JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. A6, PAGES 13,103-13,117,JUNE 1, 2000
Plasma sounding at the upper hybrid frequency
M. E. Dieckmann, • S.C.
and G. Rowlands2
Chapman
=A Ynnerman,
•
Abstract.
A sounder measuresthe density of plasmas in various parts of the
solar system. The sounder emits wave pulses into the ambient plasma and listens
to the response. Intensity peaks in the wave responseare typically related to two
mechanisms.One is provided by wavesthat are reflectedoff plasma inhomogeneities
and propagate back to the emitting antenna, where they are then detected. The
secondis provided by waves propagating with the same group velocity as that of
the receiving antenna. In the secondcasethe waves stay closeto the antenna and
thus yield a long-lasting response.Responsepeaks to soundingat the upper hybrid
(UH) frequencyhave,in mostcases,beenrelatedto reflectedwaves.In this workwe
examine if accompanyingwavescan give rise to the UH responsepeak. We examine
quantitatively how the plasma responseto sounding at the UH frequency depends
on the plasma density, on the electron temperature, and on the emissionamplitude.
For the first two parameters this is done by solving the linear dispersion relation.
The well-known property of the UH waves to change from having a zero group
velocity to propagating waves, depending on how the electron density compares to
the electron cyclotron frequency,is applied to Alouette sounderdata. It is discussed
how the change in the group velocity may affect the spectral profile of the UH
resonance.We presentresultsfrom numericalparticlein cell (PIC) simulations
which show that in the caseof nonpropagatingUH waves,energy can be coupled
into the plasma even though the vanishing group velocity of the UH waves should
not allow this.
The PIC
simulations
and sounder data from the Alouette
mission
show that in the case of propagating UH waves the responseduration to sounding
may be used to determine the electron temperature. Emission amplitudes that are
typical for plasma soundersare also shown to suppressthe generation of certain
electron cyclotron harmonic waves.
1.
Introduction
A plasma sounder measuresthe plasma densities in
parts of the solar system[D•crdau et al., 1993; Etcheto et al., 1981; Fejer and Calvert,1964]. The sounder
emits a wave pulse into the ambient plasma and listens
to the response. The peaks of the wave responseat certain frequenciesare commonly explained by two mechanisms.
One mechanism
is the excitation
of waves with
a group velocity closeto the spacecraftvelocity in the
plasma frame of reference. Suchwavesare calledaccompanying waves. For soundingfrequenciescloseto these
frequenciesthe wave energy will remain close to the
satellite. The plasma responsesampled by the receiv-
ing antennais then strongand long-lasting.The second
mechanismexplainsthe resonances
by the excitation of
obliqueechoes[McAfee,1969].Waveswith frequencies
close to resonancefrequenciesare reflected back onto
the receivingantennaby plasmainhomogeneities.
Resonancesthat are commonly attributed to oblique
echoesare the plasmafrequencycop,the harmonicsof
the electroncyclotronfrequencycoc,and the upper hy-
brid (UH) frequency
co•n= ¾/(cop2
_}_coc
2) [Benson,
1977; Le Sager et al., 1998]. The resonances
at the
frequencymaxima (the fq) of the electroncyclotron
harmonic(ECH) waveswith frequencies
aboveco•nare
attributed to accompanying
waves[Warren and Hagg,
1968;Benson,1977;Le Sageret al., 1998]. This separation is, however,not strict. In the work of Fejer and
Calvert[1964],accompanying
waveshavebeennamed
•Inst.foer Teknik och Naturvetenskap,
Universityof Linkoeping,
Norrkoeping, Sweden.
2Physics
Department,
University
of Warwick,Coventry,England,
United Kingdom.
by meansof particlein cell (PIC) simulationsand by
Copyright2000 by the AmericanGeophysicalUnion.
Papernumber1999JA900491.
0148-0227/00/1999JA900491
as a possiblereason for the responseat
In this work we investigatethe emissionof UH waves
$09.00
comparingthe simulation results with the results predicted by the solution of the linear dispersionrelation.
The aim is to see if accompanying waves could be a
mechanismfor the plasma responsepeak at co•n. A sim13,103
13,104
DIECKMANN
ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
pie antenna model is implemented as the wave source.
The linear dispersionrelation for ECH wavespropagating perpendicular to the magnetic field B with frequencies close to C•unis solved. The group velocities are
calculated, and the responsetime to sounding at C•unis
estimated assuming that accompanying waves are the
reason for the plasma response. We assume that the
most long-lasting plasma responseat C•unis given by
a plane ECH wave with its wave vector parallel to the
antenna axis. The antenna axis is aligned perpendicular to B. We assumethat the spacecraftvelocity in the
plasma frame of referenceis zero. Then the longestlasting responseis given by the ECH wave with the lowest
group velocity. We estimate the responsetime by dividing the antenna length by the group velocity of the
slowest wave. This is clearly an oversimplifiedmodel
case of a weakly magnetized plasma. Up to this point
the waves are assumed to be linear; that is, they are
solutions of the linear dispersion relation.
In principle,it is possibleto examinethe generationof
a wave packet by a pulseusing the saddlepoint method
describedby Brillouin [1960].Wavescloseto COun
have
an electrostatic and an electromagnetic wave component. It is thus not possibleto approximate the linear
dispersionrelation either by the electrostaticlimit or
by the electromagneticlimit. In addition, we cannot
assumea plasma sounder to generate linear wavesdue
to the strongemissionamplitudes(the emissionamplitudesfor the Clustersounderare 50 or 200 V [D•cr•au
et al., 1993], which comparesto temperaturesof the
magnetospheric
coldelectronpopulationof i eV [Etcheto et al., 1981]).
Instead, a one and a half dimensionalelectromagnetfor the real (three-dimensional)
soundingexperiment,
but its only purpose is to give us an estimate for the
responsetime. This estimate agreeswith the response
time provided by the PIC simulations. It is also in reasonableagreementwith the responsetimes measuredby
the Alouette
mission.
In section 2 we revise how the plasma responseto
ic PIC code is employed. It solvesthe nonlinear Vlasov
equation together with the full set of the Maxwell equations. The equations are solved in one spatial dimension for all three velocity components.The emissionis
modeled by applying electrostatic fields that oscillate
in time to a set of simulation grid cells. Section 3 describesin more detail the simulation code, the plasma
sounding
at C•unchanges
with the ratioC•p/C•c
andwith
a changingelectrontemperature. For C•p/C•c< 1.7 parameters, and the antenna.
In section 4 the case of propagating UH waves is
(O•un/C•< 2) the UH waveshave a zerogroupveloc-
ity. In the opposite case they are propagating waves investigated, which requires the plasma to be weak[Akhiezeret al., 1975].A changein the plasmaresponse ly magnetized. Then the plasma supports two wave
durationat O•p/O•- 1.7 has beenreportedby Fejer solutionsat COun.Simulationsfor two emissionampliand Calvert [1964]. In the work of McAfee[1969]the tudes are compared. It is shown that fi•r the typicalchangein the topology of the ECH wave solutionsclose ly strong sounder emission amplitudes the wave soluto O•un(andthusindirectlythe groupvelocity)hasbeen tion with the high wavenumbercannot be excited. The
used to derive the conditions for oblique echoesat O•un strong emissionamplitudes excite waves in a regime
which predicted correctly qualitative featuresof the UH
in whichelectron
trappingnonlinearly
dam•ps
electroresponseto sounding. The change in the group veloci- staticwavespropagatingperpendicular
to B [Riyopouty has, to the knowledgeof the authors, not previously los, 1986]. The observation
from the simulationsthat
been applied in the context of accompanyingUH waves, the absorption mechanism is more efficient for the wave
and we examine this here.
with the high wavenumberis also consistentwith elecIt is shown that for increasing electron temperatures tron trapping being the relevant damping mechanisthe wave group velocity strongly increases. This in- m. Direct evidencethat this is the nonlineardamping
crease may be the reason for the absenceof a plasma mechanismrequires,however,a detailed investigation
response to sounding at COunin the hot parts of the of the electronphasespacedistribution,whichis left to
magnetotail[Etchetoet al., 1981]and in the Io plasma
torus [Le Sageret al., 1998]. The absenceof theseres-
future
work.
not have any impact on the soundingexperiment. The
rest with respect to the plasma frame of reference.
