JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 93, NO. A2, PAGES858-866, FEBRUARY
1, 1988
A THEORY
FOR THE
RADIATION
AT THE
THIRD
TO FIFTH
HARMONICS
OF THE PLASMA FREQUENCY UPSTREAM FROM THE EARTH'S BOW SHOCK
Iver H. Cairns
Departmentof Physicsand Astronomy,The Universityof Iowa, Iowa City
Abstract. A theory is presentedfor the radiation at the
third to fifth harmonicsof the plasmafrequencyobservedupstream from the Earth's bow shock:the radiation is produced
by the processL + T • -• T in the foreshock,
with the initial
T • radiationbeingthe frequentlyobserved
secondharmonicra-
diation(generated
by anotherprocess)
andthe L wavesbeing
productsof the decayL• -• L + S of L• wavesgeneratedby a
has previouslybeen consideredin an astrophysicalcontextby
Colgate[1967],TakakuraandYousef[1974],Zheleznyakov
and
Zlotnik[1974],andRussell
et al. [1985].Theseauthors
primarily discussed
the ratesfor the processL + T * -• T, although
Russellet al. alsodiscussed
the equilibriumspectraexpected
for the radiation. A detailedtheory for the plasmaphysicsof
the processL + T * -• T hasrecentlybeendeveloped
by Cairns
streaming
instability.(HereL, S, andT denoteLangmuir,ion
[1986b,1987c].Thistheoryspecifies
the kinematic
conditions
acoustic,and 'transverse
electromagnetic
waves,respectively.) on the participatingL and T * waves,the maximumbrightness
The theory can accountfor the observedradiation when unusu-
allylargelevels(electric
fieldsin excess
of 10mV/m)ofsuitable
L wavesare present. Such levelsof L wavesare possible,in
principle, but have not been reported before; the radiation is
observedquite infrequently,therebyimplyinga requirementfor
unusual foreshock conditions. Predictions for the characteristics
of the source
regions
(oneto eachwingof the foreshock)
and
the bandwidthof the radiationare given. Potentialproblems
for the theory,relatingto the largelevelsof L wavesrequired
to accountfor the radiation, are discussed.
1.
Introduction
Radiation at the fundamental and second harmonic of the
plasmafrequency
(fp = •Vp/2•r)
isobserved
in solarradiobursts
[e.g.,McLeanandLabrum,1985,andreferences
therein]and
upstream
fromthe Earth'sbowshock[Dunckel,
1974;Gumett,
temperaturesof the radiationproduced,the growthrates, and
path-integratedwavetemperatures
for simplesourcemodels,as
well as discussing
the sourceof the requiredL wavesand kinematic restrictionson the growthlength.
In this paper a theory involvingthe processL + T * -• T
is proposedfor the observedthird to fifth harmonicradiation,
the conditionsfor whichthe theory may explainthe observed
radiationare determined,and the viability of the theoryis discussed.It is foundthat underrelativelyextremebut possible
conditionsthe theory can explainthe observedradiation;note
that the infrequentproductionof the radiationimpliesthat unusual foreshockconditionsare required. The frameworkof the
theory developedhere is as follows:
•
1. L wavesparticipatingin the processL + T • -• T are
either generateddirectly by a streaminginstability or by the
Langrnuir
wavedecayL -• L• + S (S denotesan ion acoustic
wave).
2.
1975; Gumett et al., 1979; Harveyet al., 1979; Hoanget al.,
1981;Cairns,1986a].Solarradioburstsarenotbelieved
to have
third or higherharmonic
components
(see,however,
Takakura
and Yousef[1974]andZheleznyakov
and Zlotnik[1974]).In
contrast,
Cairns[1986a]
hasrecently
identified
radiationat the
third,fourth,andfifthharmonics
of theplasmafrequency
(with
associated
fundamental
andsecondharmonicradiation)origi-
The initial T' radiation is second harmonic radiation
generatedby the processL + L -• T + S proceedingvia succes-
sivethree-wave
processes
L -• L• + S andL + L• -• T [Cairns
andMelrose,1985;Cairns,1986b,C87].
3. The process
L + T • -• T produces
(n + 1)th harmonic
radiation
from nth harmonic radiation
for n > 2.
4. The sourceregionsare in the foreshockwhere nonther-
natingfromthe Earth'sforeshock
plasma.Thisthird andhigher
malL andS wavesareobserved
[Anderson
et al., 1981].
harmonicradiationis observed
quiteinfrequently
(sevenevents
in two yearsof data), but whenobserved
is significantly
nonthermal(i.e.,T• >>T,} with minimumbrightness
temperatures
of 2 x l0 s K to 10m K (Table1). Eachbandof the radiation
hasa narrowwidth(<•10kHz) andis centered
within10%of
The nature and spatial variation of the distribution function of electronsin the foreshockmust be specifiedso as to
constrainthe sourceof the L wavesparticipatingin the process
L+ T • • T, the characteristics
of the sourceregion,the growth
lengthsfor the radiation, and the likely effectivetemperatures
the appropriateharmonicof the plasmafrequency.
of theobserved
L waves.FilbertandKellogg[1979]andCairns
[1986b,1987a]haveshownthat the distribution
functionof the
Cairns[1986a]
suggested
that thethirdandhigherharmonic
radiation might be producedby the coalescence
L + T * -• T of
Langrnuir
(L) andtransverse
(T andT') waves
witha separate
source
of second
harmonic
radiation[Cairns,1986b].(A theory
for the second harmonic radiation
backstreamingelectronsin the foreshockmust have a •cutoff•
feature,whichis a sourceof free energyfor L wavegrowth;Figure la shows a •cutoff • distribution.
Since additional
sources
of freeenergy,suchasa bump-on-tail
distribution(Figurelb),
will be discussed in a future
paper(I. H. Cairns,A semiquantitative
theoryforthe2fpradia-
need not be but might conceivablybe superposedon a cut-
tion observedupstreamfrom the Earth's bow shock,submitted
offdistribution
[Cairns,1987a],cutoffdistributions
describe
a
to Journalof Geophysical
Research,
1987),hereinafter
referred
to as C87.) For initial T • wavesat harmonics
of the plasma
frequency,this processnaturally produceshigherharmonicradiation due to frequencyconservationin the wave process.
