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Electromagnetic tornadoes in space
Computer Physics Comniunications 49 (1988) 61—74 North-Holland, Amsterdam
61
ELECTROMAGNETIC TORNADOES IN SPACE Ion conics along auroral field lines generated by lower hybrid waves and electromagnetic turbulence in the ion cyclotron range of frequencies Tom CHANG, G.B. CREW and J.M. RETI’ERER
*
Center for Theoretical Geoplasma Physics, MIT Centerfor Space Research, Cambridge, MA 02139, USA
The exotic phenomenon of energetic ion conic formation by plasma waves in the magnetosphere is considered. Two particular transverse heating mechanisms are reviewed in detail: lower hybrid energization of ions in the boundary layer of the plasma sheet and electromagnetic ion cyclotron resonance heating in the central region of the plasma sheet. Mean particle calculations, plasma simulations and analytical treatments of the heating processes are described.
1. Introduction Imagine tornadoes the length of the earth or larger containing whirlwinds of millions of charged particles reaching speeds of tens of thousands of kilometers per hour (fig. 1). Such are the exotic phenomena recently detected by polar orbiting scientific satellites and sounding rockets. Space physicists have measured positive ions at altitudes from one to several earth radii at the auroral and polar cusp latitudes [1,2]. These ions gyrate around the earth’s magnetic field lines at extremely high speeds while flowing upward from the ionosphere into the magnetosphere with energies ranging form tens of eV to tens of keY; and populations of these ions have been christened “ion conics”. The name “conic” refers to the fact that these distributions are strongly peaked in pitch angle, so that ions are concentrated on a cone in velocity space, indicating some form of heating transverse to the ambient magnetic field. The discovery of conics was somewhat startling, largely because no mechanism for transversely accelerating ions to what are essentially magnetospheric energies had been anticipated. the generally accepted scenario for such transverse acceleration is some sort of wave—particle interaction. In this picture, the ions are energized perpendicular to the geomagnetic field lines by energy-carrying plasma waves. (Pop*
ular candidates include lower hybrid and ion—ion hybrid waves, and electrostatic and electromagnetic waves in the ion cyclotron range of frequencies [3].) One can then account for the conic form of the distribution by realizing that the magnetic field strength decreases with altitude. Thus the adiabatic motion of the ions drifting to higher altitudes transforms the heated distribution into one that is more field-aligned, i.e., a conic. Alternatively, it has also been suggested that the transverse heating of ions could be accomplished by the presence of oblique double layers [4]. In this review, we shall consider two special types of ion acceleration events which frequently occur within the plasma sheet of the magnetosphere (fig. 2). During a magnetic substorm, the plasma sheet thins down and numerous plasma processes are induced. For example, in the boundary layer region of the plasma sheet at altitudes around iRe, strings of weak double layers have been detected along the geomagnetic field
Also at Boston College, Chestnut Hill, MA 02167, USA
OO1O-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Fig. 1. Electromagnetic tornado in space.
62
T. Chang el aL
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Electromagnetic tornadoes in space ~FIELDLI:
Fig. 2. Plasma sheet regions.
lines [5] (fig. 3). These double layers characteristically have potential drops of the order of i eV over a spatial extent of several Debye lengths and are generally observed to propagate upward along the field lines. These double layers seem to set in intermittently in space and in time and on the average are separated along the magnetic field lines at approximately 1 km intervals. The origin of these double layers is not clear but they are probably produced by some sort of current driven plasma instability [6]. It can be argued that during a magnetic substorm, upward field-aligned currents are the strongest at altitudes around one earth radius in the boundary plasma sheet region [7]. Such a region can have an altitudinal extent of over 1000 km. Thus, on the average, electron populations can be accelerated by the combined effect of these potential drops to above 1 keY. Indeed, keY electron beams have been detected streaming toward the ionosphere in the boundary plasma sheet [8]. In addition to creating the visibly observable discrete aurorae in the E-region of the ionosphere, such electron beams can also excite a number of plasma instabilities in the suprauroral region. In particular, in situ observations have detected enhanced wave intensities near and above the lower hybrid resonance frequency. (See, e.g., Mozer et al. [7] and fig. 7 in Gurnett et a!. [9a].) In addition to the observed lower hybrid waves, simultaneous in situ measurements have also detected counterstreaming electron populations [10] and ion conic distributions such as those described above. Above the kilovolt potential drops, on the other hand, upflowing keY ion beams have been detected along with enhanced emissions of electrostatic ion cyclotron modes [ii]. In the calmer central region of the plasma sheet
Fig. 3. Boundary plasma sheet (BPS) field line: (a) Region where weak double layers have been detected. (b) Region where lower hybrid waves, ion conics, and counterstreaming electrons have been detected. (c) Region where ion beams and electrostatic ion cyclotron waves have been detected. Central plasma sheet (CPS) field line: (d) Region where low frequency electromagnetic waves in the ion cyclotron range of frequencies and oxygen-dominated, shallow conics are detected.
(figs. 2 and 3), ion conics have also been observed [2]. These populations generally have velocities which are more oblique to the geomagnetic field lines than those usually detected in the boundary plasma sheet, suggesting that the transverse heating process of these ions are probably more local in origin, and they commonly have a finite drift in the field-aligned direction. Simultaneous plasma wave instrument measurements have detected intense low frequency electrosmagnetic fluctuations during these ion conic events [9b]. We consider in this review ion conic formation processes in the boundary plasma sheet region where keY electron beams are detected as well as those in the central plasma sheet.
2. Ion acceleration by lower hybrid waves in the boundary layer plasma sheet As discussed in the introduction, both lower hybrid waves and ion conics have been detected in the auroral electron beam region of the boundary plasma sheet during magnetic substorms. Lower hybrid waves represent probably the most efficient mechanism for transferring energy from electrons to ions in collisionless plasmas. These waves are
T. Chang et aL
/ Electromagnetic
broad band both in wave frequency w and wave number k. Their frequency range generally lies between the electron gyrofrequency ~ eB/m e~ and the ion gyrofrequency Q1 eB/m 1c with ~e =
=
where B is the magnitude of the magnetic field, and me and m1 are the electron and ion mass, respectively. Since the ions under consideration are of ionospheric origin, we shall assume that they are singly charged. The wavelengths of these waves usually satisfy the condition Pc ~ k~ ~ p1, where Pe and p1 are the electron and ion gyroradli, respectively. Therefore, for these waves, the electrons are strongly magnetized and essentially move along the magnetic field lines while the ions can be treated as unmagnetized. The domain of the so-called “lower hybrid plateau” is displayed in a three-dimensional frequency-parallel wave number k11 -perpendicular wave number k ~ plot, for a typical hydrogen ion—electron plasma in the suprauroral region of the magnetosphere (fig. 4) [12]. Since the plasma beta is very small in the region under consideration, these lower hybrid modes are essentially electrostatic and lie on the resonance cone of the whistler modes. We note that the wave vectors of these modes are directed at large oblique angles from the magnetic field line direction (i.e., k ~ >> k) and therefore can resonate simultaneously with >> w>>
~,
tornadoes in space
63
the field-aligned electrons and the heavier ions which gyrate ponderously across the field lines, thus allowing energy to be transferred between the two species. Chang and Coppi [13] suggested that the keY auroral electron beams in the boundary plasma sheet could excite the observed lower hybrid waves in the YLF range and that these waves, in turn, could transfer the energy of the electron beam to the ions, by accelerating them in the transverse direction to the geomagnetic field lines. Since the observed lower hybrid waves were generally broad band, a quasilinear diffusion operator was used to estimate the average energy transfer from a steady state of waves populating a portion of the field line. It was found that typically 1 eV ions could be raised in energy to tens or hundreds of eY and beyond by lower hybrid waves of moderate intensity in the boundary plasma sheet where electron beams were detected. The ions were energized primarily transverse to the field lines. Because of the “mirror” geometry of the earth magnetic field, however, some of the transverse energy gained by the ions were converted into longitudinal energy as they evolve upward along the field lines. The calculated pitch angles ranged from 90° to 140°. As the ions moved upward they eventually left the primary heating region. Thus, the heating process was found to be self-limiting and generally lasted for a period of 30—40 s to four or five minutes. An illustrative example is given in fig. 5. When the ions propagate across the region of the kilovolt potential drops, they tend to become field-aligned. The combination of the background ions with such keY ion distributions can lead to
20 A 00
the excitation of electrostatic ion cyclotron modes as those observed experimentally by Kintner 0’~..
X
-2
C -
B
I
2
2.1. Evolution of the ion distribution function
Y Fig. 4. Dispersion surface where the lower hybrid waves are located. The plasma is assumed to be consisted of electrons 3 and and hydrogenequal temperatures ions to (with 1eV)densities in a homogeneous equal to 3 magnetic x i0~cmfield of 0.21 G. X= log(k 11p.), Y= log(k1p.), and Z= w/Q.. The region of the whistler mode is denoted by A, the lower hybnd plateau” by B, and the ion Bernstein mode by C. ~ = ~LH/Z. (After H. Koskinen, ref. [12c].)
We assume that the evolution of the ion districonsideration bution function is influenced f(s, v, t)byinthe theresonant region under interaction with the lower hybrid modes the geomagnetic field B, and the dc electnc field E. Since the .
.
.
.
lower hybrid modes observed in this region are generally broad band, coherent wave-stochastic
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Electromagnetic tornadoes in space
PITCH ANGLE, 9 5
00°
80°
I
-
20°
I
400
where SE(k, w) is the electric field spectral density of the turbulence. A self-consistent calculation of SE requires the determination of the saturation mechanism of the electron-beam induced lower hybrid instability in the suprauroral region, a subject we postpone for consideration in section 2.2. We sidestep this problem for now by assuming the amplitude E,~,, of the lower hybrid waves as given. Considering only perpendicular diffusion 2 which e2~Ew~ is dominant here and in order of magnitude, we have, in the range of resonant velocities.
