~ 2 E 0 % 2 mV. However, it
is worth mentioning that the field strength depends on the
electron temperature as given by equation (4). For higher
StNOH AND CHAN: PLASMA PROCESSES DRJVEN BY TRANSVERSE ION HEATING
11,683
0 1875 3750 5625 7500
Distance S/X^t n e th >
Fig 5d. Perturbation in perpendicular temperature.
electron temperatures, a higher field strength is expected.
For example, if T e was chosen to be 10 eV, fields up to 20
mV/m are expected, and for = 100 eV, the fields scale to
be as high as 200 mV/m. Even the shock processes may en-
hance T* and hence the electric field. [ Forslund <ind Shonk ,
1970]. The electron temperature enhancement occurs when
the electrons are trapped in the potential well created by the
density enhancement. However, in the present calculations
we have assumed electrons to remain isothermal and hence
such effects are not included.
Figure 6 shows that, in the wake region of the solitary
electric field pulse, oscillating fields develop. Such fields are
clearly seen for t > 1600 and they are well developed at
i > 2000. The amplitude of the wave is seen to increase to
8 x 10~ 3 £ 0 % 2.5 mV/m. The oscillating fields are associ-
ated with vortices in 5 — Vjj phase space (Figure 3a). The
vortices can be barely seen from Figure 3. Therefore, we
have replotted them on an expanded scale in Figure 7 for f
= 2200; the vortex sise ranges between 100 to 250 A which
corresponds to the range in the wavelength of the spatial os-
cillations in the parallel fields. The vortices occur over 3750
< s/X'U < 5625, which is the spatial region in which the
oscillating fields occur at this time (see Figure 6e).
The ion-ion instability occurs when the relative velocity
11,684
SlNOH AND CHAN; PLASMA PROCESSES DRIVEN BY TRANSVERSE ION HEATING
0.005 h
K
It
1
K
1
' i = 1400
0.005
" r s
A
0.000
J ]
vj
yVWyVA
0.005
(b)
V
t
1 —
— i
2
0.005
0.000
-0.005
DISTANCE S/A*
Fig. 6. Evolution of the parallel electric fields; different panels show electric field profiles at the times indicated
in the panels.
(Vrei) between the stre&ms is limited to V Tt \ < 2 C t [c.g M
Gresillon and Doveil, 1975 ; Singh , 1978 ], where C, is the
ion-acoustic speed. In our calculation, T< = 3.3To, for which
C 0 ~ 2 Vi,, where V« is the ion thermal velocity given by
(*7o/m) 1/3 . Thus the coupling is expected to occur when
Vret < 4 Va. The space-phase plots (Figure 3a) show that
for the double streaming in the reverse shock region, this ve-
locity condition is well satisfied. On the other hand, for the
double streaming above the forward shock, the two streams
are generally too fast to drive the ion-ion interaction. How-
ever, for such fast streams ion-electron interaction may lead
to instabilities, which occur when the negative energy (slow)
mode of an ion beam is damped by the Landau damping
caused by the thermal electron population [Sinph, 1978 ]. In
the present model, electrons are assumed to obey the Boltj-
mann Law, so this kinetic instability is suppressed from the
model.
3.6. Numerics Versus Physics
Plasma problems in space involve a wide variety of scale
lengths, ranging from plasma Debye length to the geophys-
ical distances. This makes it impossible to develop self-
consistent models including both small- and large-scale pro-
cesses. Recently, large-scale semikinetic models have been
developed to study the polar wind [ Wilson et aL 1990 ;
Brown et a/., 1991 ; Bo et al, 1992 ] and the plasmaspheric
refilling [Lin et al, 1992 ; Wilson et al, 1992 ]. These mod-
els employ a particle code in which the number of particles
is limited to about 10 5 , filling a flux tube of length up to
several earth radii. Thus the models have, on the average,
about 1 particle per kilometer. In these models, electric
fields are calculated from the ion density, which is obtained
by the number of particles in numerical cells and their vol-
umes. In order to have reasonable statistics, the cell siie is
typically several tens of kilometers. Due to these reasons,
the large-scale kinetic models suppress the microprocesses,
even though the codes treat ions kinetically.
In order to demonstrate the above points on how the nu-
merics suppress the physical processes, we repeated the cal-
culations presented earlier with different grid sizes for cal-
culating the electric fields. The evolution of 5 — Vjj phase
space for different grid sizes is shown in Figures 8a to 8 f\
A s = 5A ,i for Figures 8a and 8 b; A s = 20\ d for Figures 8c
and 8d; and As = 100A^ for Figures 8e and 8/. The left- and
right-hand columns of Figure 8 show different stages of the
evolution of the ion-ion instability which occur in the per-
Singh and Chan: Plasma Processes Driven by Transverse Ion Heatino
11.685
0 1875 3750 5625 7500
Distance S/A &
Fig. 7. S - K|| plot at 1 = 2200, showing the vortices on an
expanded scale.
turbations. For As — 5A,i (Figure 8a and 86) the instability,
manifested by the vortices, is much more fine grained than
that for As = 20A,* (Figures 8c and 8d), for which the vortex
formation is quite clear, especially at t = 2200. When As is
increased to 100Aj (Figures 8e and 8/), the instability does
not occur at 1 = 1800, and at t — 2200 the vortex structure
tends to appear, but is not strong enough to fully couple the
two ion streams. When As is further increased, the insta-
bility nearly disappears, even though the ion streams have
nearly the same bulk properties, such as the flow velocity,
density and temperature.
In order to understand the above feature of ion-ion insta-
bility and its numerical realization, we discuss here briefly its
linear properties. The ion-ion instability is limited to rela-
tively long wavelengths given by A > 2irV rr t/tJ J , t , where V T ~t
and <jJ pt are, respectively, the local relative velocity between
the streams and the ion plasma frequency. In the present sit-
uation the relative velocity is about 4V r |, and u? JlX 0.4u> r ,„
(corresponding to a local density of 0.2 inside the cavity),
giving A > 80A,i,. However, the waves are strongly excited
near the lower limit on the wave lengths [see Baker , 1973],
Therefore, when As = 5A,i, the growing waves are properly
described by the numerics because there are several grid
spacings in a wavelength. In Figures 8a and 86, the size
of the vortices is about 100A,i,. When As is increased to
20A,it, the relatively short wavelength waves are eliminated
numerically and those left have a relatively long wavelength.
In Figures Sc and 8 d, the vortices are separated by about
200A,t. When As is increased to 100A,i, the instability is al-
most entirely eliminated by the numerics. However, Figure
8/does show a relatively weak vortex structure, as expected
from the fact that the long wavelength waves, not unaf-
fected by the large grid size, have relatively small growth
rates [Baker, 1973].
The above discussion shows that in order to properly
model the ion waves associated with ion streams in space, a
sufficiently small resolution depending on the ion parallel en-
ergy and plasma density, is needed. For typical energies and
densities in the auroral plasma at relatively high altitudes,
the resolution required is < 1 km. Therefore large-scale
models even though they may be kinetic, fail to treat the
microprocesses, and results from them under the conditions
of counterstreaming and double streaming must be treated
with caution.
4. Conclusion and Discussion
The main aim of this paper is to study the variety of
plasma processes which can be driven by localized trans-
verse ion heating in a diverging flux tube. Although we
have simulated here an artificial flux tube, the main motiva-
tion for this study is the transverse ion heating occurring in
the Earth’s magnetosphere, producing the well-known phe-
nomenon of ion conics. The self-consistent generation and
transport of ion conics, including the driven microprocesses,
are almost impossible to model theoretically because of the
range of scale lengths involved in space plasmas. Therefore,
in order to develop a feel for the possible processes we have
adopted an artificial diverging flux tube, in which effects of
the transverse ion heating on the plasma are simulated. As
described in the previous section, the results from this initial
study are interesting because they show that the transverse
ion heating does not just produce ion conics, it also drives
a host of plasma processes, some of which are revealed here
by the simulation. Among the important processes revealed
are the formation and dynamics of plasma density pertur-
bations, generation of parallel electric fields, multistreaming
of ions, and ion-ion interactions generating oscillating field-
aligned electric fields.