•minimum free energy' foreshock.It is assumedhere that the
backstreamingelectronsin the foreshockhave cutoff distribution functions. In section5 the effectson the theory of additional sourcesof free energyfor L wave growthsuperposed
on
The processL + T • -• T, sometimes
calledRamanscattering,
the cutoff distributions
2.
Copyright1988by the AmericanGeophysical
Union.
are discussed.
Semiclassical Formalism
for
the Process L + T • -• T
Paper number 7A9171.
The rateequation
for the T wavesis [e.g.,Tsytovich,
1966;
Melrose,1980]
014s-0227/8S/007A-m$0.00
858
Cairns:Theory for High HarmonicPlasmaRadiation
TABLE 1. BrightnessTemperatures
859
3. High HarmonicRadiation by the Processes
L + T • -• T in the Foreshock
Harmonic MinimumT, (Isotropic) LikelyMaximum
Kinematics of the L Waves
Second
Third
Fourth
Fifth
2 x 109 - 2 x 1011K
5 x l0 s- 101øK
3 x 10s - 6 x 10s K
2 x l0 s K
d•V•
•
2 x 1012 K
1011 K
6 x 109 K
2 x 109 K
The strictest kinematic conditionson L waves participat-
ing in the processes
L + T' -• T are set by the 2fp to 3fp
emission
process
[Cairns,1986b,1987c](seeappendix),since
(1) L wavesparticipating
in the 2fp to 3fp emission
process
havethe smallestrangeof allowedwavenumbers(1.1cop/c
to
4.6/4. (2)
w,v,
h,v
wavenumbers,corresponding
to the most decaysL -• L' + S
when the decayL • L' + S producesthe relevantL waves,
dskL
dSkT'(kT
kL,kT,)[NL(kL,)N•(k•)
and(3) 3fp radiationmustbeproduced
beforehigherharmonic
=
(2•r)S
(2•r)S
UTL• ,
-- NT(kT){NL(kL) +/V•(kr,)}]
(1)
wherethe quantity UTLT, is the probabilityfor the processL +
T••
T:
emissionis possible.
The kinematicsof the wave processes,together with the
structureof the foreshock
and directivityof the 2fp radiation,
constrainthe sourceof the L wavesparticipating in the processesL + T' • T. Although not well constrainedby present
IITLT•
=
he2
cop
theories(or observations),
the 2fp radiationis likelyto be di-
kL'(2•r)s(1
+ [K•. KT[2)
rected at angleslessthan about 45ø with respectto the mag-
16e0m•2 coTCOT,
ß•S(kT- kL- k•)•(coT- col- coT,)
(2)
(summing
overthestatesof polarization
of T wavesin the final
stateandaveraging
overthe initialstatesof polarization).The
deltafunctions
in (2) leadto kinematicconstraints
on the frequenciesand wave vectorsof wavestaking part in the processL + T' -• T, sincethey requireconservationof frequency
coT(kT)= coL(kL)+ coT,(k•) andconservation
of wavevector
kT = kL + k•.
Derivationsof somekinematic constraintsused
in section3 aregivenin the appendix.Notethat sincecol---cop,
neticfield[Cairns,1986a,C87].Accordingly,
if the L wavesare
generated
directlyby a streaming
instability(kL = copvb/v•2),
thenkL' k2 > 0 (with kT, = k2), requiringstreaming
speeds
,, ~
fo, = 2). A,
ing electronshave'cutott• distributionfunctions(Figurela)
[Filbertand Kellogg,1979;Cairns,1986b,1987a,b], the volumeof the foreshock
with suchspeeds
is extremelysmall(Figure 2), and it seerr•unlikelythat this volumecouldgenerate
the observedthird and higher harmonic emission. Moreover,
them is no statistically significantevidencefor such energetic
electrons
in the foreshock
[Anderson,
1981].Directgeneration
frequency
conservation
requires
cot--•(n+ 1)cop
fo•coT'
of the L wavesparticipatingin the processL + T' -• T by
The quantityNM(k) is the plaamon
or photonoccupation cutoff distributionsof electronsis consequentlynot a plausible
number(forM = L or T, respectively),
definedto be the numhypothesis;insteada scatteringprocessmust producethe reber densityof quantain the modeM within the elementalrange
quired L waves.The effectsof additionalsourcesof free energy
dSkof k. NM(k) is relatedto the energydensityof wavesin
for L wave growthon this conclusionare discussed
in section5.
themodeM, W•(k), by N•(k) = 2•rW•(k)/h•(k).
temperature
TM(k) is definedby
A wave
Cairns[1986b,C87]presented
strongevidence
that the decay L -• L' + S proceedsin the foreshockand is an essential
process
in producing
the Earth's2fp radiation.Accordingly,
it
is natural to supposethat the decayL -• L' + S generatesthe
T•(k)= h•(k)N•(k)
2•rks
(3)
where ks is Boltzmann's constant. Waves are nonthermal if
T• > T,, whereT, is theelectron
temperature
[Melrose,
1970].
L wavesrequiredin the processL + T' -• T either in one step
or in a seriesof decays(a cascade).
The decayL • L' + S proceedsonlywhenthe wavenumbers
kL andkL, satisfy[Cairns,
1986b,C87]
kL,kL, >_ko/2
(4)
(b)
(a)
CUTOFF
DISTRIBUTION
ß
"&
h
•,
f(V•)•
f (V•)
ARB. UN ITS
THERMAL
..... BUMP
-ON
-TAIL
Cu•u•-•-
FEATURE
' •/
D
DISTRIBUTION
'•
I
Vc
Vl•
Vc
_•
V•
Fig. 1. (a) A cutoffdistribution.The cutoffis imposed
by the requirement
that the electronescapeupstream
fromthe shock.(b) A cutoffdistribution
with a superimposed
bump-on-tail
feature. This featuremightbe
imposedby an unusuallyhigh flux of semirelativistic
electronsproducedovera largefractionof the bowshock's
surface.
860
Cairns: Theory for High HarmonicPlasma Radiation
•V-sw
Fig. 2. The linesof constantcutoff velocityupstreamfrom a nominal terrestrial bow shockin the ecliptic plane.
The assumed
solarwind and bowshockparameters
are vo• = 350 km s-1 , an angle of 315ø betweenvo• and
B, and a bowshockX - a - bY2 with a -- 14.6Rs and b-1 = 45 Rs. Distances
are shownin Earth radii Rs.