The propertiesof the plasmaresponse,sampledby a
onances may, however, also be related to a decreasing virtual antenna placed in the simulationbox, are invesantenna response function for increasing wavelengths tigated. Of particular interestis the questionof whether
the antenna length and the signal duration could be
[Meyer-Vernetet al., 1993].
Nonzero electron temperatures are shown to change used to determine the electronthermal velocity. It is
the frequencyat which the plasma responseto sounding shownthat this is indeedpossiblefor the simplecaseof
peaks. This changeis, however,small, so that it should a one-dimensionalgeometry and an antenna that is at
responsepeak is shiftedbelowWunfor Wp/Wc< 1.7
Section5 dealswith the caseof sounding
in a strong(stronglymagnetizedplasma). It is shiftedaboveWun ly magnetizedplasma. Soundingat COun
generatesa
in the oppositecase(weaklymagnetizedplasma).It is nonpropagating, localized standing wave. Simulations
also shown that for the plasma parameters under con- for two emissionamplitudesare compared.They show
sideration the UH responsein a strongly magnetized that the plasmaresponsepoweris quadraticallydepenplasma should be relatively narrow banded in frequen- dent on the emissionamplitudeand that it is linearly
cy whereas it should be relatively wide banded in the dependenton the emissiontime. The energytransport
DIECKMANN ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
into the plasma required to build up the standing wave
is achievedby wave forerunnersdescribed,for example,
13,105
sus /•. The horizontal solid lines show the harmonics
ofv:•.The
dashed
line
shows
•n - V/•p
2q-1- 2.1.
by Brillouin [1960]. The nonpropagating
perturbation The dots are the • values calculated by the linear dis-
is consistentwith the observationby Fejer and Calvert,
[1964]that the signalresponse
at Wunis long-lastingin
the case of a strongly magnetized plasma. The results
are then discussed in section
2. Kinetic
6.
Theory
persionrelation solver. The dashedline intersectswith
the linear dispersionrelation at points I and 2. Some
properties of these points are examined in section 4.
Point 3 in Figure I showsan fq. Thesenonpropagating
wave solutionsprovide an important classof resonances
to sounding[WarrenandHagg,1968].Someproperties
of these waveshave been investigatedby meansof PIC
The focusin ourst•udy
is onECH wavespropagat- simulationsby Dieckmannet al. [1999].
ing perpendicular to B. These wave modes are linearly
At low • in FigureI we get the (electromagnetic)
undamped. The dispersion relation has been derived
usingthe tensorcomponentsDII, Dyy, Dxy, and Dyx
in the notation of Krall and Trivelpiece[1986]. Since
the dispersion relation close to v:un is examined, the
electromagneticcomponentsof the wavescannot be ignored. In what follows, vtn is the electron thermal velocity. The phase and the group velocitiesof the waves
are referredto as Vpnand vgr. The frequencies
are normalizedto v:c;that is, • - v:/v:c.The wavenumbers
are
normalized to the inverse electron thermal gyroradius;
thatis,/z- kvtn/v:•.
2.1. The Dispersion
Magnetized Plasma
slowextraordinarymode (at low 5•) and fast extraor-
dinarymode(at high5•).Forincreasing/c
thefastextraordinary mode approachesasymptotically the ordi-
nary wavemode. For the slowextraordinarymode (in
the cold plasmalimit) we get the limits for
• -+ •n and vgr -+ 0. The samelimits for •, vg• are
approachedby a purely electrostaticECH wave in the
~
branchcontainingV:unfor k -+ 0 (warm plasma). Figure I showsthat both limits incorrectly describethe low
/z waveswith 5• .• 5•un.The slowextraordinary
mode
goesoverinto the ECH wavebranchcontaining•n and
Relation
vg•.• 0 at cb•
We focus on the UH wavesat low /z as the accom-
of a Weakly
The solutions of the linear dispersion relation in a
weakly magnetized plasma closeto chunand for vtn -
9.4x 105m/s areshown
in Figure1. Plottedis• vet-
panying waves providing the strong plasma responseto
sounding. In section 4 the numerical simulationswill
show why the UH waves at point 2 in Figure I can be
ignored.
3.5
3
(1)2.5
N
2
E
o
z •.5
......
I
10-3
Figure
1.
........
I
........
10-2
Normalized
I
.
.
i
i I i i ,1
10-1
100
wavenumber
i
i
......
101
The solution of the linear dispersion relation for undamped electron cyclotron
harmonic(ECH) wavesand O3p/O3
c > 1.7. Overplotted
are the harmonics
of v:c(solidlines)
andv:un(dashed
line).PointsI and2 showthecentral
•, } ofthewavemodes
excited
in the
simulations.Point 3 showsan fQ whichis anothertype of resonanceto plasmasounding.
13,106
DIECKMANN
ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
We calculate,for •n - 2.1, the vsr of the resonance tenna length by the wavepacker'spropagationvelocity
as a function of k for three sample vtn. We then de-
in the antenna
termine the minimum value of Vsr at • • &un and
the responseduration for the Ulyssesantennawith a
length of 72 m shouldthen be 0.85 ms. The maximum
responseduration for waves closeto 0:• should be 1
ms. The Ulyssesreceiverstarts to samplethe wavere-
its •.
For vtn - l05 m/s we determinethe mini-
frame
of reference.
For the UH
wave
mumas vat • 2.26x l04 m/s at •/min-- 2.101.For
vtn-- 5 x l0s m/s thevalues
areVgr• 2.33x l0s m/s
and•/min-- 2.106. For vtn - 106m/s the valuesare sponse5 ms [Le Sageret al., 1998]after the emission
has finished. No wave responsedue to accompanying
Vg
r • 5.92x 105m/s at •/min• 2.116.
In the limit of lowvtnweget•)min--• •uh andvgr-• 0
as we would expect. For increasing vt• both •/min and
UH waves can thus be expected.
In section 4 the emissionof a wave packet with • -
vat increase.For VthlargerthanVth• 1.5X 106m/s •u• is investigatedby meansof PIC simulations.The
the minimumin vgr disappears.In this casethe waves
emission should excite wave solutions at the intersection
with • • •u• are dominated by the slow extraordinary
points 1 and 2 in Figure 1 of & = &un with the linear
dispersionrelation solutionof the corresponding
ECH
mode which has a large vsr. For sufficientlyhigh vt•
/c,
any wave energy injected into the plasma should prop- branch.Bothwaveshavethe same& but different
agate away quickly, thus giving a very short response implying differentthresholdsfor the electricfield value
to sounding. If a sounderdoesnot samplethe wave re- requiredto trap electrons.This field thresholdhasbeen
sponseimmediately after the emissionhas finished,i.e., derivedby Riyopoulos[1986]. Accordingto equation
[1986],trappedparticle
if it hasa deadtime [D•crdauet al., 1993;Le Sageret (3) in the work of Riyopoulos
al., 1998],the plasmaresponsemay be shortenoughto islandsemergein a plasmaif the normalizedamplitude
go undetected. This may be the reason for the absence Ai = (eEoknx/•)/[mO:c2riv/(•rri)]
of a monochromatic
of any plasma responseat •n in the hot parts of the ECH wave exceedsthe normalized frequency departure
magnetotail[Etchetoet al., 1981]and in the Io plasma • = 0:/0:c- n from a harmonicof 0:•. Here E0 is the
electric field amplitude of the wave, n is the cyclotron
torus [Le Sageret al., 1998].