The generation of parallel electric fields by transverse ion
heating is a novel concept. For heating localized to a few km,
the electric field pulse near the forward shock had upward
fields, it moves upward, and has the spatial and temporal
features of weak double layers. Can such fields account for
weak double layers observed in space [Temerin et ai, 1982;
Block et ai, 1987; Bostrom et ai, 1988]? At this time this
is an open question and its answer lies in a rigorous scrutiny
of the theoretical results in view of the observed features of
the fields in space. This has not been done here.
The above feature of the plasma perturbations driven by
the transverse ion heating was previously predicted from a
hydrodynamic model for the polar wind plasma flow [Singh,
1992]. However, in that study the spatial and temporal fea-
tures were too coarse to predict the fine temporal and spatial
features of the parallel electric fields obtained here. Figure
6 shows that the spatial site of the electric field pulse is
100 m; it moves with a velocity of about 50 km/s and the
corresponding time scale of the pulse is 2 ms.
The double streaming of ions produced by transverse ion
heating is noteworthy. The double streaming occurring in
the midst of the density perturbation is the consequence of
the upward acceleration of some ions by the mirror force
while some ions are being retarded downward by the down-
ward electric field in the reverse shock region. This multi-
streaming produces ion waves generating oscillating parallel
electric fields. The role of such fields in electron heating is
mentionable. However, the present simulation model does
not allow it because electrons are assumed to obey the Boltz-
mann law.
The double streaming of ions above the forward shock is
produced by the relatively slow polar w'ind ions being over-
taken by the fast ions produced by transverse ion heating.
The latter ions have the feature of “elevated” ion conics
[Temtrin, 1986; Rorwitz , 1986; Rultqvist et ai, 1988]. The
double streams on the top of the perturbation do not ex-
cite ion-ion instability because their relative velocity is too
fast. However, the presence of relatively warm electrons may
change this by increasing the ion-acoustic speed.
In the present model, electron dynamics is highly sim-
11,686
SlNOH AND CHAN: PLASMA PROCESSES DRIVEN BY TRANSVERSE ION HEATINO
’M
25 1
18.75
S>
u
12.5
O
s
6.25
%
0
3
Oh
25
t=2200 ( d >
1 1 T
t=2200 (f)
0 1875 3750 5625 7500 0 1875 3750 5625 7500
Distance S/A & Distance S/A
Fig. 8. Competition of S - Vj| plott for different values of the spatial resolution: (a) At = SX*,, (6) At _ 20>^,
(e) At = lOOA^j.
plified through the assumption that the electrons obey the
Boltzmann law. If this assumption is relaxed, electrons are
likely to be energized by the parallel fields, especially by the
oscillatory fields driven by ion-ion instability. One of the
puzzling observations in space is the simultaneous measure-
ment of elevated ion conics and field-aligned electrons with
comparable energies [Hultqvist ct al . , 1988]. These particle
populations are observed in conjunction with electrostatic
noise in the frequency range from zero to 300 Hz. The sim-
ulations presented here show how a localized heating can
generate the elevated ion conics and the field-aligned electric
fields which are capable of heating electrons in the parallel
direction. If electron dynamics is included in the model,
possibly other wave modes through ion-electron interaction
can be driven. Simulations with full electron dynamics are
needed to see if the puzzling observations by Hultqvist et
al. [1988] can be explained by localized ion heating and the
processes driven by it.
We have quantitatively demonstrated here how large-scale
hydrodynamic and kinetic codes suppress the small-scale
features of plasma flow because of their inherent coarseness.
Small-scale simulations which keep the essential features of
the problems in space and, at the same time, have sufficient
spatial and temporal resolutions, can elucidate the impor-
tant microprocesses which effectively control the properties
of the plasma flow. This paper presents an initial attempt
towards the goal of understanding the generation and trans-
port of ion conics and associated plasma processes.
Results presented in this paper are based on assump-
tions such as (1) plasma being quasineutral, (2) electrons
are a massless isothermal fluid, and (3) the simulation is
one-dimensional. We already discussed that for the param-
eters chosen in the present simulations the assumptions 1
and 2 are justified. However, for situations involving other
set of parameters, the results presented here can be only
qualitatively correct. For example, if the heating produces
SlNQH AND CHAN; PLASMA PROCESSES DRIVEN BY TRANSVERSE ION HlATlNQ
11,687
a large flow velocity approaching the electron thermal ve-
locity, the assumption of electrons being isothermal is not
justified. This situation requires a more rigorous treatment
of the electron dynamics. We are currently investigating
such situations and results will be reported later.
The assumption of one-dimensional simulation model lim-
its the treatment of the ion-ion instability. In a multidi-
mensional situation, ion beams with relative velocities V re t
> 2 C, can couple together through ion-ion instability [see
Karimabadi et ai, 1991]. The coupling occurs through waves
propagating at oblique angles with respect to the flow direc-
tion. The fast ions above the forward shock can participate
in such instability processes. But the present simulation
model is limited due to its dimensionality.
It is worth mentioning here that the hydrodynamic models
have been used to study the transverse ion heating and their
transport [Ganguli and Palmadesso, 1987; Singh , 1992]. The
qualitative comparison of the results from the small-scale
kinetic simulation and the large-scale hydrodynamic model
shows that the latter model can not handle the phenomenon
of ion conics and its transport; the temperatures and the
flow velocity of the ion conics are grossly misrepresented.
This is true despite the fact that the hydrodynamic models
are quite sophisticated based on 16-moment approximation
[Barakat and Schunk, 1982]. The major problem lies in han-
dling the multistreaming consisting of the ion conics and the
ambient plasma. Large-scale kinetic models [Wilson et ai,
1990; Brown et ai, 1991] do allow for multistreaming, but
the problem lies in the coarse resolution and the consequent
suppression of microprocesses which can critically affect the
flow behavior. A rigorous treatment of the transport of ion
conics including its interaction with the ambient plasma re-
mains a challenge.
Acknowledgments. This work was supported by the Grant
NAGW-2903 from NASA Headquarters to the University of Al-
abama in Huntsville.
The Editor thanks D. Winske and T. Onsager for their assis-
tance in evaluating this paper.
References
Baker, D. R., Nonlinear development of the two ion beam insta-
bility, Phys. Flutds, 16, 1730, 1973.
Barakat, A. R. ( and R. W. Schunk, Transport equations for multi-
component anisotropic space plasmas: A review, Plasma phys.,
U, 389, 1982.
Block, L. P., C. G. Falthammar, O. A. Lindqvist, G. T. Mark-
lund , F. S. Moser, and A. Pedersen, Electric field measurement
on Viking: First results, Geophys Res . Lett., 14, 435, 1987.
Bostrom, R., G. Gustafsson, B. Holback, G. Holgren, H. Koski-
nen, and P. Kintner, Characteristics of solitary waves and weak
double layers in the magnetospheric plasma, Phys. Rev. Lett.,
61, 82, 1988.
Brown, D G., G. R. Wilson, J. L. Horwits, and D. L. Gallagher,
Self-consistent production of ion conics on return current re-
gion auroral field lines: A time-dependent, semikinetic model,
G eophys. Res. Lett. IS, 1841, 1991.
Chang, T. (Ed.), Ion Acceleration in the Magnetosphere and
Ionosphere, Geophys. Monogr. Ser., vol. 38, AGU, Washing-
ton, D. C., 1986.