ION
ANGLE
-
___________________________________________ I I I I I 0 0 2 4 6
(3)
D~L_~m~LHR~
ION ENERGY, 00eV Fig. 5. Illustrative example of the energy and pitch angle evolution of a hydrogen ion due to lower hybrid heating in the suprauroral region of the boundary plasma sheet based on a mean particle calculation. Initial energy of the ion was taken as 1 eV. The value I E,,~I 2/c,, LI-I was assumed to vary linearly from 1.0 (mV/m)2/Hz at the initial altitude of 500 km to 0.55(mV/m)2/Hz at the final altitude of 5000 km, where E~ and ~~LI-I are the average amplitude and frequency of the lower hybrid waves, respectively,
acceleration theories are inappropriate here. As an estimate, we follow Chang and Coppi [13] and describe the transverse heating process by the quasilinear diffusion operator. Since the distances traversed by the ions are much larger than the ion gyroradius, it is appropriate to use the guiding center approximation here. Neglecting the transverse drifts, the evolution equation may be expressed as follows:
af
~f +
v as
+
e ‘ m E
—
+
v c
—
XB
)
If ~v
(D ~ 8vj
(i)
av ~
where s is the arc length along the field line, v is the component of the velocity v along s and the quasilinear diffusion coefficient D is given by: D
—
e (m)
x
J -i-—f dco
3k kk d
3
—~
SE (k, w) tr6 ( w
—
k v), (2)
where ~ LHR = ~ (1 + e/~2~ )1/2 is the lower hybrid resonance frequency, and WI,e and ~ are the electron and ion plasma frequencies, respectively. The resonant range extends from a few times the ion thermal velocity up to a velocity V±max Ub(kII/kjjmaxLH — ub(me/mi)”, where Ub is the velocity of the electron beam. Thus ions
should stay in resonance until they reach energies near the electron beam energy. Above the upper limit v ~max’ the perpendicular diffusion coefficient D 1 should behave asymptotically as which follows from dimensional considerations. Eq. (1) with the approximation of the velocity diffusion coefficient given by (3) was solved by Retterer et al. [14] using a particle simulation procedure, in which the stochastic effects of the diffusive operator in eq. (1) were described using a Monte Carlo technique. The simulation was carned out over the altitudinal range from 1000 km to 5000 km for hydrogen ions. The initial density distribution was assumed to decrease with the second power of the altitude an approximation to Maeda’s [15] model of ionospheric density. The initial ion distribution was taken as an isothermal M~welliandistribution with an initial temperature of 1 eY. As the simulation progressed, a —
steady state was established by replacing every particle which leaves the the simulation by adistribuparticle picked at random from primordial lion. Fig. 6 illustrates the ion conics formed with scatter point plots of particle kinetic energies parallel and perpendicular to the magnetic field
T Chang et al. TIME
soc
20;oo
65
50.0 SEC
30000km
2000.0 km 200
/ Electromagnetic tornadoes in space
.
O2~0400
:
.
:
~
6~O
800
1.0
.
:.
IO~
-
I2~O 400
600
E IPERP) (eV)
Fig. 6. Monte Carlo simulation of ion conics in the boundary plasma sheet. Lower hybrid acceleration of hydrogen ions occurs over the altitudinal range from 1000 and 2000 km with E~= 50 mV/m. The lower panel gives the conic at the top edge of the acceleration region (altitude = 2000 km), while the upper panel gives the conic folded by the geomagnetic field at an altitude of 3000 km.
for a moderate wave intensity. The result is very similar to that seen by the S3-3 satellite [16]. Even under the approximation of pure perpendicular diffusion, eq. (1) is very difficult to handle analytically because of the presence of the mirror force term. When the region of transverse heating is limited to a small altitudinal range, the problem can be solved using a two-stage approximation by assuming that the upward moving component of the ambient population is first subjected to the transverse heating by lower hybrid waves. Identification of the ratio of ion thermal speed to the mean wave speed as a small parameter leads to a uniformly valid asymptotic solution of the quasilinear diffusion equation for a reasonably general distribution of wave energy in the wave vector (k) space. The ion distribution function is then allowed to drift adiabatically up the geomagnetic field line. Analytical descriptions of this type were given by Crew and Chang [17a]. Fig. 7 depicts the process of evolution of the ion distribution function for a special case based on
v
1iv0
Fig. 7. Evolution of an hydrogen ion conic based on the analytical solution using the two-stage approximation. The diffusion layer is 10 km thick. Typical ionospheric conditions are assumed and E~= 33 mV/rn. Cases (a), (b) and (c) are at altitudes 10. 2500 and 5000 km above the base of the diffusion layer.
such an analysis. Results compared well quantitatively with the Monte Carlo calculations [17b]. 2.2. Self-consistent particle-in-cell plasma simulation In the above calculations, the self-consistent evolution of the wave-spectrum has been ignored. To address this problem, a one-dimensional electrostatic, particle-in-cell simulation was performed [18]. The YLF waves are excited at higher altitudes by Landau resonance with the electron beam. As the waves propagate downward into higher densities, their wave vectors turn perpendicular to the magnetic field as the wave frequency approaches the local lower hybrid resonance frequency [19a]. For simplicity, it was assumed in the simulation that the lower hybrid waves were generated locally with a frequency near the lower hybrid resonance frequency, in this way taking advantage of the vertical inhomogeneity of the plasma but neglecting any horizontal
66
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Electromagnetic tornadoes in space
inhomogeneity and the resulting convective effects. This was justified by linear calculations [20] which showed that, for beam densities of the order of one-thousands of the ambient density, the lower hybrid waves would grow to nonlinear strength before convecting out of the auroral arc. If we look for an obliquely propagating (k 1/ k~1~ ~ 1), electrostatic mode such that ~ ~o 4~ ~e and Pc << k11 ~ p~,a linear analysis of the plasma dispersion relation yields a frequency which, without thermal corrections, depends on the wave vector only through its direction [19]: ~LH
WLHR
(i
+
m
k~
__‘-
_~—).
(4)
.
tion oblique to the magnetic field in the x—z plane is simply the projection in that direction of its velocity along the magnetic field direction, if the small polarization drift is neglected. With this approximation, the acceleration of a guiding center in the x-direction is eE(x, t) cos2O/m, implying that the effective inertia of aangles guiding in this 2O. At the ofcenter propagation model is m/cos of lower hybrid waves, the effective inertia of an electron guiding center, me/cos29, is comparable to rn, permitting realistic mass ratios, me/rn,, to be used in the simulation. Fig. 8 summarizes the energetics of a typical simulation in which the ambient electrons was
‘LI
An approximate analytical expression for the lineargrowthrate,
ii....
6xlO~
[~~
/ ~ \1/2[ y~w~-~-)
~pb
(kub — kIvtb
~)
xIc~5
>-
xexp(
—
1 2
(w_kllub)2\
k~v~b
)
w 1 ~2 ca~w -~--—exp~— / / ~1~2\
-‘‘ ~O~e
\
]
(5)
shows that the lower hybrid waves can be linearly excited over a narrow range of wavenumbers: U/u,, k S w/(ub — nvLb), where w = WLH, Wpb is the beam plasma frequency, u,, is the beam velocity, Vte~ v 1~,and Vtb are the electron, ion, and beam thermal velocities, respectively, and ~ 1—2. As mentioned previously, because of the range of frequencies and wavenumbers of the modes that we are studying, the dynamics of the ions can be treated in a good approximation as unmagnetized, while the electrons can be treated in the guiding-center-drift approximation. a one-dimensional model, particle cross-driftsIn are out of the x—z plane which contains the directions of propagation(x) and the magnetic field; in this case an electron guiding center velocity in a direc—
(9 tE tjJ
u z
4x10
z 2
3x10 x -5 2xl0~
(9 U
Z Lii (I) lii
IO~ 0
I
200
400 600 800 TIME Fig. 8. The energetics of a particle-in-cell simulation. The drift and thermal energies of the electron beam are scaled from the values Eb = 1 keY and2a/me Tb = 125 and electron nb/no = 1, eV, the respectively, velocity of the beam projected onto the x-direction is approximately 31.6 = 0.025. m1 velocity cos times the With thermal of the ambient ions. The top panel plots the ion energy ‘density versus time, while the lower panel gives the electrostatic energy density versus time in units of ~iii/i. The energies are measured in arbitrary units.
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Electromagnetic tornadoes in space
treated as a fluid. It can be seen that the total ion energy density started to rise rapidly only after the electrostatic energy had neared saturation. Following saturation, the ion energy continued to grow approximately linearly with time, but at a somewhat slower rate. The simulation was carried out for m~/cos29= m1 with ub equal to 31.6 times the thermal velocity of the initially cold ions. This meant that the linearly excited waves had phase velocities far out on the tail of the initial ion distribution function and would have been very inefficient in resonantly heating any significant fraction of the ion population. Thus, some nonlinearly excited waves must have been generated during the wave saturation process in order to obtain the type of observed ion heating. To check this, we compared the numerically calculated imtial growth rates for the waves that were observed to grow during the simulation with the linear theory. While good agreement was found for the most unstable waves, we found that there were
~
~
[:I~m~
~\
67
unstable waves with wave vectors larger than those predicted by linear theory with appreciable growth rates. These waves would have much smaller phase velocities and could resonate with a significant tail population of the initial ion distribution. This was further verified by the evolution of the wave spectrum obtained from the simulation (fig. 9). It is noted from these snapshots that at the early times, the linearly unstable modes were most energetic. At later times, however, the wave energy redistributed itself throughout k space, probably through a complicated sequence of mode-coupling processes. The spectrum seemed to reach a quasi-stationary state upon saturation, albeit with a continually heating ion population. We now consider the evolution of the ion distribution function in detail. A series of snapshots of the ion velocity distribution at different times for the simulation, overlaid in fig. 10, illustrate vividly how such type of ion acceleration takes place. We see that tails of energetic ions formed, emerging from both sides of the initial velocity distribution
____
I0~0 I
0
83
I
60 WAVENUMBER
I
80
04 I
.0204.06
.11.1
v~,
60
Fig. 9. The wave spectrum for a particle-rn-cell simulation. The drift and thermal energies of the electron beam are scaled from the values Eb =With 1 keYm and 29/,n~ Tb = 125 nb/no = 0.025. = 1,eV, therespectively, velocity of and the electron beam projected onto 1 costhe x-direction is approximately 31.6 times the thermal velocity of the ambient ions. In each panel is plotted I E(k, t) 12 versus wave number k at a fixed time. In the upper row the times are 160 and 240; in the lower row, 320 and 400 (all in units of wj~j).
.
.
.
.
.
.
Fig. 10. The ion velocity distnbution of a particle-in-cell simulation. The drift and thermal energies of the electron beam are scaled from the values Eb = i key and 29/m~ Tb = 125 = 1,eV, therespecveloctively, 0.025. projected With m1 cos ity of and the nb/~no electron= beam onto the x-direction is approximately 31.6 times the thermal velocity of the ambient ions. Each curve is a plot of the ion velocity distribution versus velocity at a fixed time. The times run from zero to 400 in increments of 40 wj~.A velocity of 0.04 corresponds to an ion energy of 230 eV.