As an illustration, we take parametersfor the plasma harmonicnumber, elm is the electrons'characteristic
and the spacecraftvelocity encounteredby the Ulysses chargeto massratio, and ri is the normalizedvelocity
sounderin the Io plasmatorus [Le Sageret al., 1998]. vk/o:• of the ith trapped particle islandcenter.We set
0:andk by• and/z.Thenthe
We set o:p/o:c- 4.9. For simplicitywe take a single Ai = •, andwereplace
for the critical electricfield in V/m is
MaxwellJan with a temperature of 10 eV for the electron expression
distribution
function.
At •n - 5.001,vg•m 1.85x 10s
m/s. The lowestvaluevgr• 1.71x 10s m/s closeto
0:•n is found at • - 5.007.
We assumethe antenna axis to be parallel to the spacecraft velocity vector in the plasma frame of reference. The antenna
axis orientation
is also assumed to be
perpendicularto B, and we assumethat the strongest
plasma responsecomesfrom ECH wavespropagating
perpendicularly to B. The emission generates wave
packets propagating away from the antenna. The antenna measuresthe potential differencebetween the an-
Ecrit
--•i (•_n)
•mriVth•c(7rri)l/2
n• -•(1)
Equation (1) determinesthe electricfield value required to trap electrons. A wave electric field exceeding this thresholdhas been shownto nonlinearlydamp
the wave [Riyopoulos,
1986]. It has been derivedfor
a monochromatic wave with 0: m n0:•. Both these as-
sumptionsare not necessarilyfulfilled in our case,but
(1) neverthelessindicatesthat the wave mode at high
• shouldbe saturatedat muchweakerfieldamplitudes
than
thewavemodeat a low•. In section
4.1,numerical
packet can only contribute to the plasma responseif the
tenna wires. The wave electric fields of the emitted
wave
wave packer's end did not leave the spacebetween the
end points of the antenna wires. This will be shownin
section 4.3 to be true for a one-dimensional
antenna
at
rest with respect to the plasma frame of reference.
simulations show that an emissionwith a strong electric
field amplitude can only generatethe ECH wavewith
the low k.
This observation
would be consistent with
theprediction
of(1)thatthehigh• waveissuppressed
Accordingto Le Sager et al. [1998], the antenna by trapping.
frame of reference moves in the Io plasma torus with
m 105m/s relativeto the ambientplasma.The UH 2.2. The Dispersion Relation of a Strongly
waves
have,fortheparameters
above,
a Vpn• 5.8x 107
Magnetized
Plasma
The linear dispersionrelation solution for a strongly
m/s, and thusthe Dopplershiftis negligible.The wave
packet that leaves the space between the antenna's end magnetizedplasma and for frequenciescloseto •n is
points last is the wave propagating in the same direc- shown
in Figure2. Here• is plottedversus
•c. The
tion as the spacecraft in the plasma frame of reference. dots indicate the 5• values calculated by the linear dis-
Its propagationvelocity is given by its vg• minus the
relative velocity of the spacecraft with respect to the
plasma frame of reference. We obtain the plasma responseduration seenat an antenna by dividing the an-
persionrelation solver. The vertical solidlinesshowthe
harmonics of co•. The dashedline shows•n - 1.9. At
low k we identify the slow extraordinary mode at low
• and the fast extraordinary mode at high •. The s-
DIECKMANN ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
13,107
2.5
c-
(D 2
(!3 1.5
E
o
z
!
10-4
10-3
i
10-2
Normalized
i
10-4
wavenumber
I
100
.....
104
Figure
2. The
solution
ofthelinear
dispersion
relation
forundamped
ECH
waves
and
COp/We
<
1.7.Overplotted
aretheharmonics
ofwc(solid
lines)
andw•,n
(dashed
line).Point
I indicates
the•, • excitedin thesimulations.
low extraordinary mode goesoverinto the ECH branch
If the resonanceat &,n is due to accompanyingwaves,
containing •n.
In section 5 we will discussthe emis- the responsetime to soundingshouldreflect this. If we
sion of a wave packet with • - •n. A wave should be assumethat the plasma responsetime for accompanying
excited at the intersection point between &•n and the waves seen by an antenna at rest with respect to the
lowestECH branch(indicatedby point 1). The mini- plasma is approximately the antenna length divided by
mumof Ivgrlat • .• •n is alwayszerolikefor point3
v•, we can relatethe response
time to v•r.
in Figure 1. Contrary to the latter resonance,however,
In the workof Fejer and Calvert[1964]the observed
point I in Figure 2 dependson vtn. The resonantwave response
timeto sounding
at &•n is .• 104wavecycles
hasan & • 1.899for vtn- 105m/s, an • • 1.894for for 5•n < 2 and 3 x 103wavecyclesfor •n > 2. For
of co•nm 1.5 x 106
vtn= 5 x 105m/s,andan& • 1.888forvtn= 106m/s. a typicalvaluein the ionosphere
The changein & is small for this interval of vtn.
rad/s [Fejer and Calvert,1964]the response
durations
are 6.7 and 2 ms, respectively(obtainedby multiplying
2.3.
the responsetime in wave cyclesby the wave period in
Comparing the Width
of the Response
Peaks
physicalunits).
For an antenna length of the Alouette sounderof 45.7
In Figure3 we plot v9r(•)/vtn for •n = 1.9 (Figure 3a) and for &•n - 2.1 (Figure 3b). The circles m [Bensonet al., 1997],the observed
response
duration
correspond
to vtn= 9.4x 105m/s, andthe crosses
cor- of 2 ms,andvtn= 105m/s, wemusthaveVg,/Vtn=
respond
to vtn- 105m/s. Whilethecircles
correspond0.23. This is the valueof the minimumVg,in Figure3b.
to magnetospheric
vtn [Roennmarkand Christiansen, Defining the width of the responsepeak as the frequency
1981]usedfor the numericalsimulationsin sections4 intervalfor which
< 0.46 (halfthe response
time
and 5, the crosses are representative for the Earth's of 2 ms) givesa spectralwidth of 0.014
ionospheric
vtn [Baumjohannand Treumann,1997].
For a response
durationof 6.7 ms, vg,/vtn mustbe
Figure 3a showsthat &•n is not a plasma eigenmode. 0.068. This low value is only achieved for • closeto
The gap between the ECH wave's maximum • and &,n
(Figure3a). Definingthe spectralwidth of the response
increasesfor increasing vtn. Figure 3a also showsthat peak as the co-intervalfor which
< 0.136 (half
the Vg• are low for & • &,n only.
Figure 3b showsthat the & at which the ECH waves
reachtheir minimumvg• is above&•n and that it increasesfor increasing vtn. While for ionosphericvtn
the responsetime of 6.7 ms) givesa spectralwidth of
4 x 10-400½.
The spectrallinefor•,n < 2 isthusnarrow
compared with the spectral line for the case •,n > 2.
Finally,for a magnetospheric
vtn = 9.4 x 105m/s
the minimumin v• is pronounced,
it practicallydisap- and for •n > 2, the minimumva,•/vtnis 0.58. The
pearedfor magnetospheric
vtn. Only for CVp/CV•
< 1.7 intervalforwhich
< 1.16is0.22co•.Themaximum
andfor • • •n canwe get Ivgl - o.
responseduration for an antenna with a length of 45.7
13,108
DIECKMANN
ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
(A)i
o
0.5
:
0
i
i
x
o
x
xO ,,o ,,o (• Ox Ox Ox
0.5
0000000(•
x
x
x x x
x
x
x
xxxxxxxxX)•<
o
z
-1
1.875
1.88
1.885
1.89
1.9
1.895
Normalized frequency
(B)1
o
oooooooooooo
xoo•
ox
xo•o•.oxoxoxøxøxøxø'
XXx XX
_
x
:0.5-
o
x
xxX
xxxxx
xx
XxxXX
o
I
0
2.1
I
2.105
I
I
2.11
2.115
,
I
2.12
I
2.125
'
2.13
I
2.135
2.14
Normalized frequency
Figure 3. The dependence
ofva•/vtnon• oftheECH waves.(a) cop/coc
< 1.7. (b) COp/COc
> 1.7.
m would be m 0.08 ms. Thus the responsewould be
broad in 5• and of a short duration.