Forslund, D. W., and C. R. Shonk, Formation of electrostatic
Collisionless Shocks, Phys. Rev. Lett, iS, 1699, 1970.
Ganguli, S. B., and P. J. Palmadesso, Plasma Transport in the
Auroral Current Region, J. Geophys. Res., 9t, 8673, 1987,
Gresillon, D., and F. Doveil, Normal modes in the ion-beam-
plasma system, Phys. Rev. Lett., 34, 77, 1975.
Ho, C. W., J. L. Horwite, N. Singh, T. E. Moore, and G. R.
Wilson, Effects of magnetospheric electrons on polar plasma
outflow: A semi-kinetic model, J. Geophys. Res., 97, 8425,
1992.
Horwits, J. L. Velocity filter mechanism for ion bowl distributions
(Bimodal Conics), J. Geophys. Res. 91, 4513, 1986.
Hulqvist, B., On the acceleration of electrons and positive ions in
the same direction along magnetic field lines by parallel fields,
J. Geophys. Res., 93, 9777, 1988.
Karimabadi, H., N. Omidi, and K. B. Quest, Two-Dimensional
Simulations of the ion-ion acoustic instability and electrostatic
shocks, Geophys. Res. Lett., IS, 1813, 1991.
Klumpar, D. M., Transversely accelerated ions: An ionospheric
source of hot magnetospheric ions, J. Geophys. Res., S4i 4229,
1979.
Klumpar, D. M. A digest and comprehensive bibliography on
transverse auroral ion acceleration in Ion Acceleration in the
Magnetosphere and Ionosphere, Geophys. Monogr. Ser., vol.
38, edited by T. Chang, p. 389, AGU, Washington, D. C.,
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Lin, J., J. L. Horwite, G. R. Wilson, C. W. Ho, and D. G. Brown,
A semikinetic model for early state plasm aspheric refilling, 2,
Effects of wave- particle interactions, J. Geophys. Res. 97, 1121,
1992.
Puri, S., Plasma heating and diffusion in stochastic fields, Phys.
Fluids, 9, 2043, 1966.
Singh, N., The ion-electron instability of ion-beam-plasma sys-
tems, Phys. Lett., 67A, 372, 1978.
Singh, N., Plasma perturbations created by transverse ion heat-
ing events in the magnetosphere, J. Geophys. Res., 97, 4235,
1992.
Temerin, M., Evidence of a large bulk ion conic heating region,
Geophys. Res. Lett., 13, 1059, 1986.
Temerin, M., K. Cemy, W. Lottko, and F. S. Moser, Observation
of double layers and solitary waves in the auroral plasma, Phys.
Res. Lett., 4S, 1175, 1982.
Whalen, B. A., W. Bernstein, and P. W. Daly, Low altitude
acceleration of ionospheric ions, Geophys. Res. Lett., 5, 55,
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Wilson, G. R., C. W. Ho, J. L. Horwit*, N. Singh, and T. E.
More, A new kinetic model for time-dependent polar plasma
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C. D. Chan awl N. Siugli. Department of Electrical ami Computer
Engineering. University of Alabama, Huntsville. AL 35899.
(Received August 19. 1993:
revised October 19. 1993:
accepted November 33. 1993.)
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. A8. PAGES 13,581-13,593, AUGUST 1, 1993
Plasma Expansion and Evolution of Density Perturbations in the Polar Wind:
Comparison of Semikinetic and Transport Models
C. W. Ho, J. L. Horwitz, N. Singh and G. R. Wilson
Department of Physics and Center for Space Plasma and Aeronomtc Research, The University of Alabama in Huntsville
Comparisons are made between transport and semikinetic models in a study of the
time evolution of plasma density perturbations in the polar wind. The situations modeled
include plasma expansion into a low-density region and time evolution of localised density
enhancements and cavities. The results show that the semikinetic model generally yields
smoother profiles in density, drift velocity, and ion temperature than the transport model,
principally because of ion velocity dispersion. While shocks frequently develop in the
results of the transport model, they do not occur in the semikinetic results. In addition,
in the semikinetic results, two ion streams, or double-humped distributions, frequently
develop. In the transport model results the bulk parameters, at a given time, often have
a one-to-one correspondence in the locations of their local minima or maxima. This is a
consequence of the coupling of the fluid equations. There is, however, no such relationship
among the moments produced by the semikinetic model where the local moment maxima
and minima are often shifted in altitude. In general, incorporation of enhanced heat
fluxes in the transport model leads to somewhat improved agreement with the semikinetic
results.
Introduction
W
yr
J5r\
Numerous models have been developed in the last
three decades to treat the outflow of plasma from the
topside ionosphere. These models fall mainly into
two categories: kinetic descriptions and hydrodynamic
descriptions. Hydrodynamic models were first formu-
lated by Banks and Holzer [1968]. In assuming an
isothermal temperature distribution, they found that
the electric field, which is determined by the electron
pressure gradient, is strong enough to accelerate H +
and He + ions to supersonic velocities. This and other
related studies [Banks and Holzer , 1969; Marubashi ,
1970] established the basic characteristics of the polar
wind, such as the ion density versus altitude and the
outflow fluxes.
Realizing that ions become collisionless and their
velocity distributions highly anisotropic at sufficiently
large radial distances, Dessler and Cloutier [1969]
proposed a single- particle evaporative polar “breeze”
model as an alternative to the hydrodynamic approach.
They argued that ion acceleration due to the po-
larization electric field occurs at altitudes where the
mean free path is large, and where the ions cannot
be regarded as interacting directly with each other.
They questioned the pressure gradient term in the
hydrodynamic equations of motion and argued that it
cannot be responsible for the acceleration of the light
ions. This led to the famous Banks-Holzer and Dessler-
Cloutier controversy which is discussed in detail by
Donahue [1971].
Since the early theoretical models of the polar wind
were established in the late 1960s and early 1970s
[Banks and Holzer , 1968; 1969; Holzer et a/., 1971;
Lemaire and Scherer , 1970; 1971], polar outflows have
Copyright 1993 by the American Geophysical Union.
v ^aper number 93JA00635.
V 0148-0227/93/93 JA-00635S05. 00
been studied through the use the hydrodynamic or
transport [Schunk and Watkins , 1981; Mitchell and
Palmadesso, 1983; Singh and Schunk, 1985; 1986; Gan -
guli and Palmadesso , 1987; Gan^u/i et al. , 1987; Gom-
bosi and Nagy , 1988], ion kinetic [Horwitz and Lock-
wood, 1985; Horwitz, 1987] and semikinetic [Barakat
and Schunk, 1983; Li et al,, 1988; Wilson et al., 1990;
Brown et al., 1991; Ho et al., 1992] models.
Transport models involve the solution of a set of N
moment equations solving for N + 1 bulk parameters.
The equation set is closed by expressing the highest
moment as an assumed function of the lower order
moments. The principal advantages of the transport
model include its efficiency in the use of computer
resources (compared to the semikinetic model) and
its ability to easily include chemical and collision al
processes. However, many problems require a detailed
knowledge of the ion velocity distribution function
beyond that which would be available from a transport
model. The ability of a transport model to accurately
describe the velocity distribution increases with the
order of the moment equations employed, but the
highest order equations can be difficult to solve [e.g.,
Gombosi and Rasmussen, 1991]. In contrast, in solving
the Boltzmann equation the kinetic model solves an
infinite hierarchy of moment equations since its results
yield the full distribution function. This however is
achieved at the expense of computer efficiency. As an
approximate solution to the Boltzmann equation one
can solve the gyro- averaged Boltzmann equation by
a hybrid or semikinetic (kinetic ions, fluid electrons)
technique.
