Numerical simulation studies on the broad upshifted maximum of ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. A7, PAGES 12,787-12,801, JULY 1, 2001

Numerical simulation studies on the broad upshifted maximum of ionospheric stimulated electromagnetic emission H. Xi and W. A. Scales Bradley Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State University, Blacksburg, Virginia

Abstract.

Stimulatedelectromagneticemission(SEE) producedby interactionsof

high-power radio waveswith the Earth's ionosphereis currently a topic of significant interest in ionosphericmodification physics. SEE is believed to be produced by nonlinear wave-wave interactions involving the electromagnetic and electrostatic plasma waves in the altitude region where the pump wave frequency is near the upper hybrid resonancefrequency. The most prominent upshifted feature in the

SEE spectrumis the broad upshiftedmaximum (BUM). From characteristicsof this feature, a four-wave parametric decay processhas been proposed as a viable

mechanismfor its production.The object of this work is to (1) investigatethe early time nonlinear development of the four-wave decay instability by using theoretical

and numericalsimulationmodels,(2) study the variation of the four-wavedecay instability spectral features for a wide range of plasma and pump wave parameters, and (3) accessits possiblerole in the productionof the BUM spectral feature. Results of this investigation show that there is good agreement between predictions of the proposed theoretical model and the numerical simulation experiments. The simulation electric field power spectrum exhibits many of the important features of the experimental observations. The numerical simulation results show that considerationof the full nonlinear developmentof the four-wave parametric instability is crucial in providing insight into the asymmetric nature of the wave frequency spectrum observedduring the experiments.

1.

Introduction

Stimulatedelectromagnetic emission(SEE) hasbeen observed during ionospheric modification experiments where a powerful high-frequency O-mode pump wave is transmitted from heating facilities located at Alaska, United States; Arecibo, Puerto Rico; Nizhniy Novgorod, Russia; and Troms•b, Norway. The SEE exhibits frequency sidebands upshifted and downshifted from the pump wave frequencywithin roughly 100-kHz bandwidth. Interest in SEE has been steadily grow-

ing sinceit wasfirst experimentallyobserved[Thiddet al., 1982]becauseit may be usedas a diagnostictool for the ionosphereand it is also a fundamental nonlin-

ear plasmaphenomenon [Stenfio,1990]that is not well understood. A classificationof SEE spectral features and the descriptionof their possiblegeneration through parametric decay instability processeswere provided by

are the downshiftedmaximum (DM), the broad downshifted maximum (BDM), and the downshiftedpeak (DP). Some of the frequency-upshifted sidebandsare the broadupshiftedmaximum(BUM) andthe upshifted maximum(UM). The BUM is one of the most prominent upshifted features in the SEE spectrum and has been the subject of intensiveinvestigationin past ionosphericmodificationexperiments[Leyseret al., 1989, 1993; Stubbe et al., 1994; Frolov et al., 1996, 1997, 1998; Wagner et al., 1999]. The BUM is a broad upshifted sideband extending more than 100 kHz above the pump frequencyand developsonly whenthe pump frequency w0 is in the vicinity of harmonics of the electron cyclotron frequency,nf•½e. It is widely consideredthat the BUM developsfor 0•0 slightly above nf•ce. However, it was also reported that the BUM could appear when the pump frequency is a few tens of kHz be-

Stubbeet al. [1984]. Someof the important SEE fea- low the gyroharmonic frequency[Frolovet al., 1996].

tures that consist of frequency-downshifted sidebands When the pump frequency is more than 25 kHz above Copyright 2001 by the American Geophysical Union.

Paper number 2000JA000322.