68
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Electromagnetic tornadoes in space
at about three times the ion thermal velocity. Some ions were then accelerated to velocities comparable to those of the electron beam. In addition to the tails, the core of the velocity distribution showed evidence of nonresonant heating. But the resonantly heated tail population accounted for most of the energy transferred to the ions. As a consequence, a significant portion of the ions in the tail region (with v1 ~ 3 v~1)were accelerated to the velocities where the linear waves were intense, (Positive and negative v~velocities, distinguished here, are generally not distinguished in space observations, v~being essentially a component of v1.) At this time, the waves had saturated and the heated ions continued to receive energy from the waves while the wave energy was being continuously replenished by the electron beam. The existence of such a quasi-stationary state along with the knowledge of the saturated shape of the wavenumber spectrum provided the basis for the earlier Monte Carlo and analytic calculations of ion evolution along the scribed in section 2.1. geomagnetic field line deAlthough the above discussion is based on the result of one-dimensional considerations, similar results are obtained from two-dimensional simulations [21]. (See also refs. [12c] and [18] for two-dimensional mode-coupling discussions.) However, nonlinear modes with wave vectors out of the plane generated during the cascading process would not grow to large amplitudes for current sheets which are thin in the y-dimension. Thus, for actual discrete auroral geometry in the boundary plasma sheet, the one dimensional results might actually better describe the physics of ion acceleration. Only simulation results using an ambient electron fluid have been reviewed above. Retterer et a!. [18] have also considered the case where the ambient electrons were treated as particles using the guiding center approximation. Similar results were obtained. In addition, it was found that high ~
2.3. A strong turbulence model In order to understand the simulation results from a more theoretical point of view, a strong turbulence model was constructed [18]. Because lower hybrid waves can be supported in a plasma treated as a multicomponent fluid, a fluid description was used in the description of the cascading process of waves in wavenumber space. In analogy to the Langmuir turbulence theory [22], the fluid quantities (both for the electron and ions) were resolved into components varying at the lower hybrid frequency and at smaller frequencies of the order of the acoustic frequencies. The resulting approximate nonlinear governing equation for the Fourier components of the amplitudes of the electric field fluctuations Ek at the lower hybrid frequency was a nonlinear Schrodinger equation: 1 BE WLH — —
2E —
2 Dk
~
E E
*
k’k”
k’
k”
+ ~
k
E
where C and D are coefficients definable in terms of the angle of propagation, and the electron and ion masses and acoustic speeds. To introduce the wave—particle interaction effects, an instantaneous Landau damping term YL( k, t ) Ek/U LH was added to the right-hand side of eq. (6) (where YL is the instantaneous Landau damping or growth rate). In addition to this equation governing the wave amplitudes, the wave-partide interaction effect on the particles was described by the quasilinear diffusion equation:
af.
a =
~
af.
-~—-
~
(7)
,
V
V
where, in one dimension, the diffusion coefficient is 2
D.(v)
=
(—~--~ (~~~ 1E 21T8(~
velocity tails of the ambient electrons were produced during the heating process. When projected along the magnetic field direction, these electrons mimic nicely the observed “counterstreaming electrons” which are sometimes detected in conjunction with the ion conics [10].
6 k+k”k”
\
rn~)J 2ir
/c
H(k)
—
kv)
L
(8) and j = e, i. Numerical solution of this set of equations have been compared with the simulation results dis-
T. Chang et aL 5
~
/
Electromagnetic tornadoes in space I0~
~
3.5x1d
~
3.0~0~ >0~
69
‘11111
02
H
w
I
~2.5xI1i
.
I-.
z
p
I
I
>-
20~O’ -Y
0
o
-i
Lii
>
z
I~
‘
I.
I
I i
.‘
101 I -.06 -.04 = 400. The dashed line
>(5
z w
0
I
I
~,
I
-.02 0 velocity .02 distribution .04 .06 gives the ion from
Vx
(/) Ui
Fig. 12. The ion velocity distribution in the theoretical model. The solid line is snapshot of the velocity distribution at w LIlt
IO~
the comparable time in the particle simulation. A velocity of 0.04 corresponds to an ion energy of 230 eV. 0*
~
0
~
00
200 TIME
300
400
Fig. 11. The energetics of the theoretical model. As in fig. 8, the top panel gives the ion energy as a function of time, while the lower panel gives the electrostatic energy. The dot-dash lines give the corresponding results when mode-coupling is not present, while the dashed line in the lower panel gives the result from the particle simulation, with its origin of time shifted to match the initial conditions of the mode-coupling calculation.
cussed earlier. As can be seen from figs. 11 and 12, general agreements are obtained. Thus it can be concluded from these comparisons that this theoretical model of strong turbulence with mode-coupling adequately describe the phenomena observed in particle simulation. While the simple quasilinear-diffusion model was adequate to describe the effect of the turbulence on the ions, a full nonlinear treatment was necessary to describe the evolution of the turbulence. 2.4. Additional remark The above discussion considered only one species of ions. If the plasma contains several
species of ions of approximately the same temperature, then it can be easily argued that only the lightest species can receive appreciable lower hybrid heating. The reason is that the heavier species has much fewer particles in the velocity range where the lower hybrid waves are intense. Nevertheless, both hydrogen and oxygen ion conics have been detected in the region under consideration. If the lower hybrid mechanism is to be the heating mechanism also for the oxygen ions, some sort of pre-heating process for the oxygen ions must have taken place before the lower hybrid heating scenario takes over. We will come back to this point later in section 4.
3. Ion acceleration by electromagnetic turbulence in the ion cyclotron range of frequencies Recent particle data collected by the plasma instruments HAPI and EICS aboard the polar orbiting satellite Dynamics Explorer 1 [2,23] re-
70
T. Chang ef aL 60
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Electromagnetic tornadoes in space DE—1
I
10
,,.,..,.I
I
I
DAY 318 NOV 14, 1981 UT234544
>• I...
~
PWI I
10
30
I
I
SO
68
-60 \~L(km/s)
1
10
100
1K
10K
lOOK
FREQUENCY (Hz)
Fig. 13. The bottom panel presents a contour diagram of the observed oxygen-dominated ion conic distribution function in the central plasma sheet, measured by the HAPI instrument on Dynamics Explorer 1 on 14 Nov. 1981 at the geocentric altitude of 2.OR E and invariant latitude of 600. The top panel presents the theoretical contours for the same event based on the Monte Carlo simulation calculation. The contours are uniformly spaced with an increment of 0.4 in the logarithm of phase space density. The density of these ions at3.the observation point is approximately 10 cm~
vealed a prevalent population of oxygendominated ion conics in the central plasma sheet region at a geocentric altitude of approximately 2 RE during an intense magnetic substorm. The observed peak energies of these conics varied from tens to hundreds of eV at pitch angles around or below 1300 (lower panel of fig. 13). The corresponding field-aligned currents measured by the magnetometer [24] and the plasma instrument HAPI exhibited irregular changes in sense of direction and their magnitudes were generally not very intense. There was no meaningful correspondence between these central plasma sheet conics and the sparsely observed, relatively low energy, field-aligned electron beams. This is more or less expected, since the central plasma sheet represents a much more quiet plasma state when compared with the boundary plasma sheet where discrete aurorae occur, The above observations seem to rule out some
Fig. 14. Typical electric field spectral density in the central plasma sheet measured by the PWI instrument on Dynamics Explorer I on 14 Nov. 1981 at the geocentric altitude of 2.OR E and invariant latitude of 600.
of the more popular heating mechamsms such as the lower hybrid interaction electroprocess discussed above wave—particle and the current-driven static ion cyclotron wave mechanism proposed by a number of previous authors [25,26]. On the other hand, the plasma wave instrument PWI on board the Dynamics Explorer I, though operating in a low resolution mode, gave clear indications of an intense low frequency noise (less than a hundred Hz) across the central plasma sheet region (fig. 14) [9b]. Based on mean-particle calculations. Chang et al. [27] demonstrated that the observed low frequency electric field turbulence in the ion cyclotron range of frequencies was sufficient to account for the transverse acceleration of ions through cyclotron resonance to the observed energy levels in the central plasma sheet region. They assumed that the ions were oxygen and that some of the observed turbulence was electromagnetic and left-hand polarized. These waves, however, are not very effective in accelerating the hydrogen ions because the wave energy falls off at the hydrogen gyrofrequency. This fact seems to explain why the observed central plasma sheet conics were oxygen-dominated [23].
T. Chang et aL
/ Electromagnetic tornadoes in space
3.1. Monte Carlo calculations Since the observed low frequency noise is broad band and the wave intensity is not very strong, it is reasonable to represent the perpendicular heating due to electromagnetic ion cyclotron resonance by the quasilinear diffusion operator. In the absence of an external dc electric field, the ion evolution equation (eq. (1)) along a geomagnetic field line may be written as:
a.!
