GEOPHYSICAL RESEARCHLETTERS, VOL. 25, NO. 3, PAGES265-268,FEBRUARY 1, 1998
Whistler mode wave coupling effects on electron
dynamics in the near Earth magnetosphere
S.K.Matsoukis, S.Chapman and G.Rowlands
Space and AstrophysicsGroup,University of Warwick, UK.
Abstract. Nonlinear electron whistler mode wave particle interactionsin the near earth magnetospheresuggest severalcandidatemechanisms
for phasespacediffusion of electronsinto the lossconeto provide a source
of auroral precipitating electrons. A unidirectional
whistler mode wave propagatingparallel to the backgroundmagneticfield and interactingwith an electron
can be shownto produceresonance(trapping of electrons) but the processis not stochastic.Chaoscan be
introducedby resonancebetweenelectronsthat mirror
many times in the earth's field, and a whistler wavefield. Here, we show that an electron interacting with
two oppositelydirected, parallel propagatingwhistler
mode waves exhibits
stochastic behaviour
in addition
to resonance.The backgroundfield is uniform, so that
stochasticity arises solely due to the presenceof the
wavefieldwithout requiringbouncemotion of electrons
between mirror points. Stochasticityfirst appearsfor
waveamplitudesnot inconsistentwith observations
at
disturbed times in the near earth magnetosphereand
thereforemay contributeto pitch anglediffusion.
suggestthat reflected bidirectional and for much of
their ray paths, parallel propagating whistler wavesare
present in the near earth magnetosphere.The electron
interaction with an omnidirectionalparallel propagating wave is not chaotic, and we shall see, for bidirectional wavesthis is still the casesufficientlycloseto resonance. Away from resonancewe find that in the presence of oppositely directed parallel propagating waves,
much of phase spacebecomesdominated by stochastic
orbits which act to enhancepitch angle diffusion.
2. Theoretical
approach
We consider the case of an electron in the presence
of a wave field gyrating around a uniform background
magneticfield Bo& with vectorpotentialA__
o - Boy•.
The wavefield B w(x, t) is the superposition
of two oppositely directed, parallel propagating whistler mode
B•+ = B•[cos(kx- wt)•9- sin(kx- wt)i]
(1)
B•, = B•[cos(-kx-wt +O)•9-sin(-kx-wt +O)•] (2)
1.
Introduction
Each one of the above right hand polarized waves is
in resonancewith electronsmoving antiparallel to the
A number of mechanisms have been proposed for
k_,and satisfyingthe resonance
condition
pitch angle scatteringof electronsby whistler mode wavenumbers
wavesin the near earth magnetosphere.As well as pro,• - •. v = nfl
(3)
cessesassociatedwith wavesthat are strongly oblique
(e.g.Inan,1992),a candidate
mechanism
iscyclotron
where
v istheparticle
velocity
wisthewave
frequency
resonance
processes
withnear
parallel
propagating
waves
and• = eBo/mis thegyrofrequency.
Forwhistler
(Kennel
andPetschek
1966;
Lyons
andWilliams,
1984;mode
waves
w < n•. Thevector
potential
fortheabove
Villalon and Burke, 1991). Stochasticityhas been in- wave field can be expressedas:
troducedby considering
the bouncemotionof electrons
interactingwith a singleunidirectionalwhistlermode
wave (Faith, 1997). Here we highlightthe possibility
that reflected,bidirectional,wavesinteract with single
-_Aw++A__•
:2•-•Wsin(kx-•)
.[sin
(•ot
+•),0- cos
(aJt
+•):•] (4)
electron motion to introduce chaos in the electron tra-
jectoriesin the whistlerwavefield.Ray tracingstudies
(Thorneand Horne, 1994; Thorneand Horne, 1996) where 0 is the initial difference in phase between the
Copyright1998 by theAmericanGeophysical
Union.
Papernumber97GL03716.
