Radio Science, Volume 25, Number 6, Pages 13111320, NovemberDecember 1990
On the efficiencyof ionospheric ELF generation K. Papadopoulos, C. L. Chang, P. Vitello, and A. Drobot Science Applications International Corporation, McLean, Virginia
(Received June 1, 1988; revised February 21, 1989; accepted November 9, 1989.)
The scalinglaws that control the efficiencyof convertinggroundbasedHF power to ELF power by using modulationof the polar electrojetcurrent is discussed.The analysisis based on kinetic calculationsof the modificationof the ionosphericconductivityby HF waves in conjunctionwith the experimentalresults reported from the Tromso Max Planck and the Alaska high power auroral simulation(HIPAS) facilities. It is shownthat the efficiencycan be increasedby more than a factor
of 104 by (1) usingphasingto sweepthe antennabeamoveran areaspanned by a maximumtilt of 35ø, on a time scale faster than the cooling rate at the heating altitude (<100/•s) and (2) localization of the heating at the E region (90100 km) where electron runaway can be induced resulting in substantial modification of the Pederson conductivity.
1.
Generation of ULF/ELF/VLF
INTRODUCTION
waves by modu
lating ionospheric currents has been confirmed for Generating ULF/VLF/ELF waves by utilizing both the equatorial and the auroral electrojet in the ionosphereas an active medium is an exciting experiments conducted in the U SSR [Migulin and prospect. The techniqueprovidesfrequency agility Gurevich, 1985; Belyaev et al., 1987], the Maxand avoids many of the geopolitical and economic Planck Tromso facility [Stubbe et al., 1982a, b; difficulties associated with large and inefficient Barr and Stubbe, 1984a, b; James et al., 1984; Barr groundbased facilities. By adjusting the HF beam et al., 1985], and the United States [Ferraro et al., geometry, the lowfrequency waves can be made to 1982; Ferraro and Lee, this issue]. Most of the propagate upward in the magnetospherefor use in active magnetospheric stimulation experiments or experiments were performed using HF frequencies downward into the Earth's ionosphere waveguide in the range of 25 MHz, while the power density at regionvariedbetween104 and for communication and geologicalprobing pur the interaction poses. One of the mechanismsdiscussedfor down 103 W/m 2. The most exhaustive studies were converting HF power to ULF/ELF/VLF power by performed at the Tromso MaxPlanck facility and interactioning with the ionosphere requires the the typical results are summarized in Figure 1. presence of ambient ionospheric currents, such as These results are in general terms consistent with the auroral or equatorial electrojets [Germantsev et results produced at other facilities, [Ferraro and al., 1974; Kotik and Trakhtengerts, 1975; Stubbe Lee, this issue; Belyaev et al., 1987] although the and Kopka, 1977; Chang et al., 1981;Ferraro et al., precise values of the detected field amplitudes de1982], while another relies on pure plasma nonlin pend on local conditions, characteristicsof the HF earities and does not require the presence of iono facility, and other specific factors. The purpose of spheric currents [Papadopoulos et al., 1982, 1983; this paper is to explore the potential and the limitaPapadopoulos and Chang, 1985; Ko et al., 1986; tions of the ionospheric generation of ULF/ELF/ Ganguli et al., 1986]. Although both mechanisms VLF waves using as a guide the current experimenhave been verified experimentally, the predom tal results in conjunction with theoretical inance of the experimentsdealt with the modulation extrapolations. Namely, starting from the current of ionospheric currents because this technique de experimental and theoretical status, we develop mands lower HF transmitter power and antenna scalinglaws and use them to explore the efficiency directionality. with which HF power can be converted to the desired lowfrequency range. It is expected that the Copyright 1990 by the American Geophysical Union. conclusionswill be useful in guidingfuture facilities and experiments. For specificity we selectedfor our Paper number 89RS03537. 00486604/90/89RS03537508.00 study the ELF frequency range (i.e., 50500 Hz), 1311
1312
PAPADOPOULOS
ET AL.: ON THE EFFICIENCY
1
OF IONOSPHERIC
the calculation
ELF GENERATION
of the modulated
current
altitude