Bf
V 1
+ =
V11
ã~
—
1
B
—
.~—
Bf
dB ( V1
a~
(v±D1 .~L’),
Bf —
\
V II
(9)
where f(s, V v1, t) is the ion distribution function, and (V II’ v1) are the parallel and perpendicular velocities, respectively. In terms of the electric field spectral density, the perpendicular velocity diffusion coefficient is found to be [28]: II’
D1
=
—~—~IEL(w = ~(s))I2,
(10)
where I EL ( ~o= ~( s)) 2 is the spectral density of left-hand polarized waves, and ~2(s) is the local ion cyclotron frequency. Eqs. (9) and (10) have been solved using the Monte Carlo method similar to that employed for lower hybrid heating [28]. Because the number of accelerated ions is small (oxygen is a minority constituent of the plasma), the ions were treated as “test particles” in externally imposed fields. To study the particular event of the lower panel of fig. 13, the ions were assumed to be thermally distributed with a temperature 0.2 eValong at ana 2RE, andoftracked initial altitude of s = l.plasma sheet geomagnetic portion of a central field line at an invariant latitude of 600, extending up to the geocentric altitude of the satellite, roughly at 2.ORE. Because of the form of the velocity-diffusion rate, however, most of the ion acceleration observed would have occurred near the altitude of the observation point, and the results of the calculation should be insensitive to the initial conditions chosen for the oxygen ions. Guided by the previous mean particle calcula-
71
tions, the spectral density of left-hand polarized waves was chosen to be one-eighth of the actual measured wave intensity. As ions passed the ob2.ORE, statistics on their velociservation point at ties were accumulated to calculate the ion-velocity distribution at this point. The results are presented in the top panel of fig. 13. Without a simultaneous measurement of the ion density in the source region, the absolute normalization was per force arbitrary. As can be seen from fig. 13, the quantitative agreement is remarkable between the calculated and observed contours. Despite the absence of a parallel electric field in this calculation, noticeable parallel acceleration was achieved. All transverse acceleration schemes must result in some parallel acceleration, because the effect of the mirror force is to convert the perpendicular energy gained into parallel energy. Although the intense, low frequency electric and magnetic field noise has been observed over the aurora! zones for many years, the nature and origin of the turbulence are still not thoroughly understood and are the subjects of ongoing research. The polarization data for the low frequency turbulence in the Dynamics Iauroral surveyzone appears to be consistent with a Explorer nonlocal source mechanism for the turbulence. But the propagation of left-hand polarized waves at frequencies near the ion gyrofrequency, with the resulting possibility of mode-conversion phenomena, remains largely unexplored. The uncertainty in the origin of the low frequency turbulence does not alter the conclusion, based on the success of the Monte Carlo calculation, that this turbulence can explain the observed central plasma sheet oxygen corncs. 3.2. Similarity solution for the central plasma conics Although the Monte Carlo model used in the previous section was based on the actual spectrum, the fit of a power law of the form I E( U) 2 a gives a convenient means of characterizing the spectrum. For the case of a power law wave spectrum and a dipole magnetic field, the steady state velocity distribution f(s, v, V 1) satisfying eq. (9) exhibits similarity scaling [29]. It can be —
—
72
T Chang et aL
/ Electromagnetic tornadoes in space
shown that there exists a self-similar solution such that the scaled ion distribution function F(x, y) v II’ V1) depends on the velocities only
3
through the scaled variables x C~v1s~ and y cx: v ~ where a = a + 1/3. Such a self-similar solution generally cannot satisfy arbitrarily imposed initial conditions. Monte Carlo calculations for the evolution of the ion distribution function indicated that the calculated results quickly reached an asymptotic state and became independent of initial conditions. Thus the electromagnetic turbulence behaves like a tornado; it picks up charged particles at lower altitudes and whirls them upward. Quickly the heated particle distribution acquires the properties endowed by the local electromagnetic turbulence and forgets entirely about its initial conditions. This peculiar asymptotic property conveniently to the type of self-similar solution describedleads above. Upon such a reformulation, the quasilinear ion evolution equation (eq. (10)) may be re-expressed as a convective-diffusion equation: B2N Bx2 with N(x, y) = xF(x, y), and • (Nu)
u~= =
=
1 —
— (~
4x2
—
+
a)xy,
a~~’2.
(11)
(12a) (12b)
The two-dimensional flow field given by eq. (12) describes the deterministic parts of the evolution (dotted curves of fig. 15). These are the effects of the magnetic mirror force and mean particle heating due to ion cyclotron resonance. The diffusion term represents transverse particle fluctuations about the mean motion due to the imposed turbulence. Thus, Eq. (11) may be replaced by a simple set of Langevin equations: = u.,, + ~, (13a) =
u~,,
(13!,)
where ~ is a random noise with Gaussian statistics, and (~) denotes derivative with respect to some pseudo-time, T. Numerical implementation of the time-asymp-
I
I
~
.
.
.
.
. ‘,
.
.
. .
.
2
:
—, .
.
.‘
.
.
.
, .
.
.
.
.
—
.
y,.:.,,.:.:..H 1
0
1
2
3
X
2/2)— Fig. 15. represent the streamlines of the convecay2. TheDotted closedcurves curves are calculated logarithmically spaced tive flow field: u~ = (1/x)—(3/2+ a)xy and u~= (3x contours for the function N(x, y) = xF(x, y) using the Langevin equations.
totic solution of eq. (13) is trivial and efficiently leads to the desired results [29]. Solid contours of fig. 15 presents the logarithmically spaced contour plot of the function N(x, y) = xF(x, y) for the same event as that was presented based on the Monte Carlo calculation (fig. 13). The result may be understood by recognizing that the flow field u contains a sink at some finite point (x 0, y0) toward which the ions are drawn, balanced by the action of random diffusion in the x-( V II-) direction which tends to disperse the particles from the sink. Fig. 16 translates the result to contours of the scaled distribution function F(x, y). It is to be noted that this asymptotic solution compares favorably with the Monte Carlo solution (fig. 13) and it can be obtain more efficiently. We conclude this section by pointing out that the stochastic property of the Langevin eq. (14) may be expressed exactly in terms of a functional path integral representing the probability density functional of the random functions x ( ‘r) and y( ‘r). Using this path integral, the moments of the distribution function may be obtained via Feynman diagram expansions. These results were presented at this International School [29,30].
/
T. Chang et aL 3 I
2
I
I
I
I
I
I
I
I
I
Electromagnetic tornadoes in space
ions can still be heated by these low frequency waves via electromagnetic cyclotron resonance to the order of 10 eV provided that there is a significant left-hand polarized component present. At
I
these energies, the preheated oxygen ions can easily be boosted to higher energies by the parametrically excited lower hybrid waves.
—
-
73
5. Swnmary ergetic ion populations in the plasma sheet of the The phenomenon of upflowing, gyrating, en-
0 I
0
I
I
I
1
I
I
I
I
2
I
I
I
3
X Fig. 16. Contour plot of the scaled ion distribution function F(x, y). The contours are logarithmically spaced at half-dccades, and the normalization is arbitrary. The parameter a = 2.0 which corresponds to a =1.7. The lowest values, ragged contours at the edges of the conic have been retained as an indication of the statistical error inherent in this technique.
4. Oxygen conies in the boundary plasma sheet In section 2, it was demonstrated that lower hybrid waves produced by precipitating auroral electron beams in the boundary plasma sheet could be responsible for the transverse heating of hydrogen ions. Because of the low thermal velocities of the cold oxygen ions, however, resonant heating of the oxygen ions by lower hybrid waves (which generally have phase velocities larger than the thermal velocities of the cold hydrogen ions) will be difficult unless there is some pre-heating mechanism which can launch the cold oxygen ions to perpendicular velocities in the range of the phase velocities of the lower hybrid waves. In situ measurements of typical electric field wave spectra in the boundary plasma sheet at low altitudes have detected both intense lower hybrid waves and broad band low frequency noise below 100 Hz [9a]. The intensity of the low frequency fluctuations seems to be significantly smaller than those detected in the central plasma sheet such as the one considered in the previous section. However, even for such low wave intensity, oxygen
magnetosphere is considered. It is shown that in the primary auroral electron-beam region of the boundary plasma sheet, lower hybrid waves could be an efficient mechanism for the transverse heating of ions of ionospheric origin (H and 0~) although for oxygen ions to be energized by such a ±
wave—particle interaction mechanism, some sort of pre-heating process would be required. Lower hybrid waves can also be the agent responsible for the generation and energization of the so-called “counterstreaming electrons” along auroral field lines. In the central plasma sheet, where intense low frequency electromagnetic turbulence are detected, it is suggested that oxygen ions can be conveniently energized to the observed energy levels via electromagnetic ion cyclotron resonance provided the wave spectra contained a small fraction of left-hand polarized component. Monte Carlo calculations provided the first successful comparison of an observed conic with a realistic theoretical model. Asymptotic properties of this sort of heating allowed the search for a self-similar solution yielding a more efficient method for calculating the detailed shape of the ion distribution function as well as some additional insight into the physics of the heating process.
Acknowledgements The authors are indebted to M. André, B. Basu, J. Burch, B. Coppi, D. Gurnett, J. Jasperse, N. Hershkowitz, M. Hudson, P. Kintner, D. Kiumpar, H. Koskinen, R. Lysak, M. Mellott, W. Peter-
74
T. Chang et aL
/ Electromagnetic tornadoes in space
son, I. Roth, E. Shelley, M. Temerin, D. Vvedensky and J.D. Winningham for collaborative research and/or useful interactions. This research is partially supported by the US Air Force Office of Scientific Research under Contract No. F49620 86-C-0128, and the Air Force Geophysics Laboratory under Contract Nos. F19628-86-K-0005 and FY7121-84-0-0006. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.
References [1] R.D. Sharp, R.G. Johnson and E.G. Shelley, J. Geophys. Res. 82 (1977) 3324. D.M. Kiumpar, ibid. 84 (1979) 4229. D.J. Gorney, A. Clarke, D. Croley, J. Fennell, J. Luhman and P. Mizera, ibid. 86 (1981) 83. A.W. Yau, BA. Whalen, W.K. Peterson and E.G. Shelley, ibid. 89 (1984) 5507. [2] J.D. Winmngham and J. Burch, in: Physics of Space Plasmas (1982—4), eds. J. Belcher, H. Bridge, T. Chang, B. Coppi and JR. Jasperse, SPI Conf. Proc. and Reprint Series, vol. 5 (Scientific Publishers, Cambndge, MA, 1984) p. 137. [3] R.L. Lysak, in: Ion Acceleration in the Magnetosphere and Ionosphere, eds. T. Chang, M.K. Hudson, J.R. Jasperse, R.G. Johnson, P.M. Kintner, M. Schulz and GB. Crew, Geophysical Monograph, vol. 38 (Amencan Geophysical Union, Washington, DC, 1986) p. 261 and references contained therein. [4] J.E. Borovsky, J. Geophys. Res. 89 (1984) 2251. [5] M. Temerin, K. Cerny, W. Lotko and F.S. Mozer, Phys. Rev. Lett. 48 (1982) 1175. H. Koskinen, R. Bostrom and B. Holback, to be published in Ionosphere—Magnetosphere—Solar Wind Coupling Processes, eds. T. Chang, G.B. Crew and J.R. Jasperse, SPI Conf. Proc. and Reprint Series, vol. 7 (Scientific Publishers, Cambridge, MA, 1988). [6] C. Barnes, M.K. Hudson and W. Lotko, Phys. Fluids 28 (1985) 1055. R.H. Berman, D.J. Tetreault, and T.H. Dupree, in: Ion Acceleration in the Magnetosphere and Ionosphere, eds. T. Chang, M.K. Hudson, JR. Jasperse, R.G. Johnson, P.M. Kintner, M. Schulz and G.B. Crew, Geophysical Monograph, vol. 38 (American Geophysical Union, Washington, DC, 1986) p. 328. [7] F.S. Mozer, C.A. Cattell, M.K. Hudson, R.L. Lysak, M. Temerin and R.B. Torbert, Space Sci. Rev. 27 (1980) 155. [8] L.A. Frank and K.L. Ackerson, J. Geophys. Res. 76 (1971) 3612. [9] (a) D.A. Gurnett, R.L. Huff, J.D. Menietti, J.L. Burch, J.D. Winmngham and S.D. Shawhan, J. Geophys. Res. 89 (1984) 8971. (b) M. Mellott and D. Gurnett, private communication.