Cutoffvelocities
greaterthan107m s-1 arefoundveryclose
to thetangential
magnetic
fieldline,i.e.,x <•1 Rs
for R < 25 Rs.
tern),thedecayL --• L•+ S proceeds
primarilyasa backscaSter
with kL, •-- kL- k0 andkL,. kL -----kz•,kz•, sincethesecon-
corresponding
to electronstreaming
speeds
(fork•, =
-
(s)
Here,k0 = 2%,V's/3V'e:
is a constant
of the plasma,and
Vs is the ion sound speed. Typical foreshockparameters
Ve - 1.5 x 10• m s-1 and Vs - 5 x 104m s-1 then imply
vb < 1.4 x 10Sm s-1. This conditionrestrictsthe sourceof
the high harmonicradiation to regionzof the foreshockwhere
v• <_1.4x 10Sms-1.
Combiningthe requirement(5) with the requirements
on
k•, in the appendix,
1.1 <__
X = k•,c/%,<__
4.56,leadsto a
minimumelectronthermal speedfor whichL • wavesproduced
by the process
L --• L• -!-S mayparticipase
in the 2f• to 3f•
process
[Cairns,1986b,1987c]
2.3x 10•
V,>_-••
V/1+ 3T•/T•
ms
-1
(6)
Emissionby the processL + T * -• T involvingsuchL wavesis
thenfavoredin higherV,, lowerTi/T, plasmas.In theforeshock,
Veusuallyrangesbetween106m s-1 and3 x 106m s-1 (7 x
104K <__
T, <__
6 x 105K) witha typical
valueof1.5x 106ms-1,
ditionsmaximizethe rate. Whenko/2 _<k•, _<3k0/2, the
decay tends to isotropizethe L wavesby diffusingthe warm
in angle rather than wave number. Consequently,the decay
mayleadto a quasi-isotropic
condensate
nearkL ~ k0 [Newman, 1985]. (Strongturbulence
effectsmay be importantfor
suchcondensatss
[e.g.,Goldman,1984];thispointis discu•d
furtherin section5.) The kinematicconditions
on the processL + T • --• T (seeappendix)do not, in principle,restrict
the participating L wavssto be from a backscaSterspectrum
with kL >• 3k0/2 or from a quasi-isotropic
condensate
with
3k0/2 >_k•, _>k0/2. However,
quasi-isotropic
L wavespectra
with kz•~ ko~ %,/c- 5%,/cshouldbeparticularly
suitablefor
emissionby the processœ+ T • • T due to the addedflexibility
to meet kinematic
restrictions on the wave vector of the L waves.
In the absence of suitable numerical studies of the evolution of
the L wavespectrain the foreshock,estimatezof the numbersof
decays
to reachwavenumbers
of orderk0 and%,/c arehelpful
in decidingwhether backscaSteror quasi-isotropiccondensate
spectraare the primary sourceof L wavesparticipatingin the
processœ+ T • --• T in the foreshock.
The minimu
TMnumbern of decays
to reachthe minimum
wavenumberko/2fromthewavenumberk• = u•/v•, (assumed
whileTi/T, usuallyliesbetween
0.1 and10with a typicalvalue
of 1/3. With X rangingbetween1.1 and 4.6, it is clearthat
larger
than3ko/2)isthenn = integer
part{%,/kovb
- •}, and
the decay L -• L * + S shouldproduceL waveswith suitable
wavenumbersto participatein the processL + T • -• T in the
foreshock.
Undertypicalforeshock
conditions,
(6) impliesthat
valuesX > 2 arefavored,
sothekL'k2 < 0 (see(A3) withk•, =
k2). Thasis, L wavesand 2f• radiationparticipating
in the
2f• to 3f• process
shouldtendto be oppositely
directed.This
bera4v•,/c
fromkl.,is rn= integerpart {•(1 -av•/c)/kov•}. ff
rn, n < 1, thenk•, = •v•/v• < 3k0/2. Valuesfor rn and n are
givenin Table2 forTi/T• = 1/3 andV, = 1.5x 106ms-1 and
V, = 2 x 106ms-1 (bracketed),
forv• in therange5 x 106ms-1
to 5 x 107m s-1. Thenumbers
of decays
rn andn increase
with
increases
in V• anddecreases
in a, vb,andTilT,. Restricting
rn
conclusionneednot be appropriatewhen the higherharmonic
radiation is producedin unusuallyhot forsshockplasmas.
implyinglimits on the sourceregionof the radiation.
For wavenumbersk•, >_ 3k0/2, corresponding
to v• _<
V•2/Vs (or v• _<5 x 107ms-1 fortheaboveforeshock
pararne-
In regions
oftheformhock
where5 x 107ms-1% v• %1.4x
lOSms
-1, thedecayL --• Lt+S tendsto convert
L waves
gener-
the minimum numberrn of decaysrequiredto reacha wavenum-
andn to be lessthan,say,5 requires
v• >• 107m s-1, thereby
Cairns:Theoryfor HighHarmonicPlasmaRadiation
Bandwidth of the Radiation
TABLE 2. Number of DecaysRequired
Frequency
conservation
and the L and T dispersion
rela-
vb, ms-1
5 x 106
=s
=
2 x 10?
(4)
(s)
(s)
861
tions(col
s = •op
s+3kLsV,
sand•ors = cop
s+ksc•, respectively)
5 x 107
implythat thefinalT waveis produced
at a frequency
o
o (o)
o
atedby thestreaming
instabilityintoa quasi-isotropic
condensatewith/cL~/co. TheseL wavesmayparticipatein the process
L+T • -, T. In regions
where
1x107ms-I <•vs<•5x107ms
-l,
L wavescapableof participating
in the process
L + T • -, T
shouldbeproduced
by boththe backscattering
stageof the decayL -, L' + $ andby the condensate
whichmayresultonce
the backscattering
stageof the decayis finished.In summary,
A!=X• •/p
V'•
(7)
abovethefrequency
(n + 1)/p for initialT' radiationat a frequency
n/p,whereX = kLc/o•p
isdetermined
in theappendix.
In thesolarwind,V,•/c• ~ 10-•, implying
thatthebandwidth
ofradiation
at the(n + 1)thharmonic
equals
thebandwidth
of
radiationat the nth harmonicto a goodapproximation
(since
X <_10 for T radiationbelowthe fifth harmonic).Hence,the
bandwidthsof the third, fourth, and fifth harmonicradiation
shouldequalthe bandwidthof the second
harmonicradiation.