0148-0227 / 01/ 20003A000322$09.00

the gyroharmonicfrequency,as it is for most experimentswhere the BUM is observed,the frequencyof the BUM peak 0dBU M followscloselythe empirical relation 0dBUM-- 20;0- n•ce [Leyseret al., 1989; Stubbleet al., 1994; Frolov et al., 1996, 1997]. Recently,exten12,787

12,788

XI AND SCALES: NUMERICAL

SIMULATIONS OF BROAD UPSHIFTED MAXIMUM

The four-wavesecond-order process[HuangandKuo, in interpretingsomeaspects ionospheric modificationfacilitiesof the Radiophysical 1994]hasprovensuccessful ResearchInstitute (NIRFI) of Nizhniy Novgorod,Rus- of the experimentalobservationsof the BUM. However,

sive SEE measurements were conducted at the SURA

sia[Frolov et al.,1996,1997,1998;Wagner et al.,1999]. the asymmetric nature of the spectrum that is a fundamental feature of the experimental observationsis still open to questionswithin the framework of past purely theoretical investigations.Furthermore, past work has not considered the full nonlinear development of the fects on the BUM behavior. Other major findings of four-wave decay instability and the important consetheseexperiments includethe two-component natureof quencesto the experimental SEE spectrum. Although the BUM [Frolovet al., 1998]andthe ionospheric self- numerical simulations may ultimately provide imporconditioningand preconditioning effectson the BUM tant contributions to the understanding of nonlinear processesand bridge the gap betweentheoretical develcomponents [Wagneret al., 1999]. The empiricalrelationCOBUM -- 2co0-- nf•cesuggests opment and experimentalobservations,there have been

Variousexperimentalconditionsand plasmaand pump wave parameters,which includethe numberof individual pumps,their powers,frequencies,polarizations, and on-off patterns, were used to investigatethe ef-

that the BUM feature is generatedthrougha four-wave interaction,involvingtwo pump photonsor upper hybrid plasmons,a decaymode at nf•ce, and the stimulated radiation at COBUM [Leyseret al., 1989, 1990; Bud'ko and Vas'kov,1992; Goodmanet al., 1993; Tri-

few studiesin the past [Goodmanet al., 1994; Scales et al., 1997; Hussein and Scales, 1997; Hussein et al.,

1998]. The purposeof this work is to useboth theoretical models[HuangandKuo, 1994]and numericalsimu-

lations to investigate the development of the four-wave decay instability that is proposed as a viable mechapathi and Liu, 1993;Huangand Kuo, 1994]. In the

generationmechanismproposedby Tripathi and Liu nism for the generationof the BUM. As an extension [1993]andGoodman et al. [1993], theBUMisproducedto previouswork [Husseinet al., 1998],hereemphasis by nonlinearmixing of lowerand upperhybrid waves will be placed on interpreting the behaviorof the BUM througha two-stepprocess.Huangand Kuo [1994] spectral features for a wide range of plasma and pump pointedout that the difficultyof usingthe two-steppro- wave parameters. Effects of varying the harmonic numcessto explainthe asymmetricfeatureof the BUM spec- ber and the frequency offset Aco0= coo- n•ce will be trum lies in the assumptionused in their theory that examined in detail. The remainder of this paper is organizedas follows. the frequency-downshiftedO-mode emissionsproduced In section2, we provide a study of the theoreticalmodel in the second step of the scattering process must be of the four-wave electrostatic parametric process for stronglycyclotrondamped.Bud'koand Vas'kov[1992]

andHuangandKuo[1994] developed theoretical modelsvarious plasmaandpumpwaveparameters. The simfor a four-wave interaction processinvolving the decay of the pump wave into a frequency-upshiftedupper hybrid wave, a frequency-downshiftedelectron Bernstein wave, and a lower hybrid wave. These authors proposed that nonlinear mixing of the upshifted electrostaticsideband with the field-aligned ionospheric irregularities produces beat currents that radiate the SEE observed

ulation

model

and numerical

results are described

in

section3. Discussionof the significanceof the resultsof the present work will be given in section4. The summary and conclusionwill finally be presentedin section 5.