profile, for any specifiedambient ionosphericelectric field (assumed or measured). This calculation
involves computation of the primary current, the polarization currents required to set up quasineu••E 1 oo trality, and the induction currents caused by the time dependence of the magnetic field. Finally, from the modulated current profile the excitation and propagationof the ULF/ELF/VLF waves in the Earth's ionosphericwaveguide can be computedby 10 2 10 3 1 04 using either the extended source technique [TripModulation Frequency [Hz] athi et al., 1982] or with an equivalent ULF/ELF/ Fig. 1. Typical experimental results in the ELF region gener VLF moment in the ionosphere the reciprocity principle [Galejs, 1968, 1971; Barr and Stubbe, ated by the Max Planck Tromso facility [Stubbe et al., 1982]. 1984a, b]. A comprehensive analysis of these factors lies beyond the scopeof this paper. Within the sincein this rangethe interpretationof the resultsis experimental and theoretical uncertainties our purnot complicated by waveguide resonances, high pose can be accomplished by starting from the order modes, and skin layer thickness. The plan of current experimental results in the frequency range the paper is as follows. The next section summa of 50500 Hz and analyzing them under the assumprizes the statusof understandingof ELF generation tion that the power generated is proportional to the by current modulation and the resulting scaling square of the dipole moment laws. Section 3 discussesthe level of conductivity M = IL (1) modification as a function of the incident power density and altitude. Contrary to previous studies which used fluid analysis to calculate the electron where I is the total modulated current contributing heating, a completely kinetic treatment is used to the ELF field on the ground and L is the linear here. Such a treatment is required for power densi size of the modulated region. All other factors tiesexceeding 103 W/m2. Basedontheanalysis of entering the efficiency calculation will be taken sections2 and 3, section4 discussestechniquesfor from the available experimental data base. For increasing the HF to ELF conversion efficiency. concretenessthe measurementsand analysis of the The final section summarizesthe results and pin Tromso results [Barr and Stubbe, 1984a] are used as our baseline input. These results are in general points highleverage research issues. agreement to the ones reported from high power 2.
IONOSPHERIC
ELF
GENERATIONSCALING
CONSIDERATIONS
Relating the amplitude of the ELF waves on the
auroral simulation (HIPAS) [Ferraro and Lee, this issue]. The range of 50500 Hz has been selected in
order to avoid effects associated with waveguide resonances which arise for frequencies above 1
ground to the extent and characteristics of the
kHz.
modified region in the ionosphereand to the design characteristicsof the HF transmitter is a complex problem. Aspects of the problem have been examined by many authors [Kotik and Trakhtengerts, 1975;Bellyustin and Polyakov, 1977; Tripathi et al., 1982; Fejer and Krenzien, 1982; Bart and Stubbe, 1984a, b]. It involves the following seriesof sequential steps. Based on the transmitter characteristics (power, gain, HF frequency, and modulation frequency) and the applicable model of the ambient ionosphere the conductivity modulation is first
The experimental results indicate that when the heating transmitter operated at a power level of 150 MW ERP the amplitude of the ELF field measured on the ground is 100 tzV/m or 1 pT. These correspond to about 10100 mW of radiated ELF power in the 200500 Hz range [Bart and Stubbe, 1984a]. The equivalent radiating horizontal dipole at 7580 km altitude in the ionosphere is IL • 35 x
computed as a function of altitude. This allows for
104 Am which corresponds to an equivalent groundbased electric vertical dipole with IL •
24 x 103Am.Thepolarization is consistent with predominanceof Hall current modulation. Compu
PAPADOPOULOS
•
ET AL ß ON THE EFFICIENCY
ELF GENERATION
1313
For instance, if we increase the peak conductivity modulationby a factor of 10 while keeping L = 20 km, the radiated power at 500 Hz would increase from 100 mW to about 10 W. Similarly, if we increasethe size L by a factor of 5 while keepingAo
loø
• 34 x 103 s! theradiated powerwouldincrease by a factorof (5)4 6 x 102.