Accordingly,
theratioof thebandwidth
to thefrequency
of the
nth harmonic radiation should decrease as n increases.
the allowed
variations
in v• andV, (andin m andn) should
permitthedecay• -, L•+ $ to produce
L waves
kinematically
compatible
withtheprocess
L + T• -, T in a sizeable
fraction Growth Len_zths
of the foreshock.
Characteristicsof the SourceRe_gions
Figure2 shows
that the Earth'sforeshock
hastwodistinct
wings.Observationally,
asfar asisknown,theproperties
ofthe
waves
andparticles
in thetwowings
oftheforeshock
aresimilar
The kinematicsof the processL + T' -• T restrict the
growth
lengthoftheT radiation
[Cairns,
1986b,1987c].
These
limitson the growthlengtharisea• follows.In a time-steady
plasma,
T wavesproduced
by theprocess
L + T' -• T propagatewith constantfrequency
0•T and growonlywhenwave
vectorandfrequency
conservation
is possible.
Usingthe L and
(R. R. Anderson
andT. E. Eastman,
privatecommunications,T dispersion
relations
andthedefinition
X = kLc/O,,p,
frequency
1984).Freeenergy
for L wavegrowth
should
be available
in
conservationgives
cutoffdistributions
in bothwingsof the foreshock
[Filbertand
Kellogg,1979;Cairns,1986b,1987a].It followsthat the two
wingsof theforeshock
should
besimilarassources
of thirdand
higherharmonic
radiation,
andhence
thattwothirdandhigher
=
+sg,
+
harmonic radiation sourcesshould exist near the Earth's bow
shock.The abovearguments
alsoimplythat twofundamental
and secondharmonic sourcesexist near the Earth's bow shock
where•,,(r) is thefrequency
of an initialTt waveat position
r. Restrictions
on the growthlengtharisefrom changesin, or
[Cairns,
1986b,C87].
In the lastsubsection
thesourceregionsof the highharmonic
radiationare shownto be restrictedto regionsof the foreshock
wherev• <_1.4x l0 sm s-•. Therearetwofurthereffectswhich
-
V
c;[.4
xI0emg[-•,• -
biasthe s•urceto regions
of highv•. Botheffectsfollowfrom
by (fo
popotiona
to NL and henceto the electricfield squaredin the coalescing • waves.First, the electricfieldin the coalescing
• waves
is limitedby the electricfieldEL in the streaminginstabilitygenerated
L waves.It is observed
[Filbertand Kellogg,1979;
EtchetoandFaucheux,
1984;Lacombe
et el., 1985]that EL decreaseswith increasingdistancefrom the foreshockboundary
or alternativelywith decreasing
electronstreamingspeed.One
thendeduces
that the sourceregionsshouldbe biasedto regions
of highv•. Althoughthe detailedspatialvariationin EL is not
I[
-
?c• I.•xl
known,the regionwith largeEL is at mosta few Earth radii
thick,thereby
indicating
v• >• 107ms-• (Figure2). Second,
-24
the levelof • wavesableto coalesce
in the process• + T • -, T
lutionof the L wavespectrumin the foreshock
arerequiredto
further restrict the sourceregion.
The sourceregionsdeterminedby the aboveargumentsare
givenin Figure3; thesesource
regions
areverysimilarto the
sourceregionsfor the second
harmonicradiationderivedby
Cairns(C87). Theseregions
should
containhighlevelsof L,
L•, andS waves,aswellasharmonic
radiation.
•
I
,q'my'
........
is expected
to decrease
with increases
in the numberof decays
requiredto producethese• wavesfromthe primary• waves,
againindicating
(Table2) v• >•107m s-•. Studies
ontheevo-
b
-60
-48
-•
-24
, ......
-12
0
12
24
Y
•.
3, •e p••
monicr•iatbn ••
•ume •eSiomof t•e th• to •ft• •th• the L wave rm•nsible •e pr•
duc• by the pr•
L• • L + S of L• wav• gen•at• by
stm•g
ins•bi•ty. E• wi• of the for•k
cont•ns
•ur• regiondefin• by the1• of str•g
vel•i• (•
su• to • thecutoffrellim) of v• = 1.4x 10s m s-• •d
= 10• m s-•.
862
Cairns:Theory for High HarmonicPlasmaRadiation
i
i
,
i 1111
I
i
i
i
i
i i i iI
i
i
i
Thesesourcemodelsindicatethat the growthlengthfor the
radiationshouldbe determinedprimarily by the velocitygradient in the foreshock.In this case,the radiationis likely to
be stronglybiasedto propagationangleslessthan 10ø from the
i i i i
magneticfielddirectionwith growthlengthss ~ 107-108m.
ioe
The effects on these conclusions of additional
sources of free
energyfor L wave growth are discussedin section5.
Vc/m•'
Maximum BrightnessTemperatures
r: 20 RE
Themaximum
brightness
temperature
of (n+ 1)thharmonic
radiationis givenby the conditiondNT/dt = 0 in (1) andthe
relation(3) as
= (.+
i0e
,
I• 2
Id'
,
I
I0
+
FornTz.>>T,, (10)reduces
to T,+ • • T,, •d it maybesh•n
th•
2
x/R E
Fig. 4. The detailedvariationin the cutoffvelocityv, as a
functionof z (seeFigure2) for R = 20 Rz and R = 30 Rs.
The componentof the growthlength s, perpendicularto the
magneticfield,As, maybe readfromthe graphusing(17).
restrictions
on, •r,(r), •op(r),V,, andX(r) across
the source
region.
Neglectingspatial variationsin the plasmafrequency,the
growthlengthI•,rl isdetermined
by thespatialvariationin the
electronstreamingspeed.Becausethe L wavesparticipatingin
the processL + T' --, T are producedby the decayL --, L' + S
of streaminginstability-generatedL waves,the restrictionson
IArl are determined
by conditions
on v• for the decayL -•
L' + S to proceed.Theseconditionson v• are
T• •n+l
>
for (n + 1)thh•nic
T.+•
(11)
r•ation generat• froma •ume of 2f•
r•a•on (n • 2).
For L* wav• generatedby the decayL • L * + S, TL, • TL
(e.g.,087), • th• thetem•rature oftheL wavesp•ticipating
in the pr•
L + T * • T • li•ted by the temperatureTL
of the L wav• generateddirectly by the stre••
instabi•ty.