3.
The
Code
and
the
Antenna
nite extention perpendicularto the simulationdirection.
Thus no electric fields other than those in the simulation
direction are directly generated. In an unmagnetized
plasma such waves would correspond to electrostatic
waves. In a magnetized plasma, however, this separa-
The emissionof waveswith • - •un is investigatedin
tion is not apparent[Krall and Trivelpiece,
1986]. As
sections4 and 5 by means of PIC simulations. The electromagnetic and relativistic simulation code solves for
one spatial and three velocity componentsof the particles and for the three vector componentsof the electric
and the magnetic field. The simulationdirection is in
discussedin section 2, the emitted wave at •un at low
the x- direction.
The external
B is in the z- direction.
The temperature for the electronsand the protons is 5
•cis expected
to haveanelectromagnetic
component.
We assume that the potential difference is shielded
over one Debye length, which is approximately one simulation grid cell. The antenna electricfieldsare nonzero
only at one grid cell to eachsideof the antennaand have
an oppositesign on these two cells. The electricfields
eV (vtn• 9.4 x 105m/s). The singleMaxwellian
elec- at the gridcellm andm+ 1 areupdatedevery(discrete)
tron and proton distributions are representedby 1024
simulation time step sat by adding an increment AE
particlesper cell each.We set copm 18.5 kHz and vary
to the self-consistentelectric field calculated by the PIC
simulation. The self-consistentmagnetic field calculat-
cocto model the casesof a strongly magnetizedplasma
and a weakly magnetized plasma. The simulation box ed by the PIC simulationis not modified. The value for
consistsof 1500 grid cells, and each cell is Ax - 7 m m is 750 in the case of an emissioninvolving one emission plate. The two values for m are 745 and 755 for
long.
the
emissioninvolving two antenna plates. A schematic
A typical emitting antenna on board a sounderconsistsof a pair of wires that is connectedto a transmitter diagramshowingthe antennaelectricfields(excluding
[Ddcrdauet al., 1993]. The exactcurrentdistribution the plasmaresponse)as arrowsis shownin Figure 4.
on the antenna is not known during the emissionprocess. In the numerical simulationswe thus employ a
simplified model for the antenna. We assume that the
antenna can be representedby oscillatingchargedensities at one or two points in space. Since the numerical simulationsmodel only one spatial dimension,the
antenna correspondsto one or two plates with an infi-
The direction correspondsto the sign of the electric
field, and the length correspondsto its absolutevalue.
Figure 4a showsthe one-plate emission,and Figures 4b
and 4c correspond to the two-plate emission.
The incrementAE is proportionalto cos(coosAt).
The emissionfrequencycoois set to co•n. The total
electricfieldthen oscillates
like sin(coosAt)
in time (the
DIECKMANN
ET AL.' PLASMA SOUNDING
AT THE UPPER HYBRID
FREQUENCY
13,109
2.5
_ (A)
•
E
1
o
zO.5
0
748
I
749
7• •3
752
2.5
2.5
•
E
I
I
750
751
Grid cell number
1
o
z0.5
0
744
0
745
746
Grid cell number
747
754
755
756
Grid cell number
757
Figure 4. A schematicpictureof the simulationantenna. (a) The electricfieldsas arrowsfor
the emissionwith onesounderplate at the simulationcells750 and 751. (b) The electricfields
at the cells745 and 746 (plate1) for the emission
with two plates.(c) The electricfieldsat the
cells755 and 756 (plate 2) for the emissionwith two plates.
and temporal distribution of the electrostaticfieldsapgration). The magnitudeof AE is chosensuchas to plied to the antennagrid cells. We employtwo different
givethe desiredantennaelectricfields(the relationbe- antennas. One involves sinusoidallyoscillating electric
tween the increment and the resulting field oscillations fieldsat the simulationgrid cells750 and 751 (oneanis nontrivial owing to the electric fieldsgeneratedby the tenna plate case)and is employedin sections4.1, 4.2,
and 5. The secondantenna involvesthe simulation grid
plasmaresponseto the antennaelectricfields'.
The antenna is assumedto be permeable for the par- cells 745, 746, 755, and 756 for the antenna with two
ticles; that is, no boundary conditions for the parti- plates(seeFigure4) and is usedin section4.3.
The electricfields (waves)arisingfrom the plasma
cles have been implemented at the antenna cells. Both
are not includedin A (x,t•). The powerspecthe permeable antenna and the electric field distribu- response
tion are unrealistic
in that the antenna electric fields
trum of A(x,t•) is givenby P(w, k). If, asa firstapproxdecreaselinearly in the simulation owing to the field imation, the plasmaresponseelectricfieldsare ignored,
oncoand
representation by linear splines. Debye shielding gives P(co,k) canbe splitup into a term depending
summation in time of the AE is equivalent to an inte-
an exponentially decreasingpotential. Sincethe generated waveshave wavelengthsthat are long comparedto
the two antenna grid cells, we may ignore any effects
introduced by the oversimplifiedantenna model.
In what follows, the physical times tr will be nor-
a term depending on k.
For simplicitywe calculateP(co) by meansof the
Fourier integral. We refer to the unnormalizedemission time as r. The emissionamplitude is zero outside
t• e [0,r]. For frequencies
closeto positivecooit is sufsin(cootr)by (1/2i)exp (icoot•).
malizedto t = tr/(27r/coo).The valuefor COo
and thus ficientto approximate
the normalization change depending on if the plasma
is strongly or weakly magnetized. The normalized emission duration is T = 40. For the considered cocthis
corresponds
to t• m 2 ms, whichis twice as long as the
emissionduration of the Cluster or the Ulyssessounder
[D•cr•au et al., 1993;Le Sageret al., 1998].
The antenna will couple predominantly to those co,k
for which the power spectrum of the antenna function
Then the power spectrum is
2
P(co)
-- •lj•0r
exp
(icoot•)
exp
(-icot•)dt•
=
sin[(coo
- co)r/2]
2
4(co0
- co)2 '
(2)
Mostwavepowerisconcentrated
in theintervalcoe[cooA (X,tr) iS large. Here A (X,tr) represents
the spatial 2•r/r, coo+ 2,r/r].
13,110
DIECKMANN
ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
Our simulation box is periodic; that is, a minimum
1.2
k exists.This is definedas Ak -- 2•r/(MAx), with M
as the number of simulation grid cells and Ax as the
length of eachcell. We estimateP(k) by meansof a
discrete
Fourier
transform:
E0.8
P(k,)- 1/M• A,•exp(-i2•k,n/M)
. (3)
n•0
/
-o0.6
N
In physical units the wavenumber is k = k•Ak. The
time seriesA• is, for the antennawith oneplate, nonzero o•y at two grid cells. We set the values as A• = 1
and Am+x • -1; that is, the amplitudes at these cells
have an oppositephase. Then the transform reducesto
•0.4
0.2
P(k,)- [exp(-i2•rk,m/M) (1- exp(-i2•rk,/M))[2
10-•
Normalized
= 4 x sin2(2•rk•/(2M)).(4)
Figure 5.