In view of the vastly different formulations of the
kinetic and hydrodynamic models applied to the same
geophysical environments by different investigators
over the past two decades, it is necessary to compare
the two approaches in such a way so as to elucidate
the differences, applicability, and limitations of the two
approaches. Except for some limited work done in the
13.581
pn mrnmn mg e blank not fumpo
13,582
Ho et al : Comparison of Semikinetic and Transport Models
early 70’s by Holzer et al, [1971] and Lemaire and
Scherer [ 1972], recently only Demars and Schunk [1992]
have compared the semikinetic with the transport
models for the steady state polar wind. Their results
showed close agreement in the density, drift velocity,
parallel and perpendicular temperatures, and paral-
lel and perpendicular heat flows from both models.
They concluded that the bi-Maxwellian based trans-
port equations are an appropriate tool for studying
space plasmas that develop non- Maxwellian features.
However, good agreement between the steady state
solutions from the two models does not necessarily
mean that they will continue to agree when time
evolving problems are considered.
The purpose of the present study is twofold. First,
and foremost, is to investigate the appropriateness
of using the transport model for dynamic situations,
especially in the collisionless domain. This part of
the study is accomplished by direct comparison of the
moments produced by a transport and a semikinetic
model. Of particular interest is the question of whether
steep gradient persistence (i.e M shocks) are unique to
the transport model. Another question involves the
consequences of phase mixing [Palmadesso et al ., 1988]
which is disallowed in the transport model because of
the truncation of the moment hierarchy but is naturally
included in the semikinetic model. Phase mixing
can be responsible for damping thermal waves. By
analyzing the degree of agreement of transport with
semikinetic models, we can assess the appropriateness
of using such transport models in global systems, where
semikinetic modeling is currently not feasible. The
second purpose of this study is to extend the work
of Singh and Schunk [1985] on the study of the time
evolution of density perturbations in the polar wind.
In the present study a more sophisticated transport
model and a semikinetic model are used to study the
same situations considered by Singh and Schunk.
Semikinetic Model
where the subscripts o represents the various param-
eters of the injected ions at the base of the flux tube,
U|| and v_l are the parallel and perpendicular velocities,,
m is the ion mass and k is Boltzmann’s constant/
For this study, we use the polar wind parameters
similar to Singh and Schunk [1985] for the injected H +
distribution functions: an upgoing drift speed (u 0 ) of
20 km/s; a density (n 0 ) of 500 ions/cm 3 ; and parallel
and perpendicular temperatures (Tjj e , Tx 0 ) of 3560 K.
The parallel force along the magnetic field line acting
on the ions is
F|| = m$j| -b*2?|| - (2)
where qi is the charge of the ion, is the gravitational
acceleration which varies as 1 /r 2 , p (= }^mv\/B) is
the ion’s magnetic moment, and E\\ is the polarization
electric field parallel to the magnetic field, B. B is
assumed to vary as r~ 3 . The term -,uVB is the
magnetic gradient or magnetic mirror force. The
assumed constancy of p determines the perpendicular
speed v±.
By assuming that the electrons are isothermal and
have zero mass, the electric field is given by the
Boltzmann relation
E\\ = —
kT e dn e
n e e dr
( 3 )
where k and e are the Boltzmann constant and the
magnitude of the electronic charge, T c is the electron
temperature taken to be the same as the ion temper-
ature at r 0 , and n e is the electron density which is
assumed to be equal to the ion density.
Transport Model
The collisionless transport equations governing the
magnetic field aligned gyrotropic motion of ions are the
equations of continuity, momentum and parallel and
perpendicular thermal energy given by the following:
The semikinetic model used in this paper is the same
as that developed by Wilson et al. [1990]. The model is
based on a hybrid par tide- in -cell approach which treats
the ions (H + ) as parallel-drifting gyrocenters injected,
at the lower boundary, as the upgoing portions of a
drifting bi-Maxwellian distribution. The electrons are
treated as a massless neutralizing fluid.
We simulate the motion of H + in a magnetic flux
tube extending from 1.47 to 10 R B . Within this
altitude range the plasma is taken to be collisionless.
The ions at the exobase (1.47 R B ) are assumed to be bi-
Maxwellian and the upgoing ions of these distributions
are injected into the simulation region. The distribu-
tion function used for injecting new ions at the base of
the flux tube is given by
dn d —nv dA
Tt + ^ (nv) = T T,
( 4 )
dv
d A
%
K m
k |
1 dn
( 5 )
<9Tji d
rr, £>V Id,
T|| ds nAds^^
„ 1 dA
+ 2 ^*7
( 6 )
(m/2wk) 3 ' 2
/o(f||>°) — no
■exp
( ™(t>n - 1
V 2fcT||,
- Wo) 2
mvj_
2*Ti e
/o(t>|| < o) = 0
0T± d dv 1 dA 1 d ,
1 dA
< ^ ± nA ds
( 7 )
where t is time; r is the geocentric distance to the
point along the flux tube, s is the distance along the
HO ET AL : COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
13.583
3 )
l)
( 7 )
the
the
tube from its lower boundary, n,v,T\\, T x , q\\ and
‘ are the number density, flow velocity, paraUel and
perpendicular temperatures and heat flows of the polar
wind ions, respectively. £y is the parallel elect ” c fie ^
(found from equation (3)), ff|| is the component of the
Lavitational force parallel to the magnetic field, m
o and Jk are the ion mass, ion charge and Boltzmann
constant, respectively. A is the cross-sectional area of
a -UX tube (A oc r 3 ). .
t his set of differential equations is solved numerically
by the flux-corrected transport technique \Borts and
Rook 19761 and are closed by using heuristic expres-
sions for the heat flow „ and 9 x, which dose ly foUow
the treatments in the solar wind studies [Metzler et al,
1979], In a collisionless plasma, the usual expression
for heat flow, given by q a - -K a vl a witn a
(.he.. « de.ole. || o, J.) » tk.
mav not be valid because the mean free path A >
IT~ l dT/ds)~ t. I n such a situation the maximum
heat flow may be given by the transport of theinia
energy (nfcT a ) by the unidirection parallel thermal
velocity = (fcT,|/2irm)i [Palmudesso etal 1988].
Accordingly, it can be shown that (N. Singh et al.,
Comparison of hydrodynamic and semikinetic models
t .r plasma flow along closed field lines, s«hnuUe
Journal of Geophysical Research , 1993), 1993]
q n = (ri a nh'T a Vthi
( 8 )
where e = -1 if dTjds > 0 and * r = 1 if ; OTjds < °-
Thus, the temperature gradient determines g
of the heat flow but not its magnitude. The factor
n gives the reduction in heat flow due to anomalous
plasma effects. In the present calculations we cannot
determine the value of tj„ self-consistently. We study
the effect of the heat flow on the results by varying
the values of n n . Gombosi and Rasmussen [199l\
,ll, in older ro gel
unctions from the 20 -moment mode , the heat flow
must be small compared to the thermal speed times the
pressure. In this paper, tj 0 = 0 represent no heat flow,
while tj„ = 1 corresponds to the theoretical maximum
Teat iow. However, since (8) is only a heuristic
equation, we will take the liberty of using values for Va
larger than unity to study the effect of large heat flow
in \ later section of this paper. Although the above
expression for heat flow is a simplification it allows
the inclusion of heat flow in the study rather easily
and produces reliable results at least for stea y
(Figure 1). As such it is used as a preliminary s y
before the full heat flow transport equations can be
im l‘r,r.1eomp» ie .he mlu of .he «> mo**
to, .he time-dependent eases. »« shall fits, eompat.
the steady state polar wind results. Figure 1 shows
the density, drift velocity, parallel and perpendicular
to the direction of the magnetic field) temperatures,
and parallel and perpendicular heat Hot., of the steady
state P polar wind solutions with boundary conditions
g“e„ fn the last section. Th. results of the two models
K good agreement in general. The drift velocity
obtained from the transport model (solid curve)
higher than that of the semikmetic model. This
discrepancy also appears later when we show the time
evolution of the drift velocity. The reader should keep
this in mind in subsequent comparisons.