The 2fp bri•tne• tem•rature Ts is a• li•t•
by TL [e.g.,
M•r•,
19•], • that (11) impli• TL • Ts • (2/n)T, •
(2/n+l)T,+•. •bles 1•d 3 in&ca• thattheob•rved2fv •d
hiker h•onk
r•iation •e cons•tent with this l•t.
Th•
conclusion
is not •ed
by the uncer•nti• in the soumesiz•
&• •d bri•tne• temperaturesT• of the obeyed radiation.
H•ever, sin• •cur•e esti••
of TL, for L * wav• co•escing
in the pr•e• L* + T * • T •e not preenfly avilahie, a more
•c•ate testof (11)canot • performed
here.
ifps
c
• >_
v•,>_2(1
+.Vsc/3if,')
(9)
wheren is the maxinmmnumberof decaysfor whichhigh levels
of suitable L wavesexist. The maximum growthlength is then
the maximum distancein the foreshockbetweenregionswith
v• bounded
as in (9). •kkiag If, = 1.5x 106m s-1, Ifs =
5x104ms -1 andn=5 (9) becomes
1.4x108ms-l>v•>
Rates
The rate dNr/dt for producing
highharmonicradiationderivedfrom(1) maybewrittenapproximately
as[Cairns,1986b,
1987c]
•dN*•+l =4.0x10
-ssfvs(Xs'
- Xl') T,,NI;
1.2 x 107 m s-1.
Assumingthat the primary L wavesare generatedby 'cut-
oIP'distributions
ofelectmus
[FilbertandKellogg,
1979;Cairns,
1986b,1987b],thecutoffspeedmaybeidentified
asthestreaming speed v•. Figure 2 showsthe loci of the lines of constant
cutoffspeedin the foreshock;the detailedvariationin the cutoff velocity with perpendiculardistance•: from the foreshock
boundary is given in Figure 4. These figu;esand the restrictionson v• imply growthlengthsperpendicularto the magnetic
n(n+ 1)•/ns+ 2n
(12)
where Xs and Xl are the maximum and minimum valuesof
in unitsof •p/½ givenby (A3) and (A4). Substituting
30 kHz into(12),onefinds
(Xs4 - X• •)
T,
dN,•+l/Nr.
=3.6
x10
-s4
n(n+ 1)v/ns+ 2n
dt
(13)
fieldof the orderof 0.5Rz ~ 3 x 106m for R <• 30Rs. For
T wavespropagating
at an angle0 (rood180ø) to themagnetic
field,thegrowthlengthis s ~ 3 x 106/Isin01m; consequently,
for 9 _<10ø, s _>107m, andfor 9 _<1ø, s _>108m.
Gradientsin the plasmadensityshouldbe ineffectivein re-
strictingthegrowthlength:equation(8) impliesthat changes
in
/p(r) oftheorderofthebandwidth
oftheT' radiation(>•0.2f•)
are required(the termsproportional
to X s havemagnitudes
lessthan 10-s in the foreshock
andare negligible).For instance,Ar ~ 10lø m for a sphericany
symmetric
solarwind
with•ov(r
) •x r-1. Similarly,
theobserved
solarwinddiscontinuitieswiththerequired
changes
in plasmafrequency
(>•5kHz)
haveinferreddimensions
of the orderof 50RE [Lacombe
et al.,
1987],therebyimplyingAr >•3 X 108m.
TABLE 3. L WaveTemperatures
[Cairns,1986b,C87]
Conditions
Overall.range
E _<10mV/m
Tr,/K
_<2 x 10søK
v• _<108ms-1
All
> 0.1sr
Mostplausiblerange
v• = 2 x c•7 m S--1
1mV/m _<E _<10mV/m
3 X 10ls K _<TL _•2 x 1018K
863
Cairns:TheoryforHighHarmonic
Plasma
Radiation
TABLE 4. Wave Temperatures
T2-- 2 x 1011
K
TL -- 10ls K
T3
T4
T$
2 x 106K
4 x l0s K
Thermal
Thermal
2 x l0s K
4 x l0s K
Thermal
Thermal
2 x 109K
2 x 107K
Thermal
Thermal
2 x 101ø
K
2 x 109K
3 x 10s K
4 x 107 K
s-- 107 m
TL = 10:ø K
s = 107 m
Tœ= 10:1K
$ = 107 m
TL = 10:1K
(or1022K)
s= 10s m
(or107m)
because
thegrowth
length
forthehigh
harmonic
Choosing
n = 2 andthevalues
of Ts in Table1 gives beappropriate
radiation
(107-10
s
m)
greatly
exceeds
that
for
the
2f
v
radiation
(dNs/dt)/N•
intherange
2x 10-10to2 x 10-s. Thecoef- (~ 106m)[Cairns,
1986b,
C871.
ficient
in(13)increases
slightly
withincreasing
n,butsince
T,
In Table4, path-integrated
brightness
temperatures
for
decreases
withn (•able4),therates(dN,+l/dt)/Nl;
tendto third
to sixthharmonic
radiation
areshown
forvarious
L wave
decrease
withn. Asshown
below,
these
ratesare'notsufficient
andgrowth
lengths.
Notethatthefourth
and
to explain
theobserved
levels
of3fv, 4fv, and$fvradiation temperatures
fifthrowsofthetableareconsistent
withthetemperatures
of
except
whenveryhighlevels
ofL waves
arepresent.
theobserved
highharmonic
radiation
(Table1). Thisconclusionis notaltered
by theuncertainty
in thesource
sizeAll,
Path-Integra_ted
Brig?tness
Temperatures
andhence
thebrightness
temperatures
of theradiation,
pro-
onlythatthe2fv andhigher
harmonic
radiation
have
Path-integrated
wave
temperatures
arederived
fromtherate vided
source
sizes
(since
T,(s)ocTsforn > 2 in (15)).The
equations
byconverting
thetotaltimederivative
d/drintoa similar
implies
that(1)L waves
withT• ~ 1021-1022
K and2fv
convective
derivative
%d/ds,
where
%isthewave's
group
speed table
withT2~ 1011K canaccount
fortheobserved
levandthedistance
s parametrizes
theraypath,andintegrating radiation
els
of
3f
v,
4f
v,
and
5f
v
radiation
in
the
allowed
path
lengths
over
s [e.g.,
Cairns,
1986b,
1987c].