100
wavenumber
The normalized power of the wave modes
Notethat k• hasan upperlimit of k•vy- M/2 (Nyquist
andfor Wp/W•>
theorem). This antennathuscouplesmostwaveenergy asa functionof k for a strongemission
1.7. The power has been integrated over time. The
dashed lines indicate the points where the solution of
the linear dispersionrelation crosses
to high k and to w • w0.
4.
Simulated
UH
Weakly Magnetized
Let wc -
~
Emissions
in a
Plasma
10 kHz and •0 -
•n
ear dispersionrelation solution intersectswith • - •n.
-
2.1. At this e-
missionfrequency•0 the (linear) plasmasupportstwo
waveswith differentk (seeFigure 1) and Ecrit- Accordingto Riyopoulos[1986],the ith trapped particle island'scenteris, for ri
i•r + 0.5(n + 0.5)•r. The centerof the zerothtrapped
particleisland(i-0) determines
the electricfieldthreshold for the onset of trapping. The emitted waveshave
• • 2. Thus the wave will first trap electrons at
Thewavepowerpeaksat afcclose
to thelow•cintersection point. The maximum is, however, shifted toward
highervalues
of •c. Thereason
is discussed
in section
4.2.
In Figure 5 one also seesweak sidelobesof the main
peakat •c• 9 x 10-2. The maximaof thesesidelobes
are located at • • 1.8 x 10-1 • • 2.7 x 10-1 and
} m 3.6 x 10-1. These sidelobesare harmonicsof the
peakat }•9x
10-2 .
No wave power can be detectedat the secondintersec-
n=2 (n is the cyclotronharmonicnumbernwc). We
thus replaceri in (1) by r0 • 3.9. The intersection tion point at k - 2.5. The antennapowerP(k) derived
points of • - 2.1 and the linear dispersionrelation have in (4) predictsthat this peak shouldhavemorepower
~
absorption
•1 - 0.035and •2 - 2.5. The criticalelectricfieldfor thanthe peakat • - 0.035. A nonlinear
~
kl is Ecrit • 2.3 V/m whereasit is Ecrit • 0.03 Y/m
mechanism must thus be present, absorbing the wave
for
energy
ofthehighfzmode.
4.1. An Emission Amplitude of 1.2 V/m
4.2. An Emission Amplitude of 0.3 V/m
The electric field amplitude for the virtual sounderis
lessthan Befit for kI -- (].035 b•]t higher than Ecr.;tfar
Now we apply Um•x - 2.1 V between the antenna
pl•f,p •.ncl f,hp .mlrr•11nc]ing
pl•.•m• Tho plpcf,
rr•.•f,•.f,]c
•2 - 2.5. Thepeakpotentialdifference
Umax
betweenfield data are Fourier-transformed over space. The powthe antenna plate and the plasma for Ax - 7 m is Umax erspectrum
asa function
of•, t isintegrated
overtime.
- 8.4 V, which is lessthan that for "real" sounders.For
example, the sounderon board the forthcomingCluster
The integrationis done over two integrationsubinter-
vals, one from t - 40 to t - 60 and one from t - 60 to t missionappliesa Um•xof either 50 or 200 V [Ddcrdau 80. Both subsetsare normalized to half the peak power
et al., 1993]betweenits antennawireswhilethe Ulysses in Figure 5 to accountfor the shorterintegrationtime.
sounder applies a Um•x of 30 V.
The power for the first subintervalis shownin Figure 6a.
The electrostatic field data provided by the simula- Plotted is the normalized power versusk. The dashed
tion are Fourier-transformed over space. The Fourier lines indicate the intersectionpoints betweenthe linear
transformis squaredto give the poweras a functionof dispersionrelation solutionand • - •n. As in Figure
k, t. The power is integrated from t - 40 to t - 80. 5, thepeakat low• is shifted
towardhigher• than
The result is shown in Figure 5. Plotted is the pow- expected. The explanation is given below.
No sidelobes can be detected in this case. The selfer, normalized to the peak power, as a function of k.
~
~
Overplotted
asdashed
linesarethefcforwhichthelin- interaction
ofthewaveat lowfzisnegligible
compared
DIECKMANN ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
13,111
distributionP(k,) in (4). This indicatesthat the peak
I
in powerat •c• 2 is stillnonlinearly
damped.
I
I
The k spectrum of the interval between t - 60 and t
i
- 80 is shown
in Figure6b. If thewavepeakat /• - 2
was damped, one may expect a further reduction in
the power for the secondinterval. This is confirmed by
Figure 6b. Plotted is the normalized wave power versus
k. The peak value at high k has been reducedfrom 0.13
._
to 0.09.
In Figures 5, 6a, and 6b the wave packet at low
.E0.05
• hasits maximumpowerat a higher• than the
intersection point between the undamped solution of
the ECH wave mode and &0. Since the upward shift
is approximately the same in Figures 5, 6a, and 6b,
I
1
10-•
Normalized
,
it is likely to be a linear effect. From (2) we have
10o
the normalized frequency spread of the sounder pulse
wavenumber •r
A& - (2•r/'r)/wc. For •- - 40 x 2•r/woand &0 - 2.1
wegetA& - 2.1/40-0.0525. If &maxis the fr_equency
0.15
ofthe(linear)ECHwaveat •cm0.08(thelowk peak),
I
then &max-&un • 0.0282, which is lessthan A&. The
reasonfor the shifted power maximum may then be that
i
/
,,
o
i
I
"o
i
i
N
,,
I
E0.05
o
z
theantenna
couples
moreeasilyto waves
witha high
The wave packet at & m &un and at large k is shifted
toward lower k than the intersection point of the linear
dispersionrelation with &un. The wave power peaks at
•cm2.23compared
to •c- 2.5oftheintersection
point.
The peak wave power is then shifted to &max- 2.1277.
Here the difference &max-&un - 0.0277 is also less than
A& - 0.0525 of the emission. Here the shift might be
a nonlinear
effect.Decreasing
•candincreasing
&max
increasesthe estimateof Ecrit in (1). The shift thus
1ø0-
10-•
Normalized
10o
wavenumber
increasesthe maximum allowed wave power. A more
detailed investigation of this shift is left to future work.
To show that the trapping of electronsby the ECH
waveat /• m 2 mightbe the dampingmechanism,
we
Figure 6. The normalizedpowerof the wavemodesas compare the wave amplitudes of this wave packet with
a function
of•cfora weakemission
andforwp/wc
> 1.7. Ecrit-- 0.03 V/m (1). To obtainthe waveamplitudeof
(a) The powerin the firstintegration
interval.(b) The the wavepacketcentered
at •cm 2, wefilterthe elecpowerin the secondintegrationinterval. The powerhas trostatic field distribution in the simulationbox in •cat
been integrated over time. The dashedlines indicate the time t - 45. The filter sets the amplitude spectrum
the points where the solution of the linear dispersion
relation crosses Wun.
forwaves
with•cotherthan•ce[1.7,2.7]to zero.The
result is shown in Figure 7. Plotted is the electric field
amplitudeof the wavepacketcentered
at /• • 2 as a
function of the grid cell number. The modulus of the
to the wave in Figure 5 becausethe emissionamplitude
waveelectricfield exceedslEerit] (the horizontallines).
is one-fourth times that in section 4.1. The peak power
of this wave is one-tenth times the peak power it has for
The wave may thus trap electrons. Filtering the data
set for the simulation with the emissionamplitude 1.2
the emissionamplitudeof 1.2 V/m and is thus higher V/m revealsthat at t - 45 no signalis presentin the
than expectedif we would assumethe waveto be linear.
Then the ratio shouldbe 1/16. The self-interactionof
the wave generatedby the emissionamplitude of 1.2
V/m transferswave energyto its harmonics,someof
which have values of &, k correspondingto transiently
damped waves. This energyis then absorbed.
We alsonoticethe presence
of a signalat •cm 2,
which is less than that of the intersection point at high
interval
•ce[1.7,2.7].