The density, drift velocity, and perpendicular tem-
perature of the transport model results are little
affected by the choice of the heat flow parameter Tj a in
equation (8). However, both a higher parallel and per
pendicular heat flow increase the parallel temperature.
We found that a value of 0.3 for both q,, and rjx gives
the closest agreement between the parallel temperature
profiles of the two models. In a later section of this
paper, we shall discuss in more detail the effects,
on the various moments, of varying the parallel and
perpendicular heat flows in a time-dependent situation.
With our particular choice of the amount of heat
flow (nii = tjx = 0-3). there is a cross-over at 5.5 K*
for the parallel heat flow profiles from the two models.
Below the cross-over the parallel heat flow in e
transport model is higher than that of the semikine ic
model Both the parallel and perpendicular heat
flows obtained from the semikmetic model increase
sharply near the lower boundary, and then decrease
with altitude above 1.7 R B . The transport model used
in this particular paper failed to produce this feature
Demurs and Schunk [1992] used a 16 -moment transport
model which produced a sharp bend m the heat flow
profiles at low altitude. This could be due to the
inclusion of collisions in their transport model and/or
their use of the full heat flow equation to solve for
and ox- Their semikinetic model did not produce
the low altitude heat flow bend when they assumed a
Maxwellian velocity distribution at the boundary, but
it did when the distribution was a bi-Maxwellian with
zero stress. It should be noted that Demurs and Schunfe
[1992] compared the steady state polar wind model to
about 2.9 R b whUe a flux tube extending to 10 R B »
used in this paper.
Expansion of the Polar Wind
into a Low Density Plasma
Satellite observations indicate that the ions in the
magnetosphere of ionospheric origin are much more
energetic than those in the ionosphere [ Baugher et al.,
1980; Horwitz and Chappell, 1979]. The energiza ion
of these ionospheric ions can be explained in terms
of various mechanisms, one of which is connected
with the outward expansion of the topside, high
latitude ionospheric plasma along
field lines [Singh and Schunk, 1982, 1986]. In is
section we study the time evolution of the polar wind
expanding into a low density region. The study will be
conducted using both the semikinetic and transport
models described earlier. Our initial conditions are
the same as used by Singh and Schunk [1985], who
assumed a sudden drop of plasma density above a
certain altitude. Note that the initial conditions we
used here (and subsequent sections) may not represent
real physical situations. We are mainly mterested m
the comparison of the results of two different models
under the same conditions. Our results, however, are
important to the study of the time evolution of density
perturbations in space plasmas in general, irrespective
of the initial boundary conditions.
13,584
Ho ET AL.; COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
(a) < b >
(e) (0
Fig. I. Comparison of the semikinetic and hydrodynamic steady state H + polar wind.
T
I
At time i = 0, the density of the steady state polar
wind was lowered to 0.5 ions/cm 3 at and above an
altitude of 9000 km (density profile t 0l Figure 2a).
The plasma was then allowed to evolve in time using
both the semikinetic and transport models. Bulk
parameters were calculated from the ion distribution
function in the semikinetic model at the same selected
times at which bulk parameters from the transport
model were output. The transport model used for
its initial conditions bulk parameters obtained from
the semikinetic model at to* Profiles of the density,
drift velocity and parallel temperature, at different
times, from both the semikinetic (dotted curves) and
transport (solid curves) models are shown in Figure 2.
The profiles in Figure 2 are separated by a time of 5
min.
The density profiles from both models can generally
be broken down into three regions, which are indicated
by a, b and c on profile ti. Region a (r < 3 R B ) is the
unperturbed polar wind solution, region b (3 - 4.5 R B )
is the polar wind expansion into a region of low density
plasma, and region c (r > 4.5 R B ) is the region of low
HO ET XL : COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
13,585
(a)
(b)
(c)
Fig. 2. Comparison of the time evolution of density,
drift velocity, and parallel temperature for H + polar wind
expansion into a low-density region, from the semikinetic
and transport models, to is the initial time, the next three
profiles represent time t=5, 10, and 15 mins respectively.
density plasma still flowing upwards. Region a expands
in altitude range as time advances because the polar
wind is being continuously supplied from below the
lower boundary. The perturbation propagates upward
while the density profile returns to the steady state
solution. As the plasma in the low density region
(region c) moves upward its density decreases because
of the divergence of the flux tube. In a 15-min period
the density at the upper boundary drops by about half.
In region c the bulk drift velocity also decreases slightly
from the steady state value because the electric field
goes from being zero to being slightly negative because
of the positive density gradient.
To understand the region of plasma expansion (re-
gion b ), it is helpful to examine the ion distribution
function. Figure 3 shows the reduced distribution
function which is the ion distribution integrated over
all perpendicular velocities and plotted in a phase
space of parallel velocity versus radial distance. This
distribution is displayed in a gray-scale format such
that darker shades represent higher density. At t = 0,
the electric field at the high /low-density boundary is
very large because it is proportional to the initial large
gradient of the density. This electric field accelerates
ions in both the high- and low- density regions imme-
diately adjacent to the density interface. These ions
flow upwards and disperse in time. As they do, the
density gradient and the large associated electric field
diminish. Also, the dispersing ions produce a region
of elevated parallel velocity, and a region where the
parallel temperature is first reduced below and then
elevated above the steady state temperature profile.
The region of elevated drift speed is simply a result of
the many high speed particles from below overtaking
the slower ions above them. The region of temperature
reduction occurs where ions are cooled by acceleration
(a)
Geocentric Distance (R E )
Fig. 3. Distribution function for H + polar wind expansion
into alow density region at (a) t=0 and (b) t=15 mins. The
phase plot is in gray scale in which a darker shade represents
a higher density.
13,586
Ho ET AL-: COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
through the large interface potential drop. The region
of temperature enhancement develops where two ion
streams exist. The altitudinal extent of each of these
regions expands in time because of velocity dispersion.
The transport model results are very similar to
the semikinetic model results in region a and c, but
there are significant differences in the transition region
b . A sharp and persistent density jump develops
at the upper edge of this region. At the same
location there is an abrupt jump in the drift velocity
and parallel temperature. This shock propagates
upwards with a speed of about 38 km/s which is
consistent with the Rankine-Hugoniot relation [ Singh
and Schunky 1985]. As this shock moves upward a local
density minimum develops below it. This region of
minimum density — where the maximum drift velocity
and parallel temperature occur — behind a forward
shock is a reverse shock [Soneff and Colburn , 1965].
Nothing corresponding to these features are seen in the
semikinetic bulk parameter profiles. They are smooth
and continuous throughout this region.
The parallel ion temperature obtained from the
transport model also has a leading elevated value and
a trailing suppressed value; however, this wave feature
moves up the flux tube more slowly than the similar
feature seen in the semikinetic temperature results.
Velocity dispersion plays an important role in creating
this difference. The first ions to reach a given altitude
are ions with high velocity. When they first arrive,
however, they make up only a small fraction of the
total number of ions present. Their contribution to the
local bulk moments become more pronounced as their
velocity, raised to increasing powers, starts to outweigh
their small relative numbers. One would then expect
to see increasing disagreement among transport and
semikinetic model moments with increasing moment
order, to the degree to which the transport model does
not properly describe the effects of velocity dispersion.
In the case under discussion here, the disagreement is
quite pronounced starting at the parallel temperature
moment.
The shock in the results of the transport model
in Figure 2 can be seen as discontinuous jumps in
the density, drift velocity, and parallel temperature.