[10] C.S. Lin, J.L. Burch, J.D. Winningham and R.A. Hoffman, Geophys. Res. Lett. 9 (1982) 925. [11] P.M. Kintner, M.C. Kelley, R.D. Sharp, A.C. Ghielmetti, M. Temerin, C.A. Cattell, P. Mizera and J.F. Fennell, J. Geophys. Res. 84 (1979) 7201. [12] (a) H. Koskinen, doctoral dissertation, Uppsala University, Sweden (1985). (b) M. André, J. Plasma Physics 33 (1985) 1. [13] ~ ~ Res. Lett.8 (1981) 1253. [14] J.M. Retterer, T. Chang and J.R. Jasperse, Geophys. Res. Lett. 10 (1983) 583. [15] K. Maeda, Planet. Spa. Sci. 23 (1975) 843. [16] P. Mizera, J. Fennell, D. Croley, A. Vampola, F. Mozer, R. Torbert, M. Temerin, R. Lysak, M. Hudson, C. Cattell, R. Johnson, R. Sharp, A. Ghielmetti, P. Kintner and M. Kelley, J. Geophys. Res. 86 (1981) 2329. [17] (a) G.B. Crew and T. Chang, Phys. Fluids 28 (1985) 2382. (b) G.B. Crew and T. Chang, in: Physics of Space Plasmas (1985—87), eds. T. Chang, J. Belcher, J.R. Jasperse and G.B. Crew, SPI Conf. Proc. and Reprint Series, vol. 6 (Scientific Publishers, Cambridge, MA, 1987) p. 55. [18] J.M. Retterer, T. Chang and JR. Jasperse, J. Geophys. Res. 91(1986)1609. [19] (a) J.E. Maggs, J. Geophys. Res. 81 (1976) 1707. (b) B. Coppi, F. Pegararo, R. Pozzoli and G. Rewoldt, Nuclear Fusion 16 (1976) 309. (c) K. Papadopoulos and P.J. Palmadesso, Phys. Fluids 19 (1976) 605. [20] J.E. Maggs and W. Lotko, J. Geophys. Res. 86 (1981) 3439. [21] J.M. Retterer, T. Chang and JR. Jasperse, presented at the Third International School for Space Simulation, Beaulieu, France, June 1987. [22] V.E. Zakharov, Soy. Phys. JETP, Engl. Transl. 35 (1972) 908. M.V. Goldman, Rev. Mod. Phys. 56 (i984) 709 and references therein. [23] W. Peterson, D. Kiumpar and E. Shelley, private cornmunications. [24] M. Sugiura, private communication. [25] R.L. Lysak, M.K. Hudson and M. Temerin, J. Geophys. Res. 85 (1980) 678. [26] M. Ashour-Abdalla and H. Okuda, J. Geophys. Res. 89 (1985) 2235. [27] T. Chang, G.B. Crew, N. Hershkowitz, J.R. Jasperse, J.M. Retterer and J.D. Winningham, Geophys. Res. Lett. 13 (1986) 636. [28] J.M. Retterer, T. Chang, G.B. Crew, J.R. Jasperse and J.D. Winningham, Phys. Rev. Lett. 59 (1987) 148. [29] G.B. Crew and T. Chang, to be published in: Huntsville Workshop on Magnetosphere-Ionosphere Plasma Models, ed. T. Moore (American Geophysical Union, Washington, DC, 1987). [30] G.B. Crew, T. Chang, J.M. Retterer and J.R. Jasperse, presented at the Third International School for Space Simulation, Beaulieu, France, June 1987.
61
ELECTROMAGNETIC TORNADOES IN SPACE Ion conics along auroral field lines generated by lower hybrid waves and electromagnetic turbulence in the ion cyclotron range of frequencies Tom CHANG, G.B. CREW and J.M. RETI’ERER
*
Center for Theoretical Geoplasma Physics, MIT Centerfor Space Research, Cambridge, MA 02139, USA
The exotic phenomenon of energetic ion conic formation by plasma waves in the magnetosphere is considered. Two particular transverse heating mechanisms are reviewed in detail: lower hybrid energization of ions in the boundary layer of the plasma sheet and electromagnetic ion cyclotron resonance heating in the central region of the plasma sheet. Mean particle calculations, plasma simulations and analytical treatments of the heating processes are described.
1. Introduction Imagine tornadoes the length of the earth or larger containing whirlwinds of millions of charged particles reaching speeds of tens of thousands of kilometers per hour (fig. 1). Such are the exotic phenomena recently detected by polar orbiting scientific satellites and sounding rockets. Space physicists have measured positive ions at altitudes from one to several earth radii at the auroral and polar cusp latitudes [1,2]. These ions gyrate around the earth’s magnetic field lines at extremely high speeds while flowing upward from the ionosphere into the magnetosphere with energies ranging form tens of eV to tens of keY; and populations of these ions have been christened “ion conics”. The name “conic” refers to the fact that these distributions are strongly peaked in pitch angle, so that ions are concentrated on a cone in velocity space, indicating some form of heating transverse to the ambient magnetic field. The discovery of conics was somewhat startling, largely because no mechanism for transversely accelerating ions to what are essentially magnetospheric energies had been anticipated. the generally accepted scenario for such transverse acceleration is some sort of wave—particle interaction. In this picture, the ions are energized perpendicular to the geomagnetic field lines by energy-carrying plasma waves. (Pop*
ular candidates include lower hybrid and ion—ion hybrid waves, and electrostatic and electromagnetic waves in the ion cyclotron range of frequencies [3].) One can then account for the conic form of the distribution by realizing that the magnetic field strength decreases with altitude. Thus the adiabatic motion of the ions drifting to higher altitudes transforms the heated distribution into one that is more field-aligned, i.e., a conic. Alternatively, it has also been suggested that the transverse heating of ions could be accomplished by the presence of oblique double layers [4]. In this review, we shall consider two special types of ion acceleration events which frequently occur within the plasma sheet of the magnetosphere (fig. 2). During a magnetic substorm, the plasma sheet thins down and numerous plasma processes are induced. For example, in the boundary layer region of the plasma sheet at altitudes around iRe, strings of weak double layers have been detected along the geomagnetic field
Also at Boston College, Chestnut Hill, MA 02167, USA
OO1O-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Fig. 1. Electromagnetic tornado in space.
62
T. Chang el aL
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Electromagnetic tornadoes in space ~FIELDLI:
Fig. 2. Plasma sheet regions.
lines [5] (fig. 3). These double layers characteristically have potential drops of the order of i eV over a spatial extent of several Debye lengths and are generally observed to propagate upward along the field lines. These double layers seem to set in intermittently in space and in time and on the average are separated along the magnetic field lines at approximately 1 km intervals. The origin of these double layers is not clear but they are probably produced by some sort of current driven plasma instability [6]. It can be argued that during a magnetic substorm, upward field-aligned currents are the strongest at altitudes around one earth radius in the boundary plasma sheet region [7]. Such a region can have an altitudinal extent of over 1000 km. Thus, on the average, electron populations can be accelerated by the combined effect of these potential drops to above 1 keY. Indeed, keY electron beams have been detected streaming toward the ionosphere in the boundary plasma sheet [8]. In addition to creating the visibly observable discrete aurorae in the E-region of the ionosphere, such electron beams can also excite a number of plasma instabilities in the suprauroral region. In particular, in situ observations have detected enhanced wave intensities near and above the lower hybrid resonance frequency. (See, e.g., Mozer et al. [7] and fig. 7 in Gurnett et a!. [9a].) In addition to the observed lower hybrid waves, simultaneous in situ measurements have also detected counterstreaming electron populations [10] and ion conic distributions such as those described above. Above the kilovolt potential drops, on the other hand, upflowing keY ion beams have been detected along with enhanced emissions of electrostatic ion cyclotron modes [ii]. In the calmer central region of the plasma sheet
Fig. 3. Boundary plasma sheet (BPS) field line: (a) Region where weak double layers have been detected. (b) Region where lower hybrid waves, ion conics, and counterstreaming electrons have been detected. (c) Region where ion beams and electrostatic ion cyclotron waves have been detected. Central plasma sheet (CPS) field line: (d) Region where low frequency electromagnetic waves in the ion cyclotron range of frequencies and oxygen-dominated, shallow conics are detected.
(figs. 2 and 3), ion conics have also been observed [2]. These populations generally have velocities which are more oblique to the geomagnetic field lines than those usually detected in the boundary plasma sheet, suggesting that the transverse heating process of these ions are probably more local in origin, and they commonly have a finite drift in the field-aligned direction. Simultaneous plasma wave instrument measurements have detected intense low frequency electrosmagnetic fluctuations during these ion conic events [9b]. We consider in this review ion conic formation processes in the boundary plasma sheet region where keY electron beams are detected as well as those in the central plasma sheet.