0094-8534/98/97GL-03716505.00
two waves and Bw each wave amplitude. The particle
velocity componentsparallel and perpendicularto the
backgroundfield are vx and vñ respectively.The phase
angles
between
vñ andthewavevectors
Bw+,B•are•p
265
266
MATSOUKIS ET AL: WHISTLER COUPLING WITH EFFECTS ON ELECTRON DYNAMICS
and •b- (2kx - 0), (seeFigure 1). A•b = •-
BwlBo=O.001
2kx is
the total difference in phase between the two opposite
propagatingwaves. The equationsof motion from the
Lorentzforcein a coordinatesystemof (v•, vñ, •b) are
dvz
= -vñfiw[sin(•) + sin(•b- (2kx - •))]
dvñ
dt
(5)
= -(v• - •)f•wsin(•b)
-(vz+ •)fiwsin(•b(2kx-
=
,.,t+ at+
-
0.1
(6)
cos()
- o))
-3
(7)
-I
0
I
2
3
Phase Angle
wheref•w = ebb. For
Figure 2. For a w = fl/2, Bw = 0.001Bo,the trajectories are regular for a particle of 100 keV.
m
One obtainsfrom (5)
dt 2
-2
+ kvñaw[sin(•b)
+ sin(•b- (2kx- 0))] - 0 (8) where f] is equal to the parallel velocity near resonance.
Equation(8) hasthe basicformof the pendulumequa- By substitutingthis value in (9) one finds that this
tion wherethe phaseanglesoscillatearoundan equilib- equationis satisfiedif (3) is obeyedand we shall see
riumposition,
withfrequency
w2 = k•wvñ. (8) canbe the resonancesappear in the numericalsolutionsof the
full equationsof motion (Figure 2 onwards).Equation
expressed
in a moresimplifiedway as
(9) hasresonances
at •b - 0, n7rand at (•b- 7) - 0,
--+
kvxaw[sin(•)+ sin(•b- 7)] - 0
(9) n7r. At thesepoints(9) reducesto the pendulumequadt 2
tion of motion of a particle in a singlewave (Dysthe,
where
if(t)= 2[(a-w)t- q]. (9)canbeexpressed
as 1970; Gendrin, 1974).The motion is integrablein this
onedegree
offreedom
system
withcoordinates
•, •b
_ d_•
tothese
resor d(•-•)
dt
-dt _ 2(•- W)
• •b--7- Close
d20
- 2kvñawsin(O)sin(•)
=0 (10)onant points couplingwill be weak and the motion is
dt 2
whereO = •2 + • - •b. Thisis of theformof a sim- near integrable.As we moveawayfrom resonance
the
ple pendulumwith a time modulatedlengthor gravita- systemnow has two degreesof freedom•b, 7(t). This
tional accelaration. The above pendulum equation can
givesthe possibilityfor chaosandenhanced
diffusionin
be derived from Hamiltonian
phasespace,and we will examinethis numericallyin
1 dO
the next section.
H - •(_•)2+2kvñaw
cos(O)sin(•)
(11)
whichsimplyhasthe formof the standardHamiltonian. 3.
We can assumethat around the resonancefixed points,
andfor sufficiently
smallwaveamplitudes
from (5)
Numerical
Results
The regular and stochasticbehaviourof the electron
(12) motion is investigatedvia effectivePoincaresurfaceof
x = • + fit
section
plots
ofv• = [•-•- (f•-w)]/k,versus
•b(see
for
example Tabor, 1989; Chen, 1992; Ram et al, 1993 and
references
therein). The plotsare madeby solvingthe
v
y
-k
k
w •
k
Figure 1. Coordinate system for a single electron
and two oppositelydirected whistler mode waveswith
differentwave phases. •b is the phaseangle (rotating
equationsof motion forward in time using a variable
order, variable stepsizeordinary differential equations
integrator (e.g. Chapmanand Watkins, 1993). The
surface of section is Vz = 0, where Vz is the z component of the perpendicularvelocity. Each trajectory
wasintegrated for 100000gyroperiods,in order to allow
particlesto fully exploreall regionsof accessiblephase
space. For stochastictrajectoriessignificantdiffusionin
phasespaceis foundto occuron tensof electrongyroperiods. All particletrajectorieshavethe sameenergyand
phaseangle•b at t = 0 whilst the initial pitch anglewas
varied from 0 to 7rgiving 180 setsof initial velocitiesso
frame)betweenthe electronperpendicular
velocityand
the magneticfield vectorof oneof the waves,and a the
pitch angle betweenthe electronparallelvelocityand
as to coverthe interval[-v•t v•t] wherev•t is the pesothe backgroundmagneticfield vector.