Clearly, the values of all three of these moments are
tightly coupled at the location of the shock. In
figure 4a one can see the density, drift speed and
parallel temperature profiles from the transport model
at t = 15 min. In addition to the correlation among
the moments evident at the shock, other instances of
correlation (such as the point where the maximum
drift speed and parallel temperature occur) can be
seen. This correlation is, of course, a consequence
of the coupled nature of the differential equations in
the transport model. The semikinetic model results
display no such correlation among the moments as
can be seen in the profiles in figure 46, This is
a consequence of phase mixing where kinetic effects
damp waves generated by the initial perturbation. In
the transport model such waves persist because the
truncated moment set does not allow phase mixing.
The temperature elevation of the semikinetic model
(e.g., between 7 and 10 R B at t = 15 min in Figure 2c) is
the “effective temperature” that results when the ions
in the low density region and the ions in the high speed
stream are counted as one population. Such “effective
(a) Transport model
^ Drift Velocity^km/s) ^
0 1000 2000 3000 4000 5000 6000
Parallel Temperature (°K)
(b) Semikinetic model
Drift Velocity (km/s)
20 25 30 35 40 45
0.01 0.10 1.00 10.00 100.00 1000.00
Density (ions/cm l * 3 )
l L I till L * 1 1 i I t I
1000 10000
Parallel Temperature (°K)
Fig. 4. Comparison of density, drift velocity and parallel temperature of the (a) transport and ( b )
semikinetic model at £=15 mins for the case of H + polar wind expansion into a low-density region. Note
that the transport model profiles have a one-to-one correspondence in their local minima and maxima.
i-'i
HO ET AL , COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
13,587
temperature” is not found in the transport results when
only one ion stream is simulated. Multistream fluid
codes can be implemented for streams originating from
specified sources, however, such codes cannot model
plasmas which develop multiple streams during the
course of the simulation, unless the locations and times
where such streams will develop can be anticipated.
Like the density profile, parallel heat flow (parallel
thermal energy per unit area per unit time) shows
a -udden drop at about 2.4 R B at t = 0 when the
p rturbation is first imposed (not shown). However,
the parallel and perpendicular heat flow per ion retain
the same profile at t = 0 as that of the steady state
polar wind because the distribution function remains
exactly the same as the steady state polar wind at
< = 0 except for a uniform number density above 2,4
R b . (In the following we shall restrict our discussion
to the heat flow per ion as it is found to be more
illuminating.) Figure 5 shows the parallel heat flow
per ion at time t = 15 min. The semikinetic profile
(dotted curve) has a negative heat flow from about 6.2
to 8.1 Rg which corresponds to the positive slope of the
parallel temperature as seen in Figure 4b. It also has
positive heat flow above and below this region where
te parallel temperature has a negative slope. One can
?e from this that the semikinetic results support the
idea that the sign of the heat flow depends on the sign
of the slope of the temperature, as used in the transport
model formulation (Equation (8)).
The parallel heat flow calculated from the transport
model (solid curve) also has a local minimum and max-
imum around 7.4 Rg. The direction of the heat flow is
determined by the slope of the parallel temperature as
required by (8). The magnitude of the local minimum
and maximum heat flow is about an order of magnitude
less than that obtained from the semikinetic model.
° 5xl0 17 1x10 16
Parallel Heal Flow iergs/cm 2 /s/ion)
Fig. 5. Comparison of the parallel heat flow of the
semikinetic and transport model at t=15 mins for the case
of H + polar wind expansion into a low-density region.
An increase of r) a in (8) will make the comparison of
the heat flow of the two models more favorable. Later
on, we shall see that a larger heat flow will result in
a better agreement of the lower moments also. The
reader should bear in mind that the value of Tj a use
here (r) a = 0.3) was chosen so that the steady state
solutions of both models would be as close as possible.
The effect of varying T) a on the transport model results,
will be discussed later.
Evolution of a Localized Density Enhancement
The Earth-space environment is a region of dynamic
plasma phenomena. Both heavy an£ light ions are
created and destroyed through photoionization and
charge exchange in the ionosphere, and are contin-
uously transported throughout the magnetosphere.
One should therefore expect to find regions in the
magnetosphere where the plasma densities are high and
regions of relatively low density. For instance, density
enhancements at high altitude could arise from electric
field heating at low altitudes [Hultqvist, 1991] followed
by the upward propagation of the hot plasma to higher
altitudes. Recently Singh [1992] has shown that plasma
enhancements and cavities can be created by transverse
ion heating via wave-particle interactions. In this
section, we will investigate the time evolution of a
localized plasma density enhancement in the classical
supersonic H + polar wind.
The density of the imposed plasma enhancement is
given at time t = 0 by
n ««*(r) = pn^rje (9)
where n vvt (r ) is the steady state polar wind density.
n ™>*( r ) is therefore a Gaussian distribution along r
with a peak value of p times n pw at r = r p . We chose p,
<r and r ;> to be 5, 1260 km and 15600 km respectively.
The plasma density enhancement has zero flow velocity
initially and has an ion temperature of 500 K for both
T|| and T ± .
The density, flow velocity and parallel temperature
of the semikinetic model at t = 0, when the density
enhancement was first introduced, are given in Fig-
ures 6a, 65 and 6c and are marked by t 0 (Dotted curve).
Again, the plasma distribution function in parallel
velocity and radial distance phase space, as shown in
Figure 7, are used to interpret the various bulk pa-
rameters. The stationary plasma density enhancement
causes the net bulk velocity to decrease to about 4 km/s
at the peak of the density enhancement, compared
to 23 km/s for the steady state polar wind (dashed
curves). The double peak in the parallel temperature
profile at t = 0 can be explained in the following way.
When nearly equal populations with a relative drift
exist, the parallel temperature will be associated with
the separation, in velocity space, of these populations.
When either population is dominant, the temperature
will be approximately that of the dominant population.
At 2 and 2.8 R B the density of the imposed plasma
population is comparable to that of the polar wind. At
the center of the imposed population, at 2.4 R B , the
imposed ion density exceeds the polar wind background
13,588
Ho BT AL : COMPARISON OP SEMIKINETIC AND TRANSPORT MODELS
(a)
(b)
(c)
Fig. 6. Comparison of the time evolution of density,
drift velocity and parallel temperature for a cold density
enhancement in the H + polar wind from the semikinetic
and transport models. The initial conditions for the drift
velocity and parallel temperature in the transport model are
calculated by using equations (10) and (11). The parallel
temperature, according to the transport model, decreases at
the location of the density enhancement, opposite to what
occurs in the semikinetic model.
density. Where these two different populations have
near equal numbers the effective temperature is the
highest. At points where one dominates the other
the effective temperature tends toward that of the
dominant population.
The density profiles obtained by the semikinetic
model (dotted curves, Figure 6a) show that the density
enhancement flattens out with time. This is due in
large part to the distribution of ion velocities (both pos-
itive and negative) in the density enhancement. This
dispersional flattening of the density enhancement can
be seen in the phase plot in Figure lb where the density
enhancement is now very elongated. The electric field
modifies the dispersion of the enhancement because
above and below the density peak it has opposite signs
(as a result of opposite density gradients). Above the
peak it is positive and accelerates the ions upward,
while below the peak it is negative and accelerates the
ions downward. The downward flowing ions increase
theMensity of the plasma at the lower boundary, and
lower somewhat the flow velocity of the plasma.
In comparing with the results of the semikinetic
model in a consistent manner, one could use in the
transport model the same initial bulk parameter pro-
files as produced by the semikinetic model. However,
in a single-fluid treatment, the initial parallel tem-
perature profile of the semikinetic model would be
interpreted as a warm density enhancement. In order
to find out how a cold density enhancement would
evolve under a hydrodynamic treatment, we use the
transport model with initial conditions established by
the usual definitions for a single fluid:
2 3 4 5 6 7 8
Geocentric Distance (R E )
Fig. 7. Distribution function for a density enhancement
in the H + polar wind at (a) t— 0 and (5) £=15 mins. The
phase plot is in gray scale in which a darker shade represents
a higher density.