2. Theory

on the ground. Huang and Kuo [1994]showedthat the first-orderprocessof Bud'koand Vas'kov[1992]was

The four-wave model[HuangandKuo, 1994]assumes a second-harmonic oscillation(2co0, 2k0• 0) associated not operative for conditions consistentwith past exper-

imental observations. Acompletely different generation withthelong-wavelength heater wave (coo, k0• 0)para-

mechanism fortheBUMfeature wasproposed byGrachmetrically decays intoafrequency-downshifted electron

[1999] thatconsiders theeffects related toacceleration Bernstein wave (wes, kes), afrequency-upshifted upper

of electronsperpendicular to the backgroundmagnetic field by high-frequencyplasma turbulence. The electron velocity distribution modified over the transversevelocities in the presenceof Coulomb collisionsmay form a

hybridwave(couh, kuh)alongwith a lowerhybridoscillation (colh, klh). The frequencyand wavenumbermatching conditionsfor this parametricprocessare kes+klh =

0, kuh-- klh-- 0, andwes+ cobh = COo -- COuh --COlh, where

velocityspacering distributionwith Of/Ovñ > 0 which the asterisk denotes the complex conjugate. The dismay be unstable to upper hybrid and electron Bern- persion relation is given by stein waves. It is possible for such an instability to be more efficient at generating upper hybrid than elec%B•uh-• •lh tron Bernstein waveswhich would result in an upshifted asymmetric spectrum. While this new model providesa promising mechanismfor the BUM feature, it requires +•[Xe(--•eB) --2Xe(•lh) + Xe(•uh)]= 0, (1) further development and will be consideredand comand •lh -- 1 + X•(Wlh)+ pared with the four-wave processin a future investiga- wheree•S,uh-- 1 + X, (W•S,uh)

ß •4{ I [•e(COlh) •e(COuh)][•e(COl •e(--CO:B

tion.

Xi(Wlh). The susceptibilityof the jth (j = e or i)

XI AND SCALES: NUMERICAL SIMULATIONS OF BROAD UPSHIFTED MAXIMUM

speciesXj is givenby

-

(2)

+

12,789

the pump frequencyand the sidebands.Additionally, comparisons with the theoreticalcalculationswith the real massratio mi/me - 29,362 havefoundthat the reduced value does not qualitatively changethe angu-

lar rangeand the wavenumberrangeoverwhichthe where/•e : 2kVosc/(Wo + •ce), •/osc: qEo/mecoo is four-waveprocessdevelops. The oscillationvelocity to electron thermal velocity ratio is Vosc/Vte= 0.35. ber, me is the electronmass,E0 is the pumpelectric This value is larger than its experimentalcounterpart

the electron oscillating velocity, k is the wave num-

fieldstrength, q is the electron charge, bj -- kñpj 2 2/2, whichistypicallyoftheorderof 10-2 , andit improves pj isthecyclotron radius,q• = (• - n•cj)/kllvt j, •cj the computationalefficiencyin the numericalsimulais the cyclotronfrequency, vtj is the thermalvelocity, tion model while not qualitatively altering important The dispersion relationhasbeensolvedfor a F• (b•) : I•(b•) exp(-bj), Z is the FriedContehnc- processes. tion, I• is the modifiedBesselfunctionof the first kind wide rangeof parameters.We first describea calcula-

of ordern, •j istheDebyelength,andkll(kx)is the tion for the casewherethe pump frequencyis abovethe

fourth cyclotronharmonic4C/ce< coo< Wuh.This case component of k parallel(perpendicular) to B. The methodof RSnnmark[1983]is usedto calculate has beenstudiedin a numberof past experiments.The whereW•hdethe plasmaelectronandion susceptibilities in equation frequencyoffsetAco0is takento be 3•Olh,

(2) aswellasthederivatives ofthesesusceptibilities. A notesthe lowerhybrid resonancefrequency.According NewtonRaphsonmethodis then usedto find the root to the frequencymatchingconditionsthe upperhybrid forequation (1). Wehaveusedtheartificialion-electronwavefrequencyis givenby Wuh= 4C/ce+ 6•O•h.Figure 1 massratiomi/m• = 400forboththe theoretical calcu- showsthe dispersionrelationfor the electronBernstein lations here and the numerical simulations in section 3. and upperhybridwavesand the locationof pumpfreis chosen to be aboutthe mean This reducedmi/m• allowsthe numericalsimulation quencycoo.Notethat coo model to be more computationallyefficient. Caution of the electron Bernstein wave frequency and the upper aspredictedby HuangandKuo has been taken to choosean appropriatevalue so that hybridwavefrequency, , the dispersion qualitative changes willnotbeintroduced intothephys- [1994].For givenvaluesof o:0andWuh icalprocesses underinvestigation. Specifically, the cho- relationin (1) is solvedto obtain the propagationanto the senvalueis still largeenoughto providesufficientfie- gle with respectto the directionperpendicular quency separations between theelectron andionplasma magneticfieldB forthe maximumgrowth0m•.x.In this theangleisfoundto be0max •,o3.0v/me/mi.The frequencies •pe,•pi, theirrespective cyclotron frequen- case, growth rate ' at 0max is calculated and plottedversus cies•p•, •pi, anda visible frequency separation between

