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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 
12 1 See paragraph 2.b. of financial data sheet 
13 / See paragraph 2.c. of financial data sheet 
/4/ See paragraph 2.d. of financial data sheet 




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The cost estimate presents applicable pricing information in the standard format adopted by the University. 


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are MFRTT as aCtuals and are increascd b y 4 0 P c rcent each fiscal year lo cover anticipated raises. These increases 

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b. Fringe benefits: 

the indirect rate k abSe " CeS ^ 35 VaCali ° n ’ *** ^ h °’ idayS ^ mC ' UdCd SalaricS a " d are char S ed as a direct «P«»e as negotiated in 


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March 1994 


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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 


O 

C 


(/) 

E 

JX 





> 

<D 



UJ 



> 

a> 


10 2 

P 

it 

h 


V t) I ' «l 



1 1 1 Tlhj- 

t ft - 


1 



1 • * 

n r i - 

E 

20 

1 * |1 

o 

O 

L 


II 

* i 

Oft? 

1 

0 

-20 

_ J L 

t=t 4 

1 1 1 1 

N 

1 

o 

f 

. 1 

L - 

i j 

UJ 

-60 

0 

.2 

.4 

.6 0.8 1 

C 

1 0.2 0.4 0.6 0.8 1 


S/ 1 0000km 


S/ 10000km 


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., 
1986. 

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, 
1978. 

Wilson, G. R., C. W. Ho, J. L. Horwit*, N. Singh, and T. E. 
More, A new kinetic model for time-dependent polar plasma 
outflow: Initial results, Geophys. Res Lett., 17, 263, 1990. 

Wilson, G. R., J. L. Horwils, and J. Lin, A semi-kinetic model 
for early-stage pi asm asp here refilling, 1, Effects of Coulomb 
collisions, /. Geophys. Res., 97, 1109, 1992. 


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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C. W. Ho, J. L. Horwitz, N. Singh, and G. R. Wil- 
son, Department of Physics and Center for Space Plasma 
and Aeronomic Research, The University of Alabama in 
Huntsville, Huntsville, AL 35899. 

(Received, August 3, 1992; 
revised February 22, 1993; 
accepted February 12, 1993)