••10 2
A critical input needed for the determination of
the factors that optimize PELF is the scalingof the conductivity modification Aoon the HF power
3
10
OF IONOSPHERIC
2
o
3
Frequency [Hz]
density S at the modified height,
1o
S = PHF/L2
Fig. 2. ELF power versusfrequencyfor the Tromsofacility as determined by the analysesof Bart and Stubbe [1984a]. The
(4)
where PHF is the ground HF power and L is the kin/L)4 to emphasize the scaling withsizeL andconductivityspot size at the appropriate height. Of course this Air. For the Tromso results this factor is unity (i.e., Air • 3 x neglectsabsorptionat lower heights,a point that we 103/s,L • 20 km). will return later on. Assumingthat ELF powerscalehasbeenmultipliedby (3 x 103sI/Act) x (20
Act  [PHF/L2]a tations based on a fluid model indicate that for an
we find from (3)(5) that
assumedambientionosphericelectricfield E,  25 mV/m the peak values of the modulated current
densityare in the rangeof 108 A/m2 and are located between 75 and 80 km in altitude. The effective horizontal radiated current moment M can
2a L4(i  a) PELF "' PHF
(6)
For a = I we find that PELF is independent of the antenna gain and scales as
be estimated by height integration as
M = IL = AjAzL2 = AcrAzEa L2
(5)
PELF"' P•F
(7)
(2) Namely, the HF to ELF conversion increases as
where Aj is the modulatedcurrent density, Az is the extent of the effective radiating layer in altitude, and Aois the modulated conductivity. For the typical experimental conditions L = 20 km (i.e., at a heaterbeam width of 15ø),Az • 10 km, and E a •
the square of the HF power. For a >> I the ELF conversion efficiency increases enormously by in
creasingthe groundbasedHF power PHF while maintaining the same antenna gain (i.e., L  constant). For a << 1, more efficient conversion re
25 mV/m, the value of IL = 35 x 104 Am quires large spot sizes. A computationof the value corresponds to a peakvalueof Ao= 34 x 103sl. of a for various altitudes and HF power densities is
This is achievedwith an incidentHF power density performed in the next section.
of 2 mW/m2 at 7580kmheight.Finally,it should be noted that the value of IL is independent of
'frequencyand the frequencydependencein the ELF power is attributed to the scaling of the excitation efficiency [Galejs, 1971]. Since the quantities that can be controlled from the groundare the value (and possiblythe altitude location) of the conductivity modulation Aoand the size L, it is instructive to cast the experimentaldata as interpreted by Bart and Stubbe [1984a] in the form of Figure 2. Notice the importantscalingvalid for each ELF frequency, i.e.,
PELF '•' M2 "' (Acr)2L4
3.
DEPENDENCE OF CONDUCTIVITY ON ALTITUDE AND POWER
MODIFICATION DENSITY
In this section we evaluate the level of cop.duc
tivity modulationat variousionosphericheightsas a function of the incident HF power density S. The calculation is "local" and neglectstransport. From the modulated conductivity we can easily evaluate the modulatedcurrent for a given ionosphericelectric field. Contrary to previous studies [Stubbe and Kopka, 1977; Tomko, 1981; Chang et al., 1981; James, 1985] which used fluid equations to compute (3) the variation of the electron temperature Te as a
1314
PAPADOPOULOS
ET AL.' ON THE EFFICIENCY
linear approximation, the present study uses the complete time dependent kinetic equation for the electron energy distribution function f(e). For an HF electric field of peak amplitude E o and frequency •oo at the chosenheight,f(e) is given by [Gurevich, 1978]
(O/Ot)f(e) = el/20/Oe[e3/2D(2fi2e)]  •(e)
(8)
OF IONOSPHERIC
ELF GENERATION
6
10•6
1Os
.
0•
t
0a
10•

•
014 •Z 1 0;3 o 
10•
1
0•2••.