Thetemperatures
TLand
s m),(2)L waves
withTl;~ 102o
K could
produce
T2areassumed
constant
while
T,+Iforn > 2 varies
along
the (107-10
nonthermal
3fvradiation
withTs~ 10s-109
K,butnegligible
raypath.
Then
one
finds
[e.g.,
Cairns,
1986-•,
1987c]
4fvand5fvradiation,
(3)theprocesses
L+ T• --•T should
notapproach
saturation
intheforeshock,
and(4)thenumber
T.+l(8) -- T.+I(0)
.--$ i
of observable
harlnonics
should
varyin response
to changes
in
8i+1
+Z H A"-$T"-i(O)TLi+I
(i+1)!
i:0
$=0
.--S
+II
i:0
8.--1
(.-
(14)
Tl; andthe growthlength.
Thebrightness
temperatures
oftheobserved
highharmonic
radiation
[Cairns,
1986a]
areapproximately
oneorder
ofmagnitudeabove
thebackground
ontheUniversity
ofIowa's
Sweep
Frequency
Receiver
(SFR)instrument;
thebackground
corm-
sponds
toa temperature
oftheorder
ofl0s K. Taking
this
factintoaccount,
Table4 implies
that (1) nonthermal
har-
and with
A.=1.3
X10
-41V/(n2
I's(Xs4
--X14)
+ 2n)(n2- 1)
andX1 andX2 given
bytheleftandrightsides
of (A3)and
(A4),respectively,
where
H A•_i= A.A.-1...As
monic
radiation
produced
byL waves
withTL<•102o
K should
be observable
withtheSFRinstrument,
and(2) at most
(15) not
fiveharmonics
shouldbe observable
withthe SFRinstrument
ifT• <•10•s K ands <•l0s m.
Taking
theratioof(14)fortwoconsecutive
harmonics
(n>_
2),one
finds
anapproximate
condition
forT,+l(S)toexceed
T,+s(s)
onkeeping
onlythelasttermin(14):
A. Ti;s/n< I
(16)
i=0
(15)withfv= 30kHz.Thiscondition
is
Thismodel
corresponds
to highharmonic
radiation
growing orT•s <•10søusing
analogous
toRussell
etal.'s[1985]
condition
z = I separating
froma source
of2]pradiation
bytheprocesses
L + T• -• T.
equilibrium
spectra
fortheprocess
L + T• --*T. Note
Assuming
Tstobeconstant
while
TstoT6varywithsshould different
864
Cairns:Theory for High HarmonicPlasmaRadiation
ticular,whenTr ~ 10•-10 •= K, thetheorycanaccount
for the
haveTr s <•10•9, consistent
withtheintensity
of theharmonic maximumlevelsof the 3fp, 4f•,, and5fp radiationobserved
and
that the valuesof Tr and s requiredto explainthe observations
radiation decreasingwith increasingharmonicnumber.
Estimatesof Tr for the observedœwaves,assumingthem to
be streaminginstability-generated
waves,are givenin Table 3
[Cairns,1986b,C87]. It is foundthat Tr <•2 x 10•ø K, with
the number of observedharmonics. When Tr decreasesbelow
10• K, the radiationlevelsdecrease
towardthe instrumental
threshold
(at mosta factorof 10belowthe maximumobserved
levelsof the radiation),andboththe numberof observed
har-
the mostprobablevalueslyingin the range10•s K to 10•s K.
Valuesof Tr ~ 10•ø K correspond
to the maximumobserved
monics and the levels of the observed harmonics should decrease.
electricfieldof theorderof 10mV/mandvb~ l0 s m s-•. For
vb< l0 s m s-z, temperatures
Tr >• 10•ø K implyL waveelectric fieldsin excess
of 10 mV/m. However,
the highharmonic
unobservable
with the SFR instrument,levelsof 3fp and per-
WhenTr <_102•K, thetheorystillpredicts
nonthermal•
but
radiation is observedextremelyinfrequently,and the L wave
hapshigher harmonicradiation.
In principle, there is sufficientenergy in the foreshock
's
streamingelectronsfor L waveswith temperaturesof the or-
electric fields when the radiation
is observed are not known.
der of 10•-10 • K to be generated.This may be shownas
Accordingly,
the possibilitythat L waveswith Tr ~ 10•-
follows.For L wavesgenerateddirectly by a streaminginstabil-
10• K (andE > 10mV/m)arepresent
whenthehighharmonic ity the energydensityin the L wavesis limited by the kinetic
radiation is observedcannotbe ruled out at the presenttime.
energydensityof the streamingelectrons,i.e.,
Therequirement
thatTr beoftheorderof 10• K whenthehigh
harmonicradiation is producedshould,however,be considered
a difficultyfor a theoryin termsof the pros
dsk
keTL(k)
•
-<n•m•v•
L + T' --, T.
4. A Theoryfor the 3fp, 4fp, and 5fp Radiation
ObservedUpstream of the Earth's Bow Shock
(•7)
Thewavetemperature
TL(k) isconstant
(= Tr) in a solidangle
All andrangeof wavenumbers
Ak(= 2kLIf•/vb)aboutkr =
o,,vv•/v•
•, andiszeroelsewhere,
whence
onederives
Theanalyses
in section
3 suggest
thefollowing
theoryforthe
3f•,, 4f•,, and 5f•, radiationobserved
upstreamof the Earth's
TL• nbm•Vb
2 Vb
4
bow shock:
-
2ke/•sV•All
1. Small wave number L wavesare producedby the decay
L --* L' + S of streaminginstability-generated
L waves.
2. Third and higherharmonicradiationis producedby the
processes
L + T • • T, with the small wavenumberL waves,
Taking foreshockparametersof fp ~ 30 kHz and n, =
fromthe2fv radiation
generated
in theforeshock
[e.g.,Cairns,
v•
3. The radiation is generatedin both wings of the fore-
shockin regions
wherev• <_3if,•/Ifs andhighlevelsof suitable
L waves exist.
4. Theory requiresthat the effectivetemperatureof the
primary L wavesexceedsthe brightnesstemperaturesT,• of the
nf•, radiationwith T,, >=T,,+• for all n >__
2.