HO ET AL : COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
13,589
Y _ noTqo -I- rtiTgi
a no + n i
where f) and f a are the average flow velocity and ion
temperature and a stands for j| or ±. The “0” subscript
denotes the polar wind while the “1” subscript indi-
cates parameters of the density enhancement. In (10)
and (11), nj = n enh - v x - 0 and T ai = 500 K.
7 *ese initial profiles are shown in Figure 6 (solid curve,
i Z 0). The velocity profile at t - 0 obtained from (10)
is very close to the semikinetic model initial profile,
however, the parallel temperature profile at t = 0 has a
minimum value at the peak of the density enhancement
which indicates that the density enhancement is cold.
It is necessary to point out that the discrepancies of
the two models at the lower boundary are due to the
difference between the way the boundary conditions
are handled in each case. In the transport model the
density, the drift speed, and the parallel and perpendic-
ular temperatures have specified unchanging values at
the lower boundary. In the semikinetic model only the
distribution of upgoing ions at the lower boundary is
held fixed. The velocity distribution of downgoing ions
t the lower boundary is determined by what happens
i the flux tube, and as a result, will change with time.
The moments found from integrations over the total
velocity distribution (upgoing and downgoing ions) will
also change. The increase in the density, the drop in the
drift speed and the rise in the parallel temperature seen
in the semikinetic results at the lower boundary result
'from part of the ions from the density enhancement
population falling out of the base of the flux tube.
In the various transport bulk parameter profiles
seen in Figure 6 a number of small scale features
develop. The number of these features increases with
time. They are also seen to move upward with
varying speeds. Although we have not done the wave
analysis of the transport model used in this paper,
we believe that these features result because of the
excitation of several fundamental wave modes by the
initial perturbation. (It is likely that these wave modes
will be different from those discussed by Gombosi and
Rasmussen [1991] because of differences between the
transport model used in this paper and the 20-moment
expansion of Gombosi and Rasmussen.) Differences
in the phase velocity of the different modes lead to
the development of increasing numbers of small-scale
features. If a transport model solving the heat flow
equations were used the solution would change no
doubt; the old wave modes would be modified and new
ones would be introduced. Since the semikinetic results
do not develop the same small scale features as are
produced by the generalized transport model used in
this paper, it is clear that most of these wave modes
are spurious. Phase mixing in the semikinetic model is
responsible for their elimination.
We have also compared with the semikinetic results,
the results from the transport model when its initial
parameter profiles are taken to be the same as those
produced by the semikinetic model at t = 0 [Ho et
all 1993]. Although in this case, an imposed cold
plasma (semikinetic) and a warm plasma (transport)
are compared, it is interesting to note that the re-
sults are closer than the case when a cold plasma
enhancement (according to equations (10) and (11))
is used. Furthermore, Ho et al. [1993] show that when
a strong heat flux was induced artificially by increasing
the value of r} (S in (8), the shocks are eliminated and
the results of the transport and semikinetic models are
much closer.
Evolution of a Localized Density Cavity
In this section, we study the time evolution of a
localized density cavity in the steady state H + polar
wind. The cavity was created by decreasing the density
of the plasma along r by
nc««( r ) = P n pw( r )«"* (_rt) ( 12 )
where p = 0.9, o and r p have the same meaning and
values as in case of the density enhancement (1260
and 15,600 km respectively). The density profile to
in Figure 8a is therefore given by
n(r) = Ttp W — n caTF (^)
where n vw is the density of the steady state polar wind.
Since the ion distribution is unchanged, the velocity
and parallel temperature at t = 0 when the cavity is
created are the same as that of the steady state polar
wind (to, Figures 85 and c).
For the semikinetic model, the cavity propagates
upward, becomes less deep, and extends over a larger
altitude range in time. The cavity propagates with
an average speed of about 30 km/s. From the ion
distribution function (Figure 9), the cavity is seen to
lean towards the abscissa in time. This is again due to
velocity dispersion and explains the spreading out of
the cavity in time.
The drift velocity and the parallel temperature pro-
files can also be readily interpreted by inspecting the
ion distribution function. For instance, at t = 15 min,
the reduced number of low-velocity ions near 5 R®
(Figure 96) causes a higher bulk velocity at that
altitude, while the loss of ions at the high-velocity end
around 7 R® causes a lower bulk velocity there. The
resultant velocity profile is a rounded double-sawtooth
structure (Figure 86, dotted curve).
At t — 15 min, the parallel velocity distribution
function at 4.5 R B and 7.5 R B is narrower than at other
altitudes which results in lower parallel temperatures
there. Note that the locations where the velocity
has a local minimum and maximum do not occur at
the same altitudes as where the parallel temperature
minima occur. (In the transport model results these
locations do line up.) Sandwiched between the two low
temperature regions is a region of higher temperature
(about 1.6 times that of the steady state polar wind)
The high temperature is a result of the double-humped
distribution formed by the cavity seen in the phase
space plot (Figure 96). These structures, in both the
velocity and parallel temperature profiles, propagate
upward in time and become less sharply defined due to
ion dispersion.
13,590
Ho ET AL COMPARISON OF SEMIKINETJC AND TRANSPORT MODELS
(a)
(b)
(c)
Fig. 8. Comparison of the time-evolution of density, drift
velocity, and parallel temperature for a density cavity in the
H* polar wind, from the semikinetic and transport models.
Here to is the initial time, the next three profiles represent
time t=5, 10, and 15 minutes respectively.
The density profiles from the transport model are
significantly different. Figure 8a (solid curves) show
that within 5 min after the local density cavity has
been created, the cavity is being filled with ions to
form two separate cavities. These cavities propagate
upwards with speeds of 22 and 33 km/s, respectively,
getting further and further apart. The velocity profiles
of the transport model also develop a double saw-tooth
structure as in the semikinetic case. However, the
velocity enhancement (lower “tooth”) and depression ?
(upper “tooth”) are separated more and more in time ||
and are linked by a region where the velocity returns to i
the unperturbed steady state value. It is important to w
note that the location of the velocity enhancements and
depressions correspond to the secondary cavities in the
transport model results while for the semikinetic model
they correspond to the inner walls of the original cavity. V
Note also that the overall discrepancy of the velocity
profiles of the two models at later time is due to the
discrepancy of the two models in flow velocity in steady
state (see Figure lb), The time-dependent behavior of
a cavity in a plasma obtained by the transport model
is similar to the results of Singh and Schunk [1985].
Figure 8c compares the parallel temperature of the
cavity in the polar wind obtained by both the semiki-
netic and transport models. In comparison to the
semikinetic model, the parallel temperature of the
transport model shows the same structure of a high
temperature region sandwiched between two low tem-
perature regions. However, the transport model paral-
lel temperature does not spread out as much and the
low and high temperature regions remain distinct with
magnitudes that decrease with time.