1
0• "'" o
10•ø
_ D(e)=al (e2E•/m)v(e)[(Wo + •)2 + v2(e)]1 (9) 60
•
•
•
•
•
•
•
'
70
80
90
100
110
120
130
140
10
50
where v(e) is the energy dependentelectronneutral altitude (km) collisionfrequency at the chosenheight and 12is the Fig. 3. Ionosphericprofile used in the FokkerPlanckruns. electron cyclotron frequency. The _+ signs correspond to o(+) and x() mode heating correspondingly. The term L(e) is an operator that represents the energy loss due to various inelastic processes. and high altitude is obvious from the distribution Its form is discussedin the appendix and includes function shownin Figure 4. Figures 4a4 c show the excitation of rotational, vibrational, and optical initial and steady state distribution functions for levels as well as ionization and attachment for N 2 ionosphericheating at 75, 90, and 100 km altitude and at valuesof S  103 W/m 2 102 W/m 2 and and02. Note that the latter processis not important 102 W/m2 correspondingly Thetimerequiredto for the power densitiesanalyzed here. Equation (8) reach steady state was in all cases much shorter was solved numerically for f(e, t) at various altithan 10 4 s. This implies that steadystate is tudes, HF power density values S and modulation reached at times much shorter than the relevant frequencies•o. The form of v(e) and the numerical method for solving (8) are given in the appendix. modulation frequencies. It is seen that at lowandlowpower density(i.e., 75 km, 103 The time dependenceof the conductivity is found altitude
W/m2)f(e)doesnotdeviatemuchfromMaxwellian
from
and the fluid descriptionis a reasonableapproxima
m2 e2(e)f(e, t)e de ne • •Q2+p(e) m2/ ,0,2 +D /x2(e)f(e, t)e1/2 de ne
ITp(t)= •
1/2
(10a)
tion. This, however, is not true for the'other two caseswhere the highenergytails of the distribution functions become the dominant part.
Figures56 showthe time dependenceof the Hall and Pedersen modulation
•h(t) = •
(10b)
for the above three cases.
At 75 km and 103 W/m2 the Hall conductivity
modulationis about7 x 103s] which is consistent with the value quoted in section 2 for the Tromso o'z(t)= • f(e t)e de (10c) experiments.The Pedersonconductivity modulam •• ' tion is significantlysmaller.This is reversedfor the 90 and 100 km cases at 102 W/m2. The Hall whereOp,oh, andoz arethePederson, Hall, and parallel conductivities and n is the electron density. conductivity modulation becomes progressively As noted in the introduction a kinetic analysis is an smaller and is negligible at 100 km. Furthermore, absoluterequirement for exploring highpower den the level of the Pedersonconductivity modulationis sities. The initial f(e, t = 0) was taken as Max about3 x 104s 1 at 90 km and 6 x 104 s 1 at 100 wellian at Te • 0.025 eV. For the studiesreported km. It is clear that if the size L, which is controlled belowwo = 1.8x 107rad/swhichcorresponds toa by the antennagain was the samefor all three cases and the increasein the power density was entirely heater frequency of 2.8 MHz. We report below resultsfor daytimeionospheric due to an increasein PHF by a factor of 10, the conditions corresponding to altitudes between 70 radiatedPELF would increaseby a factor of 20 and
ne 2j. 1
1/2
and 100 km. The ionospheric model used is shown in Figure 3. The need for a kinetic description at high power
100 for the 90 and 100 km cases over the 75 km case.
Notice that since steady state is established for all cases much earlier than the lowfrequency oscilla
PAPADOPOULOS
ET AL.'
ON THE
EFFICIENCY
OF IONOSPHERIC
ELF GENERATION
1315
(a) I •% Initial
•
•_
'" O.O 0.3
0.6
0.9
].2
(c)
].5
t.8
2.1
2.4
2.7
3.0
0.0
0'.3
0'.6
0'.9
]'.2
Energy(eV)
]'.5
t'.8
2•.1
2'.4
,
2.7
I
3.0
Energy(eV) Fig. 4.