10• m-s and beamparameters
of All ~ 0.1, If•/v• ~ 0.2,
andnb/n• ~ 10-s-10-•, (18) impliesTr <• 10•-10• K for
2x10•ms-• Tr <10•s-10•4Kforv•~5x10•ms-•,and
Tr <=10ss-10
• K forv• • 10sins-•. The, tem•ratur• of
10•-10 • K corr•nd to, de•nding on the stre••
s•,
10-s%to 1•% of them•mum •ble
L wavetemperature.
Therefore,onec•n•
a priori rule out the •ib•i•
of L wav•
with TL • 10•-10 • K pr•uc•g the obs•vedhi• h•onic
r•a•on.
The theory r•uk•
the •ce
re.ohs for the r•iation to
5. Thebandwidth
ofthenthharmonic
(forn >__
3) should have(1) stm•ng s•
v• •t•n
10•m s-• •d 1.4x
equalthe bandwidthof the 2fp radiation.
l•ms
-•
•d
(2)
L
wav•
wi•
TL
-10•-1•
• K (ora sparky
Pointsin favorof thistheoryareasfollows:
(1) Strongeviaveraged
T•,
{T•},
of
1•-10
•
K)
for
gr•
len•hsof the
denceexiststhat the decayL --, L • + S proceedsin the foreorderof 10v-10sm. Th• conditi• •e •stent:
Fimt, the
shock
andproduces
L• waves
withTr, >_10-sTr [Anderson
et
al., 1981;Lacombe
et al., 1985;Cairns,1986a,C87].(2) L waves
suitablefor the processesL + T • --, T shouldbe produced
by the decayL --, L • + S in a significantregionof the fore-
shockfor a widerangeof foreshock
conditions.
(3) The brightnesstemperatures
of the observed
nfp radiationare consistent
with the theoretical
restrictions
in paragraph
4 above.(4) The
processes
L + T • --• T canproducen)'pradiationwith the observedbrightnesstemperaturesand numberof harmonicsin the
allowedgrowthlengthsfor L waveswith Tr ~ 10a•-10aaK.
5.
Discussion
c•cula•om in the prece&ngp•Mraph indkate th• L wav•
withTL • 10•-1• • K •e •ible onlywh•e v• > 10Vms-•,
•d •cond, gr•th len•hsof 10V-l• m •e •ible onlyforradi•ion direaedat extremely
sm•l ••
(10ø) to them•netic
field, therebyrequi•ng propag•ion • regionsof the foreshock
wherev• > 10Vms-• (cf.Fibre 2).
A s•ulative mMel for the s•t•ing on of the high h•monic r•iation may • conaruc•d • follows:sup•
that
(1) the bow sh•k pr•uces si•ific•y
se•rel•i•c
enh•c•
fluxesof
el•tmnswi• 0.1c• vii • c (1 keVto a•ut
1 MeV) from a l•ge fr•on of its suff•e under•me condi•ons,(2) the• energetic
el•tmns su•r•
a bum•on-t•l
fe•ure onto the nor•ly
print
cutoff diaribution of elec-
The Earth's3fp andhigherharmonicradiationis observed trons(e.g.,Fibre lb), •d (3) pr•y
L wav• •e gener•ed
infrequently:
seveneventsin twoyearsof data [Cairns,1986a]. pref•enti•ly by the bumFon-t•l fe•ure. This exotic fo•
Accordingly,a theory whichaccountsfor the observedradiation
in terms of typical foreshockL wave levelsand plasmaconditions is certainly incorrect. A successfultheory must instead
identifysomeforeshock
parameter(s),
suchas the L wavesor
someplasmaparameter(s),
as beingatypicalwhenthe radiation is observed,and showthat changesin this parametermay
turn the radiation off and on. In the theory suggestedin section 4 the •switch• parameteris identifiedas the temperature
Tr of L wavesableto coalesce
in the process
L-i-T• • T. In par-
sh• •ht
• ex•t•
to • a •tter •urce of high h•onic
r•a•on
th• the nor• for•k
f• the followingre2ons.
Fkst, the vol•e of sp•e in which pr••
L waves•e gen•a•d with ph• s•s
•t•en
0.1c •d c should• enh•d
gr•tly comp•
with the no•
fore•k.
Second,L warm
with kL • k0 • •v/c- •v/c (suitablefor the ••cence
L + T • • T) should• gener•edin a si•ific•tly l•ger volume of the fomsh•k. It is •ible
th• direr p•cip•ion
of the pri•y
L wav• in the co••ce
L + T • • T •t
Cairns:Theory for High HarmonicPlasmaRadiation
865
of theradiationsoasto achieve
growthlengthss ~ 107-10s m
L waves.This is as it shouldbe: the high harmonicradiation is
observedinfrequently,thereby requiringa successful
theory to
should be weakened. Fourth, the foreshockvolume in which
identifysomeforeshock
parameter(s)
asbeingexceptional
when
primaryL waveshavetemperatures
TL of the orderof l0 ss10s6K (see(18))should
besignificantly
increased,
therebypre-
the high harmonicradiation is .observed.The availableobser-
sumably increasingthe size of the region containing /, waves
mined. The switchparameterproposedhere is TL: when T•, is
with TL ~ 10sx-10ssK capableof participatingin the process
of the orderof 10sx-10ssK in the sourceregion,correspond-
/, + T' -• T. In summary,this exotic foreshockshouldcontain greatervolumesof spacecapableof generatingthe observed
of 10 mV/m, thenthe highharmonicradiationis observed.A
radiation
speculativescenariofor this switchingprocessis suggested.This
becomesignificant. Third, the requirementson the directivity
than the nominal foreshock.
vationaldatadonot allowtheswitchparameter(s)
to be detering to electric fields in excessof the maximum observedfield
Recentsolutions
[Newman,
1985]of the Zakharovequations requirement
for fieldsin excess
of 10mV/min the source
region
for Fitzenreiteret al.'s [1984]foreshock
electrondistributions is a possibledifficultyfor the theory. It shouldbe noted, howshowthe decay/, -• L' + S producinghighlevelsof smallwave
number/, waveswith kL ~ k0. These wavesmay participate
in the processI, + T' .-, T, therebygivingsomequalitative
supportfor the theory suggested
here. Moreover,Newman's
solutionsshowrapid transitionsinto the strong Langrnuirtur-
bulence
regime(see,forexample,
thereviewbyGoldman[1984])
unlessthe Langrnuirinstability growth rates are artificially reducedby a factor of 20. This is of interestbecauseL waveswith
TL ~ 10sx-10
ssK are likelyto be withinthe strongturbulence
regime. Further work is necessary
to determinethe importance
of strongturbulenceeffectson the process/,+ T' -• T in this
situation.