Thepeakat thelow•cintersection
pointhasincreased
its power. The overall change in power is, however,
not large enough comparedto noise levels to identify
the reason for this since the peak covers only a small
interval
4.3.
in k.
The Measurement
of Vth
In section2 the dependence
of vat of the UH wave
•c.Thissignalisnotsignificantly
stronger
thanthat at
on
vth
at
low
k
has
been
investigated.
In this section
•c= 0.035aswewouldexpectfromthe antennapower
13,112
DIECKMANN ET AL' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
0.08
•
Umax = 8.4 V relative to the surrounding plasma. Between both antenna plates, Umaxis thus 16.8 V. Both
antennaplates are separatedby 10 simulationgrid cells,
and the antenna length L is thus 70 m, which is comparable to that on board the Ulyssesspacecraft.
The potential differencebetween the antenna plates
0.04
=•_••
0.02
II,I
is shownin Figure 8 as a function of the normalized
time.
• -0.02
• -0.04
-0.06
-ø'%0
i
i
1000
Grid cell number
Figure
?.
The electric field amplitude of the wave
corresponding
to thesignalat • m2 in Figures
6aand
6b for the time t=45.
Overplotted as horizontal lines
are -]-Ecrit ß
the numerical simulations revealed that emissions with
a strong electric field amplitude cannot generatethe
The dashed vertical
line indicates
the end of the
emission. Shortly after the emissionstarts, the potential differenceis approximatelythat givenby the forced
oscillation.After a few emissionperiodsthe plasmaresponsereducesthe potential differenceto a peak value
of 10 V. After the emissionhas finishedthe peakpotential differencedropsto • 2 V, which is still considerably
higherthan the noiselevelsforthe simulation(• 0.1V).
The antennaresponsespectrumis obtainedby Fourier transforming the antenna's potential differenceover
time and squaring the result. This givesa power spectrum as a function of •. The window length is 40 time
units. The integration interval starts first at t = 40,
and the start is then shifted toward higher valuesof t.
This gives an &, t spectrogram. The two •- bins with
the highestwave power are extracted and plotted as a
linearwaveat high/c.Theantenna
lengthfortheplas- functionof•- t - 40,wheret refersto thebeginning
of
ma sounderis known. The wave responseduration can
the integration
interval.A time•- 0 thuscorresponds
be measured. If the responseis due to accompanying to an integration interval from t = 40 to t = 80. The
waves(that remainin the antenna'svicinity),onemay power is normalized to the average power that would
be tempted to relate the antennalength divided by the
be present in the correspondingbin if no emissionwas
response
duration
tothevgrofthewaveat low•c.Then
present.
v9r at & • &un would determinevtn.
What is not knownis to what extent we can apply the
conceptof the group velocity calculatedfor undamped
linear ECH waves.Figure 5 showedthat the wavepacket is not centered at the intersection point of
and the undamped linear dispersionrelation solution
for the correspondingECH wave branch. The group
The •- bin with the most wavepoweroverthe longest
time interval in the simulation is the one containingthe
frequencymaximum of the ECH branch betweenpoints
1 and 2 in Figure 1. This maximum correspondsto a
nonpropagating wave. Its normalized power as a func-
tion of • is plottedin Figure9. This confirms
that,
velocity
for k m 0.035(the intersection
point)exceeds
that at k m 0.08 (the signal'spowerpeak) by lessthan
15%.Thegroupvelocityat • m0.08exceeds
theminimum groupvelocity closeto &unby lessthan 5%. These
velocitydifferencesare probably negligiblecomparedto
modificationsarising from uncertaintiesin the satellite
motion
or non-Maxwellian
features
in the electron
dis-
In addition, the absorption of wave energy by the
trapped electronsand by wave-wavecouplingmay heat
the plasma. This heating will increasethe noiselevels
and may give rise to plasma instabilities which could
affect the plasma response.For the consideredsimulation parametersthe electronkinetic energyin the simulation box has increasedby only 0.4%. If, like for the
real sounder,electronsare also allowedto movealong
,
,
10
Jl
jJJjJJjJJJJJJlj
JJllJJJJ
I Jt
-10
-15
0
20
40
60
80
1O0
the magnetic field lines, an efficient relaxation mechaNormalized
time t
nism for the electrondistribution function is provided.
For the consideredplasma and emissionparameterswe Figure 8. The potential difference between the anmay thus ignore heating.
tenna plates as a function of time. Plotted is the poThe virtual antenna is now changedto an antenna tential in volts as a function of the normalized time t
with two plates on oppositepotentials. Each plate is on for COp/W•
> 1.7.
DIECKMANN
ET AL'
PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
3500
13,113
A secondpossibility could be that the wave packet's
end is not precisely defined. The wave packet encounters an unperturbed plasma in the direction it propagates. The wave amplitudesgradually increase.In the
3000
work of Brillouin [1960], it has been shownthat the
amplitude increase is accomplishedby wave forerunners. These wave forerunners are typically transiently damped modes of the plasma. They do not neces-
2500
sarily have the group velocitiescalculatedfor the undampedplasmaeigenmodes;
that is, componentsof the
wavepacketcan propagateconsiderablyfaster or slower
than the main signal. This, together with the gradual
2000
increase of the electric field amplitude, complicates a
06
150
2'
Normalized
,i
6
strict definition of the wave packet'sfront. The end of
the wavepacketis equivalentto the switchingon (after
sometime) of a secondwavewith the samefrequency
8
time t-40
as the first wave but with a phase shifted by •r. Af-
Figure 9. The power in the bin • - 2.39 as a function
ter some time the electric fields of the first and second
of the normalizedtime •. The poweris normalized
to
waves cancel out to zero. The wave packet's end will
the noisepowerin this bin andOVp/OVc
> 1.7.
thus consist of the transient
wave modes of the second
wave. It is thus equally complicatedto define the end
of a wave packet. One can therefore not expect the
wave amplitudesto return immediately to noiselevels
after the main signalhas left the spacebetweenthe antenna plates. For a more profounddiscussionof wave
at least for this simulation, the responsemaximum is
given by an accompanying wave since the wave power stays close to the antenna. The power slowly drops
forerunners,
seethe work of Brillouin [1960].
becausethe (finite width) &- bin also containspropagating wave solutions. This resonancehas been excited
by the spread in & of the emission,i.e. by the sidelobes
5.
Simulated
UH
Emissions
in a
Strongly Magnetized Plasma
of P(ov)in (2).
The second largest responseis provided by the bin
centered at •n.
The power normalized to the noise
We now consider the case of only one nonpropagat-
ing waveat &•n. Let w•-
11.5 kHz (&•n - 1.9) and
powerin this &- bin is plottedversus• in Figure10. &0 - &uh. Two simulationsare performed. One applies
The powerstronglydropsuntil • m 2.7, whichis indicated by the dashedvertical line. Afterwards,the signal
intensity starts to oscillate on a lower level than the initial intensity but still on a higher level than the noise
background.
The physical parameters under considerationare
50
45
vtn• 9.4 x 105m/s, coc= 104rad/s,and&•n = 2.1.
Theminimum
vatat & m&•nandat low/c(/c- 0.055)
is vat .• 5.4 x 105m/s. Thusit shouldtakethe UH
wave 2.7 wave periods to propagate over one antenna
length of 70 m (thus leavingthe interval betweenthe
antennaplateswhereit canbe detected).This matches
the time interval in Figure 10 up to which the signal
:•25
intensity strongly decreased. The accompanyingwaves
thus account for a substantial fraction of the plasma
15
•20
response.
Thepowerlevelsafter• m 2.7aresignificantly
higher than the noise levels without any emission.