(continued)
(b)
104W/m 2. The resultsfor the Hall and Pederson conductivities are shown in Figures 7 and 8. The boundary altitudes were chosen in a way that they reflect the range of variation of the Pedersen and
.(2_
Hall .(2_
conductivities
with
altitude.
A more
detailed
analysis of the consequencesof the results shown in Figures 7 and 8 and of the efficiency optimization at intermediate altitudes will be presented elsewhere. We restrict our discussionhere to the consequences of the general trends derived from Figures 7 and 8.
SteadyState
These
are:
1. For low altitudes (7075 km) the Hall conductivity provides the dominant contribution. The 0.0
0.3
0.6
0.9
].2
].5
t.B
2.1
2.4
2.7
3.0
Energy (eV) Fig. 4.
Initial and steady state electron distribution œor(a) 75
km, S = 103 W/m2' (b) 90km, S = 102 W/m2' and(c) 100kin, S = 102 W/m2.
tion period, the values of Art are independentof the ELF frequency. In order to determine the scalingof the conductivity modification with power density and altitude we conducted a survey of the level of steady conductivity modification for altitudes (70 and 100 km) and values of S in the range of
value of Aoh increases very weakly with power density (a < 1/2) and saturates at a value of S about
103 W/m2. Furtherincrease in thepowerdensity does not produce any increase in the modulated current density. According to the discussion after (7), optimization of the conversion efficiency requires increase in L under constant S. 2. For high altitude (>90 km) heating the modification of the Pederson conductivity dominates.
Thevalue of/kO'p increases asS2(i.e.,a = 2),upto
powerdensityof 102 W/m2 andsaturates slowly afterward. There is an obvious premium in increas
ing PHF, sincein this casePELF'" P•F, while keepingS  102 W/m2.
1316
PAPADOPOULOS
ET AL.'
ON THE
EFFICIENCY
OF IONOSPHERIC
ELF
GENERATION
(a)
(c)
o
o
o.ooom.do,s o.dozoo.do•so.4or, o o.,3• o.4o•oo.b,oso.b,2o o.b,•s o.o,so
Time (sec)
o.o•o.oo,s 0.0030o.oo4s
Time (sec)
Fig. 5.
(continued)
(b)
interesting feature is a direct result of using the kinetic approach to estimate Hall conductivity. Specifically, at highaltitude and high heating power, the energy integral in (10b) gives an overall positive contributionto rrh as the electron distribution shifts toward the high energy end. In contrast, the conventional fluid approach always predict a decreasein oh since the collision frequency •,(Te) increases with the fluid temperature Te. Before closingthis section we should caution the reader in the interpretation of the above results. In order to have uniformity in the results and be close to the benchmark case, the survey shown in Figure 4 was performed using the same frequency o•o = o.oom•.do,s 0.4030o.do4so.dor, o o.b•
o.•
o.b,os0.4,20 o.•,,sso.o,so
Time (sec) Fig. 5.
Hall conductivity modification for the cases of Figure 4.
1.8 x 107for all the threealtitudesexamined(75, 90, 100 km). This frequency corresponds to a criti
calelectron density n•.= 105electrons/cm 3 which correspondsto an altitude of 105 km for the model ionosphereused here. The electron densitiesat 75,
90, and100km (Figure3) are 3 x 102electrons/ 3. Maximum value of Air is achieved for highaltitude heating.
cm3 8 x 103electrons/cm3 and 8 x 104 electrons/ cm3. Since the code does not include collective effects, the 100 km results are close but within the
Note that in Figure7, the IAo•lcurveat 100km validity range of the model. Furthermore, as long as
shows a dipnearthepowerlevelof 0.4W/m2.This
the half width of the heater is smaller than 26ø,
implies a sign changeof Air h from minus to plus as the heating power increases at high altitude. This
effects of resonance absorption can also be neglected even for the 100 km case. However, we
PAPADOPOULOS
ET AL.'