0.03 4.8

0.027

4.6

0.024

UH Mode

4.4

0.021

•0=4D,ce+3rOlh

4.2

0.018

0.015

3.8

0.012•

EB Mode 1%

3.6

Growth

I!

3.4-

I

•

I

!

I

I I I

,-

0.009

Rate

0.006 - 0.003

ß

I

•

30

I

0.4 ..............

0.2

0.6

0.8

0

1

•.J_Pe Figure 1. Plot forthe growthrate' in equation(1) versus wavelength kñpeandthe dispersion relationfor the upperhybrid(UH) and the electronBernstein(EB) modesfor the casecv0-

4•ce+ 3Crib, O•3.0v/me/miandVosc/Vte = 0.35.

12,790

XI AND SCALES: NUMERICAL

SIMULATIONS

OF BROAD UPSHIFTED

MAXIMUM

0.03 o

0.025

Third

...... ........ ......

Foudh Fifth Sixth Seventh

3.6

3.8

0.02

:•0.015 0.01

0.005

.4

2.6

2.8

3

3.2

3.4

4

8/(me/mi )1/2 Figure 2. Growthrate"/versus 0 forthecases v:0= n•ce-]-3V:lh(n= 3,4,5,6,7) andVosc/Vte • 0.35. Here "/is the maximumgrowthrate for each0.

wave number kñpe. The result in Figure I showsthe wavelengthrange for the four-waveprocess.The maximum growthoccursfor kñpe •_ 0.3, where"//V:lh--• 0.01. A number of dispersioncalculationshave been performedto examinethe effectsof plasmaand pump wave parameterson the developmentof the four-wavedecay

offset Aw0. Figure 6 showsthat increasingthe pump amplitude causesan increase in the growth, while the wavelengthrange is essentiallyunchanged.The values of the amplitude in Figure 6 are chosenso that the differencebetween adjunct pump power levels is 3 dB.

process. The first calculations consider the effects of changing the cyclotron harmonic number. The result

wave processis of particular importance. In the theoretical calculationsof Figure 7, it is observedthat "/max

The effectof the temperatureratio Te/Ti on the four-

in Figure 2 showsthe growth rate "/versus the propa- decreases as we increaseTe/Ti. In section4, we will gation angle 0 for the casesv:0 = nf]ce -]- 3V:lh,where discusshow this observation helps us understand the n = 3, 4, 5, 6, 7 and Av:0 = 3C•lh. It can be seen that developmentof the four-wavedecay after saturation bethe growth rate increaseswith the harmonic number, causeTe/Ti is increasedsubstantiallyat that time. accompaniedby a reduction in the angle of maximum growth toward propagationperpendicularto the mag- 3. Simulation Model and Results

netic field. Notice that the maximumgrowthrate "/max

occurs when0 •_ 3.0V/me/mifor all the casescalcu- A periodicone spacedimensionand three velocity diparticle-in-cell(PIC) simlated here. For the real massratio mi/rr•e = 29,362, mension(1D3V) electrostatic this impliesthat we can achievethe maximum growth ulation modelusingstandardtechniques[Birdsalland rate when 0 m 1 ø. The relation between the harmonic Langdon,1991]is usedto investigatethe nonlinearevonumberand the maximum growthrate is approximately lution of the four-wave decay instability. The model has three velocity dimensionsthat allow for the propalinear as shownin Figure 3 for Vosc/Vte = 0.35. Increases in the frequency offset Av:0 lead to decreasesin the maximum growth rate as indicated in Figure 4. Figure 5 showsthe sametrend where both the frequency offset and the harmonic number are varied.