2. Ion acceleration by lower hybrid waves in the boundary layer plasma sheet As discussed in the introduction, both lower hybrid waves and ion conics have been detected in the auroral electron beam region of the boundary plasma sheet during magnetic substorms. Lower hybrid waves represent probably the most efficient mechanism for transferring energy from electrons to ions in collisionless plasmas. These waves are
T. Chang et aL
/ Electromagnetic
broad band both in wave frequency w and wave number k. Their frequency range generally lies between the electron gyrofrequency ~ eB/m e~ and the ion gyrofrequency Q1 eB/m 1c with ~e =
=
where B is the magnitude of the magnetic field, and me and m1 are the electron and ion mass, respectively. Since the ions under consideration are of ionospheric origin, we shall assume that they are singly charged. The wavelengths of these waves usually satisfy the condition Pc ~ k~ ~ p1, where Pe and p1 are the electron and ion gyroradli, respectively. Therefore, for these waves, the electrons are strongly magnetized and essentially move along the magnetic field lines while the ions can be treated as unmagnetized. The domain of the so-called “lower hybrid plateau” is displayed in a three-dimensional frequency-parallel wave number k11 -perpendicular wave number k ~ plot, for a typical hydrogen ion—electron plasma in the suprauroral region of the magnetosphere (fig. 4) [12]. Since the plasma beta is very small in the region under consideration, these lower hybrid modes are essentially electrostatic and lie on the resonance cone of the whistler modes. We note that the wave vectors of these modes are directed at large oblique angles from the magnetic field line direction (i.e., k ~ >> k) and therefore can resonate simultaneously with >> w>>
~,
tornadoes in space
63
the field-aligned electrons and the heavier ions which gyrate ponderously across the field lines, thus allowing energy to be transferred between the two species. Chang and Coppi [13] suggested that the keY auroral electron beams in the boundary plasma sheet could excite the observed lower hybrid waves in the YLF range and that these waves, in turn, could transfer the energy of the electron beam to the ions, by accelerating them in the transverse direction to the geomagnetic field lines. Since the observed lower hybrid waves were generally broad band, a quasilinear diffusion operator was used to estimate the average energy transfer from a steady state of waves populating a portion of the field line. It was found that typically 1 eV ions could be raised in energy to tens or hundreds of eY and beyond by lower hybrid waves of moderate intensity in the boundary plasma sheet where electron beams were detected. The ions were energized primarily transverse to the field lines. Because of the “mirror” geometry of the earth magnetic field, however, some of the transverse energy gained by the ions were converted into longitudinal energy as they evolve upward along the field lines. The calculated pitch angles ranged from 90° to 140°. As the ions moved upward they eventually left the primary heating region. Thus, the heating process was found to be self-limiting and generally lasted for a period of 30—40 s to four or five minutes. An illustrative example is given in fig. 5. When the ions propagate across the region of the kilovolt potential drops, they tend to become field-aligned. The combination of the background ions with such keY ion distributions can lead to
20 A 00
the excitation of electrostatic ion cyclotron modes as those observed experimentally by Kintner 0’~..
X
-2
C -
B
I
2
2.1. Evolution of the ion distribution function
Y Fig. 4. Dispersion surface where the lower hybrid waves are located. The plasma is assumed to be consisted of electrons 3 and and hydrogenequal temperatures ions to (with 1eV)densities in a homogeneous equal to 3 magnetic x i0~cmfield of 0.21 G. X= log(k 11p.), Y= log(k1p.), and Z= w/Q.. The region of the whistler mode is denoted by A, the lower hybnd plateau” by B, and the ion Bernstein mode by C. ~ = ~LH/Z. (After H. Koskinen, ref. [12c].)
We assume that the evolution of the ion districonsideration bution function is influenced f(s, v, t)byinthe theresonant region under interaction with the lower hybrid modes the geomagnetic field B, and the dc electnc field E. Since the .
.
.
.
lower hybrid modes observed in this region are generally broad band, coherent wave-stochastic
64
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Electromagnetic tornadoes in space
PITCH ANGLE, 9 5
00°
80°
I
-
20°
I
400
where SE(k, w) is the electric field spectral density of the turbulence. A self-consistent calculation of SE requires the determination of the saturation mechanism of the electron-beam induced lower hybrid instability in the suprauroral region, a subject we postpone for consideration in section 2.2. We sidestep this problem for now by assuming the amplitude E,~,, of the lower hybrid waves as given. Considering only perpendicular diffusion 2 which e2~Ew~ is dominant here and in order of magnitude, we have, in the range of resonant velocities.
ION
ANGLE
-
___________________________________________ I I I I I 0 0 2 4 6
(3)
D~L_~m~LHR~
ION ENERGY, 00eV Fig. 5. Illustrative example of the energy and pitch angle evolution of a hydrogen ion due to lower hybrid heating in the suprauroral region of the boundary plasma sheet based on a mean particle calculation. Initial energy of the ion was taken as 1 eV. The value I E,,~I 2/c,, LI-I was assumed to vary linearly from 1.0 (mV/m)2/Hz at the initial altitude of 500 km to 0.55(mV/m)2/Hz at the final altitude of 5000 km, where E~ and ~~LI-I are the average amplitude and frequency of the lower hybrid waves, respectively,
acceleration theories are inappropriate here. As an estimate, we follow Chang and Coppi [13] and describe the transverse heating process by the quasilinear diffusion operator. Since the distances traversed by the ions are much larger than the ion gyroradius, it is appropriate to use the guiding center approximation here. Neglecting the transverse drifts, the evolution equation may be expressed as follows:
af
~f +
v as
+
e ‘ m E
—
+
v c
—
XB
)
If ~v
(D ~ 8vj
(i)
av ~
where s is the arc length along the field line, v is the component of the velocity v along s and the quasilinear diffusion coefficient D is given by: D
—
e (m)
x
J -i-—f dco
3k kk d
3
—~
SE (k, w) tr6 ( w
—
k v), (2)
where ~ LHR = ~ (1 + e/~2~ )1/2 is the lower hybrid resonance frequency, and WI,e and ~ are the electron and ion plasma frequencies, respectively. The resonant range extends from a few times the ion thermal velocity up to a velocity V±max Ub(kII/kjjmaxLH — ub(me/mi)”, where Ub is the velocity of the electron beam. Thus ions
should stay in resonance until they reach energies near the electron beam energy. Above the upper limit v ~max’ the perpendicular diffusion coefficient D 1 should behave asymptotically as which follows from dimensional considerations. Eq. (1) with the approximation of the velocity diffusion coefficient given by (3) was solved by Retterer et al. [14] using a particle simulation procedure, in which the stochastic effects of the diffusive operator in eq. (1) were described using a Monte Carlo technique. The simulation was carned out over the altitudinal range from 1000 km to 5000 km for hydrogen ions. The initial density distribution was assumed to decrease with the second power of the altitude an approximation to Maeda’s [15] model of ionospheric density. The initial ion distribution was taken as an isothermal M~welliandistribution with an initial temperature of 1 eY. As the simulation progressed, a —
steady state was established by replacing every particle which leaves the the simulation by adistribuparticle picked at random from primordial lion. Fig. 6 illustrates the ion conics formed with scatter point plots of particle kinetic energies parallel and perpendicular to the magnetic field
T Chang et al. TIME
soc
20;oo
65
50.0 SEC
30000km
2000.0 km 200
/ Electromagnetic tornadoes in space
.
O2~0400
:
.
:
~
6~O
800
1.0
.
:.
IO~
-
I2~O 400
600
E IPERP) (eV)
Fig. 6. Monte Carlo simulation of ion conics in the boundary plasma sheet. Lower hybrid acceleration of hydrogen ions occurs over the altitudinal range from 1000 and 2000 km with E~= 50 mV/m. The lower panel gives the conic at the top edge of the acceleration region (altitude = 2000 km), while the upper panel gives the conic folded by the geomagnetic field at an altitude of 3000 km.
for a moderate wave intensity. The result is very similar to that seen by the S3-3 satellite [16]. Even under the approximation of pure perpendicular diffusion, eq. (1) is very difficult to handle analytically because of the presence of the mirror force term. When the region of transverse heating is limited to a small altitudinal range, the problem can be solved using a two-stage approximation by assuming that the upward moving component of the ambient population is first subjected to the transverse heating by lower hybrid waves. Identification of the ratio of ion thermal speed to the mean wave speed as a small parameter leads to a uniformly valid asymptotic solution of the quasilinear diffusion equation for a reasonably general distribution of wave energy in the wave vector (k) space. The ion distribution function is then allowed to drift adiabatically up the geomagnetic field line. Analytical descriptions of this type were given by Crew and Chang [17a]. Fig. 7 depicts the process of evolution of the ion distribution function for a special case based on
v
1iv0
Fig. 7. Evolution of an hydrogen ion conic based on the analytical solution using the two-stage approximation. The diffusion layer is 10 km thick. Typical ionospheric conditions are assumed and E~= 33 mV/rn. Cases (a), (b) and (c) are at altitudes 10. 2500 and 5000 km above the base of the diffusion layer.
such an analysis. Results compared well quantitatively with the Monte Carlo calculations [17b]. 2.2. Self-consistent particle-in-cell plasma simulation In the above calculations, the self-consistent evolution of the wave-spectrum has been ignored. To address this problem, a one-dimensional electrostatic, particle-in-cell simulation was performed [18]. The YLF waves are excited at higher altitudes by Landau resonance with the electron beam. As the waves propagate downward into higher densities, their wave vectors turn perpendicular to the magnetic field as the wave frequency approaches the local lower hybrid resonance frequency [19a]. For simplicity, it was assumed in the simulation that the lower hybrid waves were generated locally with a frequency near the lower hybrid resonance frequency, in this way taking advantage of the vertical inhomogeneity of the plasma but neglecting any horizontal
66
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Electromagnetic tornadoes in space
inhomogeneity and the resulting convective effects. This was justified by linear calculations [20] which showed that, for beam densities of the order of one-thousands of the ambient density, the lower hybrid waves would grow to nonlinear strength before convecting out of the auroral arc. If we look for an obliquely propagating (k 1/ k~1~ ~ 1), electrostatic mode such that ~ ~o 4~ ~e and Pc << k11 ~ p~,a linear analysis of the plasma dispersion relation yields a frequency which, without thermal corrections, depends on the wave vector only through its direction [19]: ~LH
WLHR
(i
+
m
k~
__‘-
_~—).
(4)
.
tion oblique to the magnetic field in the x—z plane is simply the projection in that direction of its velocity along the magnetic field direction, if the small polarization drift is neglected. With this approximation, the acceleration of a guiding center in the x-direction is eE(x, t) cos2O/m, implying that the effective inertia of aangles guiding in this 2O. At the ofcenter propagation model is m/cos of lower hybrid waves, the effective inertia of an electron guiding center, me/cos29, is comparable to rn, permitting realistic mass ratios, me/rn,, to be used in the simulation. Fig. 8 summarizes the energetics of a typical simulation in which the ambient electrons was
‘LI
An approximate analytical expression for the lineargrowthrate,
ii....