ON THE EFFICIENCY
OF IONOSPHERIC
ELF GENERATION
1317
(a)
'13 c
O
o
ul
o
•.oa:•.•o•s o.•o•o o.•o,s o.,•or, o o.•s
o.•
o.hos o.h2o o.h•s o.o•so
o.ooo•.•otso.•o•o o.•o,s o.•or, o o.•,s o.;•o o.hos o.•2o o.&•s o.o•so
Time {sec)
Time (sec) Fig. 6.
(continued)
(b)
where b is the quiver energy of the electrons in the highfrequency field, with an effective frequency
Weft= Wo + f•, definedas
• m(eEo/mw elf)2 Our resultscan be generalizedto any frequency by noting that the solution of (8) is selfsimilar with respectto the value of b. Thus for a frequency o) > wo a power density is higher by a factor (w _+
fi) 2/(wo +1•)2 thanfor w = wo willberequired.
6
Altitude
,.oa•.•ots o.,•o•oo.•,s o.•,o o.,•
Time (sec)
Fig. 6.
=
5
o.•,o o.,hoso.,h2oo.h•s o.o,so
4
Same as Figure 5 for the Pedersonconductivity.
laOhl 10 (1/sec) 103 2
10
should note that for the high altitude cases(>90 cm)
the resultsare applicableto frequencieshigherthan the 2.8 MHz by a simple scalinglaw based on (8) and (9). Note that for Wo > 2.8 MHz and h > 90 km, + f•)2 >> v(e). Therefore (0)o 1
D(e) = • gv(e)
o
101 0'4
10
3
10
2
10
1
10ø
S (wattsire 2)
Fig. 7.
(11)
100 km
1
Hall conductivity modulationversus S for 70 and 100 km.
1318
PAPADOPOULOS
ET AL'
ON THE
EFFICIENCY
6
GENERATION
mW/m2, theresultant conductivity modification ac
Altitude
cording to Figures 7 and 8 increases by a factor of 100 over that for lowaltitude heating. According to
4
(3), thiswill producea factorof 104 morepower
(1/sec) 103
than the 510 roW, which results in 50100 W in
2
ELF even in the absence of sweeping. The system
efficiency increases by a factorof 103givingan overallconversion efficiency of 105. Incorporating
1
0
0
4
10
3
10
2
10
1
10
S (watts/m 2)
Pederson conductivity modulation versus S for 70 and
EFFICIENCY
OF HF TO ELF
In this section we combine
a similar sweep as for the lowaltitude heating will result in further increase of the efficiency by a factor of 400 giving an overall efficiency better than 4 x
103 . The practicaldifficultyin realizingthe highaltitude scheme, especially under day time conditions, is the fact that self absorption at lower heights could prevent the achievement of power densities
100 km.
4.
ELF
achieved with power density of the order of 10
5
Fig. 8.
OF IONOSPHERIC
CONVERSION
the results of Bart and
Stubbe [1984a] as shown in Figure 2 with the results reported in section 3 and use them to determine the HF to ELF conversion efficiency and techniquesby which it can be optimized. On the basis of Hall conductivity modulation of the polar electrojet at 7075 km altitude, Bart and Stubbe [1984a] estimate a power conversion efficiency of 510 mW per MW of HF at 200 Hz. This implies an overall
conversion efficiency of about108 , compared with 106 conversion achievedby the Wisconsin Test Facility. For lowaltitude heating the results of section 3 indicate that increasing the power does not have any significant effect in the ELF power. However, as shown in (3) for constant power den
of 10mW/m2 at 95100km altitudefor frequencies of 2.8 MHz used in our calculations. Such heating can be achieved by using higher frequencies, beating two HF waves at the local plasma frequency, or using short pulses that allow the power to "sneak through" to high altitudes. These possibilities are currently under study. The enormous increase in efficiency achieved in highaltitude heating places a large premium in realizing them even if their efficiency is by an order of magnitude lower. 5.