gation of lower hybrid waves as well as electron Bernstein and upper hybrid waves. The model geometry is shownin Figure 8. An external uniform oscillatingelec-

tric field E with amplitudegivenby E = E0 cos(v:0t)is Also noted in Figure 4 is that the angle of maximum used to represent the long-wavelengthpump wave and growth rate movesaway from the direction perpendicu-

is applied uniformly acrossthe simulation box. The syslar to the magneticfield when increasingthe frequency tem length is 1024AD, where AD is the initial electron

XI AND SCALES: NUMERICAL SIMULATIONS OF BROAD UPSHIFTED MAXIMUM 0.03

0.025

0.02

0.01

0.005

2

3

4

5

6

7

8

n (Harmonics) Figure 3. Dependence of ? on the harmonicnumbern for the cases•0 - n•ce + 3UZlh; 7 is obtained for each n at a specificpropagationangle 0.

0.05

i

(.0uh 0.045

o•o

0.04

nQce

Ao0

Amo=1.5(01h Ao0=2Olh Am0=2.5mlh Ao0=3Olh Amo=3.5mlh

Ao0 m

m

m

0.035 0.03

80.025 O.02

0.015 0.01 0.005 •!-•

1.5

2.5

..

•

•

3

3.5

4

O/(me/mi )1/2 Figure 4. Growthrate7 versus 0 for thecases •o = 4iqce + •]h

Vosc/v•= 0.35.

(m= 1.5,2,2.5,3,3.5)and

12,791

12,792

XI AND SCALES: NUMERICAL SIMULATIONS OF BROAD UPSHIFTED MAXIMUM 0.06

i

i

i

I

i

Third

...... ........ - - o

0.05

Foudh Fifth Sixth Seventh

0.04

,••0.03 0.02

O.01

ß5

2

2.5

3

3.5

4

4.5

Figure5. Growth rate• versus frequency offset forthecases coo - nf•ce+mcolh (n -- 3, 4, 5, 6, 7 andm - 2, 2.5,3, 3.5,4) andVos½/vte - 0.35;ffisobtained foreachcase at a specific propagation angle 0.

0.02

i

i

i

I

• osc/• te=O .2475

0.018

........t)osc/t) te=0.35

0.016

'"

0.014-

t)osc/t) te:0.495

; •

0.012S 0.01-

0.008-

,

I

I

I

-

i i

-

•

i

-

i

i

_

i - -i .

•

0.0060.004-

0.002

-

i i

-

•

-

/' ...: . / '"' !"

0 '""':'" 0

I

i-'-•

0.2

-

"'-

0.4

0.6

0.8

1

:,Pe Figure 6. Growthrate ? versus pumpamplitude for the cases coo - 4f•ce+ 3cobh andVosc/Vt• 0.2475, 0.35, 0.495. Here T• = T/and vt•- const.

XI AND SCALES' NUMERICAL

0.02

I

SIMULATIONS I

I

OF BROAD UPSHIFTED

I

I

I

12,793

I

........ ......

0.018

MAXIMUM

Fourth Fifth Sixth

0.016 0.014 0.012

0.01 0.008 0.006

0.004 0.0O2

Oo

5

10

15

20

25

30

35

40

T/T. e

i

Figure 7. Dependence ofgrowthrate"•ontemperature ratioTe/Tiforthecases coo = nf•ceq-3Wlh and Vosc/Vte - 0.35 for n - 3, 4, 5.

n(x)

Deybe length,with a uniformdensityof 400 particles per grid cell for eachspecies.The total numberof particles in the simulation is 819,200. The grid cell size is

equalto /kD. As wasstatedearlier,mi/me = 400 and Vosc/Vte = 0.35. The electronsand ionsinitially have Maxwellian velocity distributionswith Te = Ti. The angleof propagation0maxdescribedin the previoussection is used for the simulation. The simulation usually

runsfor up to 1.8x 106time stepscorresponding to the earlystages(up to a fewtenthsof seconds) in the SEE experiments.The systemlengthof 1024ADcorresponds to a systemlengthof the orderof tens of metersin the F regionwhere )•D is of the order of 1 cm. We assume that the heated electrons do not have the time to travel x lm

E - Eo½oSot)

out of the turbulencelayer during the simulationtime. Thereforea periodicboundaryconditionis usedhere. The early time behaviorof the four-waveprocessin a small area of the F regionis our primary consideration in the numerical

simulations.