6xlO~
[~~
/ ~ \1/2[ y~w~-~-)
~pb
(kub — kIvtb
~)
xIc~5
>-
xexp(
—
1 2
(w_kllub)2\
k~v~b
)
w 1 ~2 ca~w -~--—exp~— / / ~1~2\
-‘‘ ~O~e
\
]
(5)
shows that the lower hybrid waves can be linearly excited over a narrow range of wavenumbers: U/u,, k S w/(ub — nvLb), where w = WLH, Wpb is the beam plasma frequency, u,, is the beam velocity, Vte~ v 1~,and Vtb are the electron, ion, and beam thermal velocities, respectively, and ~ 1—2. As mentioned previously, because of the range of frequencies and wavenumbers of the modes that we are studying, the dynamics of the ions can be treated in a good approximation as unmagnetized, while the electrons can be treated in the guiding-center-drift approximation. a one-dimensional model, particle cross-driftsIn are out of the x—z plane which contains the directions of propagation(x) and the magnetic field; in this case an electron guiding center velocity in a direc—
(9 tE tjJ
u z
4x10
z 2
3x10 x -5 2xl0~
(9 U
Z Lii (I) lii
IO~ 0
I
200
400 600 800 TIME Fig. 8. The energetics of a particle-in-cell simulation. The drift and thermal energies of the electron beam are scaled from the values Eb = 1 keY and2a/me Tb = 125 and electron nb/no = 1, eV, the respectively, velocity of the beam projected onto the x-direction is approximately 31.6 = 0.025. m1 velocity cos times the With thermal of the ambient ions. The top panel plots the ion energy ‘density versus time, while the lower panel gives the electrostatic energy density versus time in units of ~iii/i. The energies are measured in arbitrary units.
T Chang et a!.
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Electromagnetic tornadoes in space
treated as a fluid. It can be seen that the total ion energy density started to rise rapidly only after the electrostatic energy had neared saturation. Following saturation, the ion energy continued to grow approximately linearly with time, but at a somewhat slower rate. The simulation was carried out for m~/cos29= m1 with ub equal to 31.6 times the thermal velocity of the initially cold ions. This meant that the linearly excited waves had phase velocities far out on the tail of the initial ion distribution function and would have been very inefficient in resonantly heating any significant fraction of the ion population. Thus, some nonlinearly excited waves must have been generated during the wave saturation process in order to obtain the type of observed ion heating. To check this, we compared the numerically calculated imtial growth rates for the waves that were observed to grow during the simulation with the linear theory. While good agreement was found for the most unstable waves, we found that there were
~
~
[:I~m~
~\
67
unstable waves with wave vectors larger than those predicted by linear theory with appreciable growth rates. These waves would have much smaller phase velocities and could resonate with a significant tail population of the initial ion distribution. This was further verified by the evolution of the wave spectrum obtained from the simulation (fig. 9). It is noted from these snapshots that at the early times, the linearly unstable modes were most energetic. At later times, however, the wave energy redistributed itself throughout k space, probably through a complicated sequence of mode-coupling processes. The spectrum seemed to reach a quasi-stationary state upon saturation, albeit with a continually heating ion population. We now consider the evolution of the ion distribution function in detail. A series of snapshots of the ion velocity distribution at different times for the simulation, overlaid in fig. 10, illustrate vividly how such type of ion acceleration takes place. We see that tails of energetic ions formed, emerging from both sides of the initial velocity distribution
____
I0~0 I
0
83
I
60 WAVENUMBER
I
80
04 I
.0204.06
.11.1
v~,
60
Fig. 9. The wave spectrum for a particle-rn-cell simulation. The drift and thermal energies of the electron beam are scaled from the values Eb =With 1 keYm and 29/,n~ Tb = 125 nb/no = 0.025. = 1,eV, therespectively, velocity of and the electron beam projected onto 1 costhe x-direction is approximately 31.6 times the thermal velocity of the ambient ions. In each panel is plotted I E(k, t) 12 versus wave number k at a fixed time. In the upper row the times are 160 and 240; in the lower row, 320 and 400 (all in units of wj~j).
.
.
.
.
.
.
Fig. 10. The ion velocity distnbution of a particle-in-cell simulation. The drift and thermal energies of the electron beam are scaled from the values Eb = i key and 29/m~ Tb = 125 = 1,eV, therespecveloctively, 0.025. projected With m1 cos ity of and the nb/~no electron= beam onto the x-direction is approximately 31.6 times the thermal velocity of the ambient ions. Each curve is a plot of the ion velocity distribution versus velocity at a fixed time. The times run from zero to 400 in increments of 40 wj~.A velocity of 0.04 corresponds to an ion energy of 230 eV.
68
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Electromagnetic tornadoes in space
at about three times the ion thermal velocity. Some ions were then accelerated to velocities comparable to those of the electron beam. In addition to the tails, the core of the velocity distribution showed evidence of nonresonant heating. But the resonantly heated tail population accounted for most of the energy transferred to the ions. As a consequence, a significant portion of the ions in the tail region (with v1 ~ 3 v~1)were accelerated to the velocities where the linear waves were intense, (Positive and negative v~velocities, distinguished here, are generally not distinguished in space observations, v~being essentially a component of v1.) At this time, the waves had saturated and the heated ions continued to receive energy from the waves while the wave energy was being continuously replenished by the electron beam. The existence of such a quasi-stationary state along with the knowledge of the saturated shape of the wavenumber spectrum provided the basis for the earlier Monte Carlo and analytic calculations of ion evolution along the scribed in section 2.1. geomagnetic field line deAlthough the above discussion is based on the result of one-dimensional considerations, similar results are obtained from two-dimensional simulations [21]. (See also refs. [12c] and [18] for two-dimensional mode-coupling discussions.) However, nonlinear modes with wave vectors out of the plane generated during the cascading process would not grow to large amplitudes for current sheets which are thin in the y-dimension. Thus, for actual discrete auroral geometry in the boundary plasma sheet, the one dimensional results might actually better describe the physics of ion acceleration. Only simulation results using an ambient electron fluid have been reviewed above. Retterer et a!. [18] have also considered the case where the ambient electrons were treated as particles using the guiding center approximation. Similar results were obtained. In addition, it was found that high ~
2.3. A strong turbulence model In order to understand the simulation results from a more theoretical point of view, a strong turbulence model was constructed [18]. Because lower hybrid waves can be supported in a plasma treated as a multicomponent fluid, a fluid description was used in the description of the cascading process of waves in wavenumber space. In analogy to the Langmuir turbulence theory [22], the fluid quantities (both for the electron and ions) were resolved into components varying at the lower hybrid frequency and at smaller frequencies of the order of the acoustic frequencies. The resulting approximate nonlinear governing equation for the Fourier components of the amplitudes of the electric field fluctuations Ek at the lower hybrid frequency was a nonlinear Schrodinger equation: 1 BE WLH — —
2E —
2 Dk
~
E E
*
k’k”
k’
k”
+ ~
k
E
where C and D are coefficients definable in terms of the angle of propagation, and the electron and ion masses and acoustic speeds. To introduce the wave—particle interaction effects, an instantaneous Landau damping term YL( k, t ) Ek/U LH was added to the right-hand side of eq. (6) (where YL is the instantaneous Landau damping or growth rate). In addition to this equation governing the wave amplitudes, the wave-partide interaction effect on the particles was described by the quasilinear diffusion equation:
af.
a =
~
af.
-~—-
~
(7)
,
V
V
where, in one dimension, the diffusion coefficient is 2
D.(v)
=
(—~--~ (~~~ 1E 21T8(~
velocity tails of the ambient electrons were produced during the heating process. When projected along the magnetic field direction, these electrons mimic nicely the observed “counterstreaming electrons” which are sometimes detected in conjunction with the ion conics [10].
6 k+k”k”
\
rn~)J 2ir
/c
H(k)
—
kv)
L
(8) and j = e, i. Numerical solution of this set of equations have been compared with the simulation results dis-
T. Chang et aL 5
~
/
Electromagnetic tornadoes in space I0~
~
3.5x1d
~
3.0~0~ >0~
69
‘11111
02
H
w
I
~2.5xI1i
.
I-.
z
p
I
I
>-
20~O’ -Y
0
o
-i
Lii
>
z
I~
‘
I.
I
I i
.‘
101 I -.06 -.04 = 400. The dashed line
>(5
z w
0
I
I
~,
I
-.02 0 velocity .02 distribution .04 .06 gives the ion from
Vx
(/) Ui
Fig. 12. The ion velocity distribution in the theoretical model. The solid line is snapshot of the velocity distribution at w LIlt
IO~
the comparable time in the particle simulation. A velocity of 0.04 corresponds to an ion energy of 230 eV. 0*
~
0
~
00
200 TIME
300
400
Fig. 11. The energetics of the theoretical model. As in fig. 8, the top panel gives the ion energy as a function of time, while the lower panel gives the electrostatic energy. The dot-dash lines give the corresponding results when mode-coupling is not present, while the dashed line in the lower panel gives the result from the particle simulation, with its origin of time shifted to match the initial conditions of the mode-coupling calculation.
cussed earlier. As can be seen from figs. 11 and 12, general agreements are obtained. Thus it can be concluded from these comparisons that this theoretical model of strong turbulence with mode-coupling adequately describe the phenomena observed in particle simulation. While the simple quasilinear-diffusion model was adequate to describe the effect of the turbulence on the ions, a full nonlinear treatment was necessary to describe the evolution of the turbulence. 2.4. Additional remark The above discussion considered only one species of ions. If the plasma contains several
species of ions of approximately the same temperature, then it can be easily argued that only the lightest species can receive appreciable lower hybrid heating. The reason is that the heavier species has much fewer particles in the velocity range where the lower hybrid waves are intense. Nevertheless, both hydrogen and oxygen ion conics have been detected in the region under consideration. If the lower hybrid mechanism is to be the heating mechanism also for the oxygen ions, some sort of pre-heating process for the oxygen ions must have taken place before the lower hybrid heating scenario takes over. We will come back to this point later in section 4.
3. Ion acceleration by electromagnetic turbulence in the ion cyclotron range of frequencies Recent particle data collected by the plasma instruments HAPI and EICS aboard the polar orbiting satellite Dynamics Explorer 1 [2,23] re-
70
T. Chang ef aL 60
/
Electromagnetic tornadoes in space DE—1
I
10
,,.,..,.I
I
I
DAY 318 NOV 14, 1981 UT234544
>• I...