The principaldifficultywith the theory lies in the lack of
observational
data for L wavesin the foreshock
with TL ~ 10sx-
ever, that changesin the sourcemodel,in particular the growth
lengths,may reducethe levelsof L wavesrequiredto account
for the observedradiation. For L waveswith TL < 10sx K the
theorypredictsthat nonthermal3fp radiationwith a brightness
temperature
of lessthan 10xøK shouldbe produced.However,
whenTL <•10søK, thisradiationshould
be unobservable
with
the SFR instrument.
Further observationalwork, particularly on the levels of
/, wavesand plasmacharacteristics
when the radiation is produced,is requiredto test the theory developedhere. Moreover,
the existenceof potential problemsfor the theory developedin
this paper indicatesthat alternative theoriesfor the third to
fifth harmonicradiation shouldbe investigated.
Appendix: Kinematicsof the ProcessL + T * -• T
l0ssK. Themaximum
observed
L waveelectric
fieldof 10mV/m
doesimplyTL ~ 10søK for vb ~ 10s m s-x. However,for
vb<• 10sins-z, L waveswith EL > 10 mV/m are required
for temperatures
TL ~ 10sx-10
ssK. In particular,waveswith
frequency
co(n)- ncop
(n neednotbe an integer).Squaring
cot
TL ~ 10sxK for v• = 5x 107ms-x (1x l0sms -x) have
~
mV/m (4o mV/m).
not
and using frequencyconservationand the dispersionrelations
in •able 3, one finds
Let the T * wave coalescingwith the L wave have angular
knownwhetherL waveelectricfieldsin excess
of 10mV/m are
kTs = (ns+ 2n)cop'/½
s+ 3(n+ 1)k,r,
SV',S/c
s
presentin the foreshockwhen the high harmonicradiation is
observed.
Accordingly
, thetheoryadvanced
in thispaperforthe
high harmonicradiation cannotbe ruled out with the available
wave data. However,if L waveswith electricfields in excessof
(A1)
assuming
kLs <<copy1/,
s. Comparing
(A1) with the equation
for kTs derivedfromwavevectorconservation
gives
10mV/mareneverobserved
in theforeshock,
theobserved
high
1
harmonicradiation cannot be readily accountedfor in terms of
the process/, + T' -• T.
Another possibledifficulty for the theory is in explaining
(2n+ 1
2v/n'
- 1)\ x )- X[13(n+
1)14'/c']
(A2)
why the brightness
telnperatures
of the observed
fp and 2fp
whereKL. K,, = KL- k,,,/kLk,.,is the cosineof the anglebe-
radiation
are smallcompared
with 10sxK, i.e., Tx <• 10xsK
and Ts <• 2 x 10xsK, whenthe highharmonic
radiationis
observed.
It is possible
[Cairns,1986b,C87]that reduction
of
tweenkL andk,•, andX = kLC/cop
(note:k,•= ¾,s _ 1cop/c).
growthlengthsfor the radiationby scatteringin densityinhomogeneities,reabsorptionof radiation by the backreactionsto
the emissionprocesses
outsidethe sourceregions,or volumeweightedaveragingof the sourceemissionmay restrict Tx and
Kinematic conditions on X and KL. K,arise
tions0 _<[KL. K,
restrictions
of 10sx-10
ss K. Alternatively,
it is possible
that the observed
valuesof Tx andTs implythat TL is not of the orderof 10sxK.
This matter is left to future investigators.
Conclusions
A theory for the 3fp, 4fp, and 5fp radiation observed
upstreamfrom the Earth's bow shock, involving the process
L + T' -• T actingon a sourceof 2fp radiationin the two
wings of the foreshock,is presented.The/, wavesare generated
primarily by the decay /, --, L' + S of streaminginstabilitygenerated/, waves. The theory requires/, waveswith temper-
aturesTL ~ 10sx-10ss
K in the sourceregionsto accountfor
the levels and nunt•r
of harmonics
of the observed radiation.
Such/, wavetemperaturesare theoreticallypossiblein the for•
shock, but are not consistentwith the mast intenseobserved
0 the
are
1-s
+1)•S/c
s +
-•/n
s-1+V•n
+3(n
2n3(n
+1)(2n
1)lf,
S/c
s)<X
T2 to their observed values rather than the saturation values
6.
from the restric-
< 1. For L waveswith KL-K,_>
<• 2n
+1
-
1 - 3(n + 1)V',S/c
s
(A3)
pamingsmoothlyinto thosefor KL' Kn < O,
+ 1)V,S/c
s<X
•1- 3(n
2n+l
<(¾nS-l+v/ns+2n-3(n+l)
-
1 - 3(n + 1)V•S/½
s
(A4)
Neglecting
theV,S/cs terms,theserestrictions
for n = 2 are
1.10 < x
< 4.56
866
Cairns:
TheoryforHighHarmonic
Plasma
Radiation
withKl;. K• = 0 forX = x/•, andforn = 3 are
Fitzenreiter,
R. J., A. J. Klimas,andJ. D. Scudder,
Detection
of
bump-on-tail
reduced
electron
velocitydistributions
at the
electron
foreshock
boundary,
Geo_oh_vs.
Res.I•tt., 1!, 496,
.05 < x < .70
1984.
with Ki;.K•
restrictions
= 0 forX = V•. In the limit n --, oo the
are
1 _<X,, <_2n
(A7)
Forrs_<4 theallowed
L wav•numbers
aresmall,
i.e.,kz,,,.,
to 9•/c.
Goldman,
M. V., Strong
turbulence
ofplasma
waves,
Rev.Mod.
Phys.,56, 709, 1984.
Gumerr, D. A., The Earth as a radio source:The nonthermal
continuum,
J. Oeophys.
Res.,80, 2751,1975.
Gurnett,D. A., R. R. Anderson,
F. L. Scarf,R. W. Fredericks,
and E. J. Smith, Initial resultsfrom the ISEE-1 and ISEE-2
Acknowledgments.
I thankS. R. Spangler
forhelpfulcommentsonearlierversions
of thispaper.Financialsupportfrom
NASA grantNAOW-831is acknowledged.
The Editor thanks H. P. Freund and D. L. Newman for their
assistance
in evaluatin•thispaper.
References
Anderson,
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