Severaldiagnosticsare usedto analyzethe development of the four-wave process. The most important

isthesimulation electric fieldpower spectrum IE(co)l 2. The PIC simulationssolvefor the electricfield E(x, t)

z

Bo

Figure 8. Schematic1D3V particle-in-cellmodel for studyingthe four-waveprocessaround elctron gyroharmonic frequenciesn•ce.

which varies in both time and space. To calculatethe

spectrum,we considera numberof fixedspatialpoints in the simulationbox. The correspondingtime seriesof the simulation electric field at each point is then used to calculatethe frequencyspectrum. This spectrumis

12,794

XI AND

SCALES:

NUMERICAL

SIMULATIONS

OF BROAD

UPSHIFTED

MAXIMUM

-25

•

UHWave1

-3O

-35 i

-

!

I

q•

.,. -..

(a) !

-

i

. .

.

•.

,, ! I

_

! I

-60

ß

-i•! I

-65

' 51•0

0

10'00

15'00

20'00

I

2500

3000

%ht 90

Electrons Ions

80

70

>,

60 (b)

20

10

0

500

1000

1•5•) • -'

2000

2500

3000

%ht

Figure 9. Resultsshowingthe time evolutionof the peak amplitude of both the upper hybrid and electronBernsteinwavesand the kinetic energyhistoryfor w0 - 4•ce + 3W]hand Vosc/Vte-- 0.35. Note that the electron Bernstein wave damps after saturation and the electronheating saturates near the same time

as the sidebands.

calculatedby taking a fast Fourier transform of the time upper hybrid waves grow exponentially at essentially seriesand forming the magnitude squared. The spectra the same rate during the time period 300 < wlht < 650. at several different spatial points are averagedtogether A calculation of the simulation growth rate during this to reduce the noise level. period has shown a quite good agreernent with the theoretical prediction in Figure 1. It has been shown that 3.1. Temporal Behavior of the Four-Wave the wavelength of these waves typically agrees with Instability the theoretical predictions as well by using an inter-

The important result of a typical simulation run of the nurnerical model is shown in Figures 9-13. Figure 9a showsthe temporal evolutionof the maximum arnplitude of the electron Bernstein and upper hybrid waves during the simulationfor the caseof 0duh= 4f]ceq-6Wlh

ferogramcalculation[Husseinet al., 1998]. After time wlht --• 650, the growth of the wavessaturates. It can be seen that after saturation, the amplitude of the upper hybrid wave is nearly constantfollowingan initial overshooting. This overshootingon millisecondtimescales

and coo- 4f]ce+ 3V•]h(Ave0-- 3V•lh).The simulation is consistent with previous experimental observations runsfor up to 1.2 x 106time stepscorresponding to an [Sergeevet al., 1997]. The electronBernsteinwave end time w•ht = 3000 and roughly four growth periods behavior has an important difference in that the amofthefour-wave interaction process toallowfora steady plitude of this wave decays significantly after saturastate to take place and sufficientfrequencyresolution. tion. Near the end of the simulation, the electron BernThe pump wave is turned on at time 0Slhl•: 300 to ob- stein wave has an amplitude that is 10-15 dB below serve the growth of the four-waveinstability above the the upper hybrid wave. Figure 9b shows that there simulation thermal noise level. It is observed that after is significant electron wave-particle heating along the the pump field is turned on, the electron Bernstein and magnetic field associated with the four-wave instabil-

XI AND SCALES: NUMERICAL

Low Fre_quency Spectra

2x10-5'

SIMULATIONS

High FrequencySpectra

UPSHIFTED

MAXIMUM

12,795

be discussedshortly. Also, our simulations use periodic boundary conditions which elevate the tempera-

(a)O