~
PWI I
10
30
I
I
SO
68
-60 \~L(km/s)
1
10
100
1K
10K
lOOK
FREQUENCY (Hz)
Fig. 13. The bottom panel presents a contour diagram of the observed oxygen-dominated ion conic distribution function in the central plasma sheet, measured by the HAPI instrument on Dynamics Explorer 1 on 14 Nov. 1981 at the geocentric altitude of 2.OR E and invariant latitude of 600. The top panel presents the theoretical contours for the same event based on the Monte Carlo simulation calculation. The contours are uniformly spaced with an increment of 0.4 in the logarithm of phase space density. The density of these ions at3.the observation point is approximately 10 cm~
vealed a prevalent population of oxygendominated ion conics in the central plasma sheet region at a geocentric altitude of approximately 2 RE during an intense magnetic substorm. The observed peak energies of these conics varied from tens to hundreds of eV at pitch angles around or below 1300 (lower panel of fig. 13). The corresponding field-aligned currents measured by the magnetometer [24] and the plasma instrument HAPI exhibited irregular changes in sense of direction and their magnitudes were generally not very intense. There was no meaningful correspondence between these central plasma sheet conics and the sparsely observed, relatively low energy, field-aligned electron beams. This is more or less expected, since the central plasma sheet represents a much more quiet plasma state when compared with the boundary plasma sheet where discrete aurorae occur, The above observations seem to rule out some
Fig. 14. Typical electric field spectral density in the central plasma sheet measured by the PWI instrument on Dynamics Explorer I on 14 Nov. 1981 at the geocentric altitude of 2.OR E and invariant latitude of 600.
of the more popular heating mechamsms such as the lower hybrid interaction electroprocess discussed above wave—particle and the current-driven static ion cyclotron wave mechanism proposed by a number of previous authors [25,26]. On the other hand, the plasma wave instrument PWI on board the Dynamics Explorer I, though operating in a low resolution mode, gave clear indications of an intense low frequency noise (less than a hundred Hz) across the central plasma sheet region (fig. 14) [9b]. Based on mean-particle calculations. Chang et al. [27] demonstrated that the observed low frequency electric field turbulence in the ion cyclotron range of frequencies was sufficient to account for the transverse acceleration of ions through cyclotron resonance to the observed energy levels in the central plasma sheet region. They assumed that the ions were oxygen and that some of the observed turbulence was electromagnetic and left-hand polarized. These waves, however, are not very effective in accelerating the hydrogen ions because the wave energy falls off at the hydrogen gyrofrequency. This fact seems to explain why the observed central plasma sheet conics were oxygen-dominated [23].
T. Chang et aL
/ Electromagnetic tornadoes in space
3.1. Monte Carlo calculations Since the observed low frequency noise is broad band and the wave intensity is not very strong, it is reasonable to represent the perpendicular heating due to electromagnetic ion cyclotron resonance by the quasilinear diffusion operator. In the absence of an external dc electric field, the ion evolution equation (eq. (1)) along a geomagnetic field line may be written as:
a.!
Bf
V 1
+ =
V11
ã~
—
1
B
—
.~—
Bf
dB ( V1
a~
(v±D1 .~L’),
Bf —
\
V II
(9)
where f(s, V v1, t) is the ion distribution function, and (V II’ v1) are the parallel and perpendicular velocities, respectively. In terms of the electric field spectral density, the perpendicular velocity diffusion coefficient is found to be [28]: II’
D1
=
—~—~IEL(w = ~(s))I2,
(10)
where I EL ( ~o= ~( s)) 2 is the spectral density of left-hand polarized waves, and ~2(s) is the local ion cyclotron frequency. Eqs. (9) and (10) have been solved using the Monte Carlo method similar to that employed for lower hybrid heating [28]. Because the number of accelerated ions is small (oxygen is a minority constituent of the plasma), the ions were treated as “test particles” in externally imposed fields. To study the particular event of the lower panel of fig. 13, the ions were assumed to be thermally distributed with a temperature 0.2 eValong at ana 2RE, andoftracked initial altitude of s = l.plasma sheet geomagnetic portion of a central field line at an invariant latitude of 600, extending up to the geocentric altitude of the satellite, roughly at 2.ORE. Because of the form of the velocity-diffusion rate, however, most of the ion acceleration observed would have occurred near the altitude of the observation point, and the results of the calculation should be insensitive to the initial conditions chosen for the oxygen ions. Guided by the previous mean particle calcula-
71
tions, the spectral density of left-hand polarized waves was chosen to be one-eighth of the actual measured wave intensity. As ions passed the ob2.ORE, statistics on their velociservation point at ties were accumulated to calculate the ion-velocity distribution at this point. The results are presented in the top panel of fig. 13. Without a simultaneous measurement of the ion density in the source region, the absolute normalization was per force arbitrary. As can be seen from fig. 13, the quantitative agreement is remarkable between the calculated and observed contours. Despite the absence of a parallel electric field in this calculation, noticeable parallel acceleration was achieved. All transverse acceleration schemes must result in some parallel acceleration, because the effect of the mirror force is to convert the perpendicular energy gained into parallel energy. Although the intense, low frequency electric and magnetic field noise has been observed over the aurora! zones for many years, the nature and origin of the turbulence are still not thoroughly understood and are the subjects of ongoing research. The polarization data for the low frequency turbulence in the Dynamics Iauroral surveyzone appears to be consistent with a Explorer nonlocal source mechanism for the turbulence. But the propagation of left-hand polarized waves at frequencies near the ion gyrofrequency, with the resulting possibility of mode-conversion phenomena, remains largely unexplored. The uncertainty in the origin of the low frequency turbulence does not alter the conclusion, based on the success of the Monte Carlo calculation, that this turbulence can explain the observed central plasma sheet oxygen corncs. 3.2. Similarity solution for the central plasma conics Although the Monte Carlo model used in the previous section was based on the actual spectrum, the fit of a power law of the form I E( U) 2 a gives a convenient means of characterizing the spectrum. For the case of a power law wave spectrum and a dipole magnetic field, the steady state velocity distribution f(s, v, V 1) satisfying eq. (9) exhibits similarity scaling [29]. It can be —
—
72
T Chang et aL
/ Electromagnetic tornadoes in space
shown that there exists a self-similar solution such that the scaled ion distribution function F(x, y) v II’ V1) depends on the velocities only
3
through the scaled variables x C~v1s~ and y cx: v ~ where a = a + 1/3. Such a self-similar solution generally cannot satisfy arbitrarily imposed initial conditions. Monte Carlo calculations for the evolution of the ion distribution function indicated that the calculated results quickly reached an asymptotic state and became independent of initial conditions. Thus the electromagnetic turbulence behaves like a tornado; it picks up charged particles at lower altitudes and whirls them upward. Quickly the heated particle distribution acquires the properties endowed by the local electromagnetic turbulence and forgets entirely about its initial conditions. This peculiar asymptotic property conveniently to the type of self-similar solution describedleads above. Upon such a reformulation, the quasilinear ion evolution equation (eq. (10)) may be re-expressed as a convective-diffusion equation: B2N Bx2 with N(x, y) = xF(x, y), and • (Nu)
u~= =
=
1 —
— (~
4x2
—
+
a)xy,
a~~’2.
(11)
(12a) (12b)
The two-dimensional flow field given by eq. (12) describes the deterministic parts of the evolution (dotted curves of fig. 15). These are the effects of the magnetic mirror force and mean particle heating due to ion cyclotron resonance. The diffusion term represents transverse particle fluctuations about the mean motion due to the imposed turbulence. Thus, Eq. (11) may be replaced by a simple set of Langevin equations: = u.,, + ~, (13a) =
u~,,
(13!,)
where ~ is a random noise with Gaussian statistics, and (~) denotes derivative with respect to some pseudo-time, T. Numerical implementation of the time-asymp-
I
I
~
.
.
.
.
. ‘,
.
.
. .
.
2
:
—, .
.
.‘
.
.
.
, .
.
.
.
.
—
.
y,.:.,,.:.:..H 1
0
1
2
3
X
2/2)— Fig. 15. represent the streamlines of the convecay2. TheDotted closedcurves curves are calculated logarithmically spaced tive flow field: u~ = (1/x)—(3/2+ a)xy and u~= (3x contours for the function N(x, y) = xF(x, y) using the Langevin equations.
totic solution of eq. (13) is trivial and efficiently leads to the desired results [29]. Solid contours of fig. 15 presents the logarithmically spaced contour plot of the function N(x, y) = xF(x, y) for the same event as that was presented based on the Monte Carlo calculation (fig. 13). The result may be understood by recognizing that the flow field u contains a sink at some finite point (x 0, y0) toward which the ions are drawn, balanced by the action of random diffusion in the x-( V II-) direction which tends to disperse the particles from the sink. Fig. 16 translates the result to contours of the scaled distribution function F(x, y). It is to be noted that this asymptotic solution compares favorably with the Monte Carlo solution (fig. 13) and it can be obtain more efficiently. We conclude this section by pointing out that the stochastic property of the Langevin eq. (14) may be expressed exactly in terms of a functional path integral representing the probability density functional of the random functions x ( ‘r) and y( ‘r). Using this path integral, the moments of the distribution function may be obtained via Feynman diagram expansions. These results were presented at this International School [29,30].
/
T. Chang et aL 3 I
2
I
I
I
I
I
I
I
I
I
Electromagnetic tornadoes in space
ions can still be heated by these low frequency waves via electromagnetic cyclotron resonance to the order of 10 eV provided that there is a significant left-hand polarized component present. At
I
these energies, the preheated oxygen ions can easily be boosted to higher energies by the parametrically excited lower hybrid waves.
—
-
73
5. Swnmary ergetic ion populations in the plasma sheet of the The phenomenon of upflowing, gyrating, en-
0 I
0
I
I
I
1
I
I
I
I
2
I
I
I
3
X Fig. 16. Contour plot of the scaled ion distribution function F(x, y). The contours are logarithmically spaced at half-dccades, and the normalization is arbitrary. The parameter a = 2.0 which corresponds to a =1.7. The lowest values, ragged contours at the edges of the conic have been retained as an indication of the statistical error inherent in this technique.
4. Oxygen conies in the boundary plasma sheet In section 2, it was demonstrated that lower hybrid waves produced by precipitating auroral electron beams in the boundary plasma sheet could be responsible for the transverse heating of hydrogen ions. Because of the low thermal velocities of the cold oxygen ions, however, resonant heating of the oxygen ions by lower hybrid waves (which generally have phase velocities larger than the thermal velocities of the cold hydrogen ions) will be difficult unless there is some pre-heating mechanism which can launch the cold oxygen ions to perpendicular velocities in the range of the phase velocities of the lower hybrid waves. In situ measurements of typical electric field wave spectra in the boundary plasma sheet at low altitudes have detected both intense lower hybrid waves and broad band low frequency noise below 100 Hz [9a]. The intensity of the low frequency fluctuations seems to be significantly smaller than those detected in the central plasma sheet such as