SUMMARY
AND
CONCLUSIONS
We presented a detailed kinetic study of local heating of the ionospheric plasma by modulated radiowaves, of the resultant conductivity modulation and of the associated
modulated
current
den
sity and thereforeconstantAo,PELF'" L4. An sity in the presence of an ionospheric electric field increase in area while maintaining approximately in the ELF region (50500 Hz). Regimes where the the same power density can be achieved by using conductivity modulation is a strong or weak funcphasing to sweep the antenna beam over an area tion of the incident HF power density were identispanned by a maximum tilt of 0•, in eachdirection fied. Combining these results with the recorded at a rate faster than the cooling rate, which for 80 observationsin this ELF region and assumingthat km is approximatelyfew microseconds.PELF will to zero order the ELF power density is proportional 4 we foundthatthe HF to ELF power increaseaccordingto (3) by a factor of (tan (Om + to (Ao')2L 0o)/tan00)4. Taking0o = 7.5øandOm= 35ø, we conversion efficiency can be increased by more find an increase on the ELF power by a factor of than 2 orders of magnitude if the heater could be 2 x 103.Thusa facilityof thetypeof Tromsoor swept over a 35øcone on timescalesfaster than the HIPAS equipped with fast sweeping over a cone of plasma cooling rate (10/•s) for lowaltitude heat35ø can produce 510 watts of ELF power and can ing. Heating techniquesthat can preferentially dehavean efficiency of 105 whichis betterthanthe posit their energy at higher altitude (90100 km) efficiency of the Navy ELF Wisconsin test facility. where the dominant modulated current is the PedAlternately, higher efficiency as well as higher erson current, can further increase the efficiencyby partof thisincrease canbe PELF can be produced by highaltitudeheating. If a factorof 104 although HF heatinglocalized in the 95100 km region can be negatedby the potential inefficiency of highaltitude
PAPADOPOULOS
ET AL.:
ON THE
EFFICIENCY
heating as well as by a more inefficient coupling to the waveguide. We are currently examining these issues theoretically and expect to resolve them in combination with the Penn State HIPAS experimental campaigns.
OF IONOSPHERIC
ELF
GENERATION
1319
where oa is the attachmentcross sectionand N o is the 02 number density. Ionization
Si
f(e, ep)epf(ep)Cri(ep) dt•p
mu
APPENDIX
The inelastic term •(e) in the FokkerPlanck equation (equation (8)) is a summation of the various inelastic
collision
contributions
•(E) = Sr 3Sv,o 3Sa 3Si
(A1)
Detailed expressions of these inelastic contributions are given as follows [Gurevich, 1978]: Excitation
of rotational
ef(e)cri(e)l
(AS)
where e i is the ionization energy and •i is the ionization
cross section.
The collision frequency P(e) used in (9) contains the elastic
collisions
of the electrons
with
the neu
tral molecules. As such, the P(e) can be written as
levels
p(e)= • Nmv•m(e)
(A6)
IH
Sr =• 2v20v v2Rr(v)
•vv+ vf
(A2)
where
Rr(v) = 8Boo'oNm/mV
O'o= 8rrQ2ao2/15 ao = h2/me 2
Bohrradius
Q = 1.04
for N2
Q = 1.80
Acknowledgments. The work was supported by the Office of Naval Research through Penn State subcontract ONRTPSU
for 02
Bo = 2.48 x 104 eV
for N2
Bo = 1.79x 104 eV
for 02
SAIC 0677388.
mv
•'. [(e+ e/•)f(e+ e/•)o/•(e + k
 •f(e)•(e)]
(A3)
where k standsfor different levels, okisthe energy of the k state, and okis the cross section. Attachment
S. = Novcraf
and Lee of Penn State
REFERENCES
Excitation of vibrational and optical levels 2Nm
Discussions with Ferraro
and Ossakow of the Naval Research Laboratory are gratefully acknowledged.
N m is the number density of N 2 and 02, and Sr is summingover both N 2 and 02 species.
Sv,o =
where N m is the neutral density at the chosen height, •m is the momentum transfer cross section, and the summationis carried over both O2 and N2 species. Numerically, a finite difference scheme for initial value problems of FokkerPlanck equations has been applied to solve the governing equation (equation (8)). Details of the numerical method can be found in the work by Chang et al. [1970].
(A4)
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