This could be causedby wave componentsof the main
signal reflected back to the antenna location by the
noise fluctuations
of the PIC
code. The low numbers
50
I
2
Normalized
4
time t-40
6
8
of
particles per cell in a PIC simulation cause statistical
fluctuations in the plasma density. These density fluctuations could then play a role equivalent to the density
fluctuations in a "real" plasma which are held respon-
siblefor obliqueechoes[McAfee,1969].
10
Figure 10.
The power in the bin & - 2.10 as a func-
tion of the normalized
time•. The poweris normalized
to the noise power in this bin. The dashed line shows
the time it takes the resonant wave to propagate over
oneantennalengthand O•p/O•> 1.7.
13,114
DIECKMANN ET AL' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
an electricfield of 1.2 V/m to the antennagrid cellsaccounting for a potential differencebetweenthe antenna
and the surroundingplasma of 8.4 V. The secondsimu-
lationuses1/x/• timesthosevalues.WeFouriertransform in spaceover the electricfield data. We integrate
the /c-spectrum
overtheinterval/c• [0.02,
0.1]. The
normalized power is shownin Figure l l as a function of
•. The dashedline showsthe powerfor the strongemission amplitude, and it is normalized to its peak value.
The solid line showsthe power for the weak emission
amplitude, and it is normalized to half the peak value
of the dashed line to account for the different
0.4[
emission
0:>
amplitudes.
Both curvesgrow linearly at the samerate and reach
the samepowerafter the emission'send (• = 40). Then
the power remains constant, indicating that no damping mechanismis present. Increasingthe emissionam-
i J,;
'
ß/
&,/
00
500
'
,
I,
1000
15oo
Grid cell number
plitudeby a factorof x/• increases
the wavepowerby a
Figure 12.
The wave power in the bin • - 1.87
as a function of the spatial grid cell number and for
factor of 2. This shows that the emission is in a linear
cop/COc
< 1.7. The dashedline showsthe powerfor
regime.
To understandhow the energyis distributed in space,
we Fourier transform over time. The integration time
interval ranges from • = 40, i.e., • = T, up to • = 106.
We examine the power in the two bins with the highest
the strong emissionamplitude, and the solid line shows
that for the weak emission amplitude. Both curves are
normalized to the peak value of the dashedline.
power,whichare thoseat • • 1.90 and (the next lower antenna. This follows from the observation of a wave
bin) at • • 1.87. The distributionof the wavepower power minimum at the antenna'slocation. A second
as a function of the grid cell number for the stronge- indicator is discussedbelow. As expected from Figure
mission amplitude and the weak emissionamplitude in 11, the power for the dashedline is approximatelytwice
the bin at • • 1.87 is shown in Figure 12. The pow- as high as the powerfor the solid line. The generated
er is normalized to the peak power of the dashed line. wave packetsare not symmetricwith respectto the anThe antenna is marked with vertical lines in Figures 12 tenna's location. A possiblereasonfor the asymmetry
and 13. Two wave packets propagate away from the is the dispersivecharacter of the excited wave modes.
The energy of the wave packet is spread over a large
!.2
,_0.8
50.8
,
I
t'
o
ø0_0.6
i
I
•0.6
N
I
i
i
co0.4
i
o
1
jI
0.2
i
00
,
20
i
40
Normalized
I
60
time t
i
1 ooo
Grid cell number
õoo
80
1500
Figure 11. The powerof the upperhybrid (UH) wave Figure 13.
The wave power in the bin • - 1.90
as a function of the normalized time t. The dashed line
as a function of the spatial grid cell number and for
shows the power of the wave mode in the case of the
cop/COc
< 1.7. The dashedline showsthe powerfor
strongemission
and COp/COo
< 1.7. It is normalized
to the strong emissionamplitude, and the solid line shows
its peak value. The solid line showsthe curve for the
weak emission amplitude. The curve is normalized to
half the peak value of the dashed line.
that for the weak emission amplitude. Both curves are
normalized to the peak value of the dashedline in Figure
12.
DIECKMANN ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
intervalin 5•,•. The wavepacket's
shapein spaceand
time strongly depends on the phase and amplitude relations of the individual wave components. Simulation
noise will modify these relations; thus the envelope in
x, t of the wave packet will change.
The distribution
for the bin at 5• •
1.90 is shown in
Figure 13. The power is normalized to the peak power of the dashed line in Figure 12. The dashed line in
Figure 13 showsthe power distribution for the strong
emissionamplitude, and the solid line showsthe one for
the weak emissionamplitude. The overall power for the
dashedline is approximately twice the one for the solid
line, but the power is distributed differently, possiblyfor
the samereasonas in Figure 12. The pronouncedmaxima and minima indicate a stationary standing wave
40L ....
!!; '
35
..' , ß
.•_,30
, ß•,
oil
Ht (Jill• I Li
0
50
1O0
150
200
250
Distance from antenna
becausevpn • 0 and becausethe extremaare fixed.
13,115
300
The absenceof pronouncedpower minima in Figure 12
is an indicator that the waves in Figure 12 propagate;
that is, they are not standing waves.
The power of the standing wave decreasesfor increasingdistancesfrom the antenna, indicating that the
wavesare spatially damped. Sincethe plasma response
has been shown to be linear, this damping should also
be linear in which case the nonpropagating waves are
Figure 15. A contour plot of the spatial distribution
of the wave power in the bin 5• • 1.90. The logarithmic
power is plotted versusthe grid cell number relative to
weaklydamped(transient)wavesolutionsof the linear
The structuresin Figures 12 and 13 have propagated
to almost the same distance even though the range of
dispersion relation.
the antenna
and versus the normalized
simulation
time
t. The contourlinesare 10-4'1 10-3'9 10-3'7 10-3'5,
10-3'25 and 10-3
•
ß
ComparingFigures 12 and 13 suggeststhat a space- vat of the wavesin both •- bins differs. To identify a
craft moving at a velocity with respect to the plasma correlation between both structures, we plot the conthat is lessthan vat of the propagatingwavesin Figure tour lines of the time-integrated power for both bins as
12 would not detect any responseat & • 1.87. It may a function of time. The full time integration window
detect the powermaxima in spaceat • • 1.90 (Figure is over 66 emissionwave periods. The interval starting
13) providedthe wavesare stableon a timescalethat is point is shifted from t - 0 to t - 40. The same bins as
long comparedto the time it takes the satellite to reach for Figures 12 and 13 are taken. The powercontainedin
the bin at • • 1.87 is shownin Figure 14. The power is
the first power maximum.
normalized to the maximum power found in the bin at
• • 1.9 at the antenna's location. The power is plotted
versus the normalized time correspondingto the integration interval starting point and versusthe distance
in simulation grid cellsfrom the sounder. The power of
both wave packetspropagating away from the antenna
is averaged. The letter L indicatesthe regionswith the
lowest power whereas the H indicates the region with
the highest power. The power increasesfrom the regions denoted by L to the inside of the structures. One
ßN 20•lr•
J
noticesa structurepropagatingaway from the antenna.
Figure 15 showsthe signalin the bin at • • 1.9. Plotted
is the power, normalized to the same value as that
10
in Figure 14, versusthe normalized time corresponding
to the integration interval starting point and versusthe
distance in grid cells from the antenna. The contour
lines are the same as those in Figure 14. The power
Distance from antenna
maximum closeto the antenna is strong and nonpropagating. It developsa minimum at the antenna's location
Figure 14. A contour plot of the spatial distribution
at a normalized time of t - 35. The power distribuof the wavepower in the bin • ,• 1.87. The logarithmic
tion
at time t - 40 is shownin Figure 13. Apart from
power is plotted versusthe grid cell number relative to
the maximum at a distanceof • 50 grid cells, a second
the antenna. and versus the normalized
simulation time
t. The contour lines are 10-4'1 10-3'9 10-3'7 10-3'5
maximum builds up at a distanceof 150 - 250 grid cells.