Effect of Heat Flow on the
Transport Model Results
The values for T 7 |j and r}± used for the heat flow in
(8) for all the cases we have studied so far is 0.3. This
(a)
60
2 3 4 5 6 7 8
(b)
60
Geocentric Distance (R E )
Fig. 9. Distribution function of a density cavity in the
H + polar wind at (a) i=0 and (b) t=15 mins. The phase
plot is in gray scale in which a darker shade represents a
higher density.
u
a
Fig.
for <
mo(
difl>
of t
vab
and
stat
neg
the
excl
cau
to \
at s
per]
par
was
whi
HO ET AL. COMPARISON OF SEMIKINETIC AND TRANSPORT MODELS
13,591
2000 2500 3000 3500
Parallel Temperature (K)
Fig. 10. Steady state H + polar wind parallel temperature
for different heat flows. The dotted curve is the semikinetic
model results and the other curves are obtained by using
i .fferent values of qy and qx (as indicated) in equations (8)
( , the transport model.
value gave the best comparison between the semikinetic
and transport models at steady state. For steady
state, the amount of heat flow was found to have a
negligible effect on all of the bulk parameters except
the parallel temperature. Figure 10 shows that the
exclusion of heat flow in the transport model (q a = 0)
causes the parallel temperature (dotted-dashed curve)
to be lower than the semikinetic model (dotted curve)
at steady state. We found that both the parallel and
perpendicular heat flow can increase the polar wind
parallel temperature. When qy = li Ty (dashed curve)
was brought close to the curve of the semikinetic model,
while rjx = 1 alone yields an even higher Ty (dashed-
dotted-dotted-dotted curve). When both qy and qx
equal one the highest Ty (solid curve) results. It is
about 500 K higher at the upper boundary than the
case without heat flow. The fact that q± can affect Ty
can be seen from equation (6), in which the last term
converts transverse energy to parallel energy by means
of the mirror force. In comparison with the q\\ term,
9x has a larger effect on Ty because the term which is
dependent on qy in equation (6) can be broken down
into a negative and positive term. The negative term
decreases Ty for increasing fly, while the positive term
is proportional to <?qy /ds, and has a magnitude smaller
than the term which depends on q±.
Although the amount of heat flow has effects only on
the parallel temperature at steady state, we found that
it can greatly affect various other bulk parameters in a
time-dependent situation. By increasing the heat flow
the sharpness of the shocks is reduced and smoother
bulk parameter profiles are produced. This can be seen
from Figure 11 which shows the density and parallel
temperature for different heat flow parameters qai at
a time of 15-min after the density enhancement was
imposed on the steady state H + polar wind. When qy
and qx both equal 1 the shocks produced by the density
enhancement are reduced in comparison to the case
when there is no heat flow (dashed curve in comparison
to dashed-dotted curve, Figure 11). We have seen from
Figure 5 that the heat flow from the semikinetic model
can be about an order of magnitude larger than that
of the transport model when qy and qx is taken to be
0.3. By using large values of qy and qx (7.5 for the
solid curves in Figure 11), the heat flow obtained from
the transport model is increased by 25 times, and the
magnitudes of the heat flow from the two models are
closer.
In allowing qy and qx to be larger than 1 we have
violated the original assumption that the heat flow
cannot be larger than the pressure times the thermal
speed [ Gombosi and RcLSTnusstTi, 1991]. However, since
(8) is only a heuristic formula, there is in practice
10 100 1000100 1000 10000
Density (cm' 3 ) Parallel Temperature ("K)
, , Density and parallel temperature from the semikinetic model (dotted curve) and the transport
r^del with qj and qx given the va’ue of 0 (dot-dashed curve), 1 (dashed curve), and 7.5 (solid curve).
^hese profiles are for a density enhancement case at t-15 min.
13,592
Ho et al Comparison of Semikinetic and Transport Models
no limit on the magnitude of and t }±. It is
shown in Figure 11 that the sharp gradient structures
of the transport model profiles are reduced when
increasing values of ri\\ and tj± are used. The results
obtained by the transport model for large heat flow
are closer to those of the semikinetic model. There
are situations when the hydrodynamic shocks can be
totally dissipated by a large heat flow, this is found in
the case of a warm plasma imposed in the polar wind
[Ho et al, 1993],
The values of the heat flow parameters T 7 || and 77 j.
which were chosen to give a favorable comparison be-
tween the transport model and the semikinetic model
at steady state have been shown to be too small for
a evolving cold plasma density enhancement. This
implies that a more sophisticated form of heat flow
equations such as the full heat flow transport equation
may be needed for a more accurate comparison. The
results obtained in the present study, however, give
strong evidence that heat flow, or even higher-order
moments, are able to reduce the sharp gradient features
of the transport model profiles. This should be true
regardless of the form of the heat flow equation being
used.
Discussion and Conclusion
Closing the set of equations in the transport model
by use of an heuristic heat flow expression, we have
shown, as have Demars and Schunk [1992], that the
transport and semikinetic models agree reasonably well
up through the heat flow moments, in steady state
with supersonic flow. However, for time-dependent
situations, drastic disagreements occur, even for the
lowest-order moments. One of the main differences
between the two models is the development of shock
fronts in the transport model. The semikinetic model
produces smooth profiles in general, and the initial per-
turbation in the density and the other bulk parameters
smooths out and diminishes in magnitude with time,
returning rapidly to the steady state solution. Another
difference between the results of the two models is that
the correlation between the location of local maxima
and minima seen in the results of the transport model
are not seen in the semikinetic model. Additionally,
the transport model may, under certain circumstances,
develop various small scale features which are not seen
in the semikinetic results. One of the main reasons
for these differences is that the semikinetic model
properly includes the effects of velocity dispersion
up through the higher velocity moments. It also
includes the process of phase mixing, which is a thermal
wave damping mechanism [Palmadesso et al. , 1988],
which acts to smooth profiles and eliminate small-scale
features.
In examining the general structure of various bulk
parameters obtained by the two models, the fact that
the semikinetic results are smoother as a result of
velocity dispersion and phase mixing leads to the
argument that the shocks seen in the transport model
results are an artificial consequence of the lack of
these processes in the transport model. Without
the cross boundary relief that these two processes
provide, the density, velocity and temperature of two
adjacent regions can maintain very different values
(i.e., a shock front). This view is supported by the fact
that the results of the transport model are smoother
when a higher heat flow is introduced artificially.
One may argue that the discrepancies between the
semikinetic and transport models may be due partly
to the inability of equation (8) to properly describe the
heat flow, and that therefore, the heat flow equations
should be included in the transport model equation
set. As discussed by Palmadesso [1988] and Gombosi
and Rasmussen [1991], such a higher order model
would still generate spurious waves since it lacks the
higher moments needed to include full phase mixing.
However, the solutions from such a model would differ
somewhat from the transport model results presented
in this paper, and might be closer to those of the
semikinetic approach.
Much of the difference between the results of these
two models is due to the fact that the transport model
encounters difficulty in handling multi-streaming ion
distributions. Although transport equations can be
formulated to simulate multiple ion streams, this ap-
proach is useful only when the origin of the ion streams
are known in advance. In many time-dependent
situations, processes in the evolving system generate
separate streams. The semikinetic model handles the
development of these streams naturally.
One of the attractive features of the semikinetic
model is that the additional information contained in
the velocity distribution function makes it very easy
to understand why certain features are seen in the
bulk parameter profiles. For example, the increase
in bulk velocity in a certain region is usually due to
the presence of high-velocity ions, as in the plasma
expansion into a low density region case, or due to the
reduction of low-velocity ions as in the propagation of
an ion depleted region case. On the other hand, the
decrease of bulk velocity could be due to the presence of
a second stream of low- velocity plasma, as in a density
enhancement case, or the reduction of high-velocity ion
as in the density cavity case. In the same way, the
elevation of the ion temperature in various regions is
often due to the presence of a second stream of ions and
the depression of the temperature can be a consequence
of the narrowing of the velocity distribution in these
regions or the presence of a dominant low temperature
population.
t
Acknowledgments. This research is supported under NASA
grant NAG8-134, NAG8-822, NAG8-239, NAGW-1554 and
NGAW-2903.
The Editor thanks C. E. Rasmussen and another referee
for their assistance in evaluating this paper.
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(Received, August 3, 1992;
revised February 22, 1993;
accepted February 12, 1993)