MATSOUKIS
ET AL: WHISTLER
COUPLING
WITH
EFFECTS ON ELECTRON
belowthe typical gyrofrequency
of 2 kHz at L-6 (Parrot et al, 1994), the plasmafrequencyat L-6 is around
8 kHz. Electron energiesinvestigatedare in the range
4-100 keV. We find that the particle dynamics is sensitive in the variation of the wave amplitude as well as
the particle energy.
The changein the dynamicsof electronsof a givenenergy as we increasethe wave amplitude is illustrated in
figures2-4, which showPoincareSOS for a 100keV elec-
267
Bw/Bo=0.008
nant v= (seeFigure2). Severalrunswith differentwave
amplitudes, particle energiesand wave frequencieshave
been performed to investigatethe level of stochasticity
in the particle dynamics. Unnormalized, the selected
parameters correspond to in the near earth magnetosphere;the range of wave frequenciescoversthe region
DYNAMICS
0.15
0.1
•o.:
-0.1
-0.15
-3
-2
-1
0
I
2
3
Phase Angle
tron. For Bw/Bo - 0.001 the behaviouris essentially
regular, that is, it is predictedby the simplependulum Figure 4. Enhanceddiffusionfor Bw = 0.008Bo for a
equationof motion. As we increaseBw/Bo to 0.005 100 keV electronand a wavefrequencyof f•/2.
(Figure 3) regionsof stochasticityappear, theseallow
waveamplitudescorrespond
to 100sof pT for L = 6 (ie
particlesto explore all phaseanglesand a range of pitch
taking 2kHz gyrofrequency)and are within the range
angles(that is, a rangeof v= - v cos(a)). Onsetof weak
of strongemissionsseenat disturbedtimes (Parrot et
stochasticity can be seen just around the separatrices
al, 1994) and individual wavepacketsnear the magneof the resonances.At Bw/Bo - 0.008 (Figure 4) we
topause(Naganoet al, 1996). The changein the deseestochasticityover significantregionsof phasespace.
tailed dynamics of the particles with frequencyis comThe particle motion is still regular closeto resonance,
plex and will be adressedin a future paper. Here we
and particles cannot diffuse between the regions surnote that the waveamplitude at which stochasticityfirst
rounding the resonancesassociatedwith the forward
appears in phase spaceis similar to within an order of
and backwardpropagatingwaves.If weincreaseBw/Bo
magnitude. We also anticipate that the more realistic
further then phase spacediffusionbecomesglobal and
situationwhereseveralwavemodesare considered
(and
stochastic particles can move acrossv• - 0.
that the wavenumbersof the oppositely directed waves
Changeof the relative phasebetweenthe wavesdoes
are not the same)will increasethe complexityof the denot affect this behaviour qualitatively, simply moving
tailed dynamics;this is beyond the scopeof this initial
the location of the resonancesas we would expect from
study and will be adressedin future.
equation(9)(seeFigure5).
Finally, Figure 6 showsthat the particle energy osThis behaviour is qualitatively the same for differcillates with time so that the net energy changeis zero.
ent electronenergiesin that there will be a threshold Hence, this Wave Particle Interaction mechanismwill
valuefor Bw/Bo whenstochasticity
appears;however lead to pure pitch angle diffusion rather than particle
the value of this thresholdis energydependent. These
energization.
BwlBo=0.005
0.15
-2
-1
0
I
2
3
Phase Angle
Figure 3. Diffusion around the separatricesfirst appears for Bw -• 0.005Bo, shownfor electronenergyof
100 keV and wavefrequencyof f•/2.
-3
-2
-1
0
I
2
3
Phase
Angle
Figure 5. Effect of varying the phasebetweenthe two
waves,for a 100 keV electron,wavefrequencyw - f•/2,
Bw - 0.008Bo and ½ - •2'
268
MATSOUKIS
ET AL: WHISTLER
COUPLING
WITH
EFFECTS
ON ELECTRON
DYNAMICS
Acknowledgments.
This work was conducted whilst
S.K.Matsoukis was partially supportedby PPARC.
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