10-3'25 and 10-3
The wave front of this maximum is strongly correlated
0 501O0150
dO0
250
30C
•
ß
13,116
DIECKMANN
ET AL.' PLASMA SOUNDING AT THE UPPER HYBRID
with that in Figure 14, which is in agreementwith Bril-
FREQUENCY
ma response.For this purposea set of one-dimensional
PIC simulationshavebeenperformed.
louin [1960]. The emissionwith a finite time duration electromagnetic
For weakly magnetizedplasmaswe have shownthat
generates a wave packet with a frequency spread. The
individual componentspropagate away from the anten- a strongemissionleadsto nonlineardampingup to the
of the linear wave at high k. As a likely
na with their vat. At a spatialpositiondifferentfrom suppression
the antenna position the wave componentswill arrive at
different
times.
Since the wave front is associated with
a signalgrowingin time, it will have a frequencyspread.
candidate, electron trapping has been suggestedhere
for two reasons.First, the valuesfor Ecrit estimatedfor
the onsetof trapping(equation
(1)_)havebeenexceed-
Thus the emissionat • m 1.9 (a nonpropagating
wave) ed by the electricfield of the high k wave. Second,the
generatesa waveforerunnerat a frequency• m 1.87 (a maximum electricfields also changed,as a function of
asexpected
fortrapping(equation
(1));
propagatingwave). After havingpropagateda certain •, qualitatively
distance into the plasma, the forerunner couplesback
wave energy into the nonpropagatingwave.
6.
aim of this work has been to understand
a large•. Foremission
amplitudes
thatwerelowerthan
the onesfor the real soundingexperiment,the simulations showedthat the UH wave at high k could not be
excited.This justifiedthe focuson the UH waveat low
Discussion
The
that is, the dampingis more effectivefor the wavewith
some
k in section
2.
fundamental propertiesof wavesin a warm plasma close
A different damping mechanism,wave-wavecoupling,
to co•n. This has been done with hindsight to the plas- affecting the wave at a low k, was lesseffectivein supma sounding experiment since there counis an impor- pressingthis wave which is thus expectedto provide,in
tant resonance. Results known from linear kinetic themost cases,the plasma responseto sounding at wuh.
ory have been applied to experimentalobservations.A
The PIC simulations
have revealed the extent to
quantitative explanation has been given for the absence which the V•r given by the linear dispersionrelation
of a plasma responseat wunin the hot parts of the mag- solution can be used to determine vtn. The attempt
netotail and in the Io plasma torus. It has been shown to relate the antenna length divided by the plasma rethat the theory of accompanyingwavescan explain the sponseduration(whichrequiresthe plasmaresponse
to
changeof the plasma responseduration to soundingat be dueto accompanying
waves)to V•r hasbeenproven
•un if the ratio •p/•c crosses
a valueof 1.7. This ob- successful.The case considered,i.e., a one-dimensional
servation can also be explained by the theory of oblique antenna at rest with respect to the plasma frame of
echoes[Benson,1982]. Explainingthe changein the reference, is, however, a very idealized case that may
responsedurationby a changein vat can, however,also not be comparable to the real sounding experiment.
predict the order of magnitude of the responseduration. By comparing the UH responsedurations given by the
For Wp/Wc< 1.7 the UH wavesdo not propagate.If Alouette sounder with the responsedurations expected
the plasma responsein this casewere due to accompa- from linear theory, we have, however, shown that the
nying waves, one would expect a long-lastingresponse responseduration at Wunmay indeed yield an estimate
to sounding. Then, however, it is not clear how the of vtn.
energy is transferred into the surroundingplasma. If
Soundingat co•nin a stronglymagnetizedplasmahas
we would relate the energy propagation velocity to the been shownto create a nonpropagatingstandingwave.
wave'sV•r, this wouldbe impossible.Previously,the We have suggestedthat the buildup of the structure
presenceof wave forerunners has not been considered. is achievedby forerunners. As a consequence,
the enThe frequency spread of the emitted wave packet as ergy remains confined since the only undamped wave
well as the possibleinvolvementof transiently damped solution at this frequency does not propagate. Emiswave modes in the emission processmay allow an ener- sionswith two different electric field amplitudes showed
gy transfer with a nonzerovelocity into the plasma by that, f•r t,h• p!a.sma a.nd wa.ve pnrnrn•t.•r.q
cnnqicl•r•cl
t.h•
means of the forerunners. At the same time, the only power transferred to the plasma increasesquadratically
waves at this frequency that are not spatially damped, with the emissionamplitude and linearly with the time.
the waveswith V•r - 0, do not propagate.The struc- This indicates that the plasma responsein this caseis
linear.
ture would then remain spatially confined.
We have shown that thermal effects alter the plasThe simulation results showed, for both magnetima resonancefrequency closeto Wuhprovided that the
zations, that accompanyingwaves can accountfor a
resonanceis caused by accompanying waves. The reso- plasma responseto soundingat w•h. The dependence
nancefrequency
shiftsdownward
for a ratio of Wp/Wc
of
Vat(W)for waveswith w • wu• furthermoresuggest-
lessthan 1.7, and it shifts upward in the oppositecase.
Both modifications are, at least for the plasma parameters consideredin this work, too small to seriouslyaffect
the interpretation of the sounderresults.
As a next step, we investigatedhow the inclusionof
both linear and nonlinear effects may affect the plas-
s that the frequencyspread of the peak also depends on the magnetization. While the peak shouldhave
a narrow frequencyband in strongly magnetizedplasmas, it shouldhave a wide frequencyband in weakly
magnetizedplasmas.This is true providedthe plasmas
are sufficientlyhot. For the cold plasmasfound in the
DIECKMANN
ET AL.: PLASMA SOUNDING AT THE UPPER HYBRID FREQUENCY
13,117
ionosphere
the minimum.in vat closeto c'•n wouldget Etcheto, J., H. de Feraudy, and J. G. Trotignon, Plasma
pronounced
for Wp/Wc
> 1.7, andthusthe UH response resonancestimulation in spaceplasmas,Adv. SpaceRes.,
peak would be sharp.
Acknowledgments. Mark Dieckmannhasbeenpartiallysupported
throughout
thisworkby a Warwickfellowshipandby the CNRS Orl6ans.
The workhasbeenfinishedundersupportof the Europeannetworkgrant
FMRX-CT98•168 andof a Universityof Linkoepingfellowship.Sandra
C. Chapman
wassupported
by PPARC.The authorswouldlike to thank
theNSC in Linkoeping,Sweden,for the generousprovisionof computer
time on its Cray T3E. The authorswould also like to thank V. V.
Krasnosel'skikh
and P.M. E. D6cr6au for the discussionscontributingto
this work
Hiroshi Matsumoto thanks T. Ono and another referee for their
1, 183-196, 1981.
Fejer, J. A., and W. Calvert, Resonance
effectsof electrostatic oscillationsin the ionosphere,J. Geophys.Res., 69,
5049-5062, 1964.
Krall, N. A., and A. W. Trivelpiece,Principles of Plasma Physics,405 pp., San FranciscoPress,San Francisco,
Calif., 1986.
Le Sager,P., P. Canu, and N. Cornilleau-Wehrlin,Impact
of the Ulyssesvelocityon the diagnosisof the electron
densityby the Unified Radio and PlasmaWave sounder
in the outskirts of the Io torus, J. Geophys.Res., 103,
26,667-26,677, 1998.
assistance
in evaluatingthis paper.
McAfee, J. R., Topsideray trajectoriesin an anisotropic
plasma near plasma resonances,J. Geophys.Res., 74{,
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M. E. Dieckmannand A. Ynnerman, ITN, University of Linkoeping,
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(ReceivedDecember18, 1998; revisedNovember25, 1999;
acceptedNovember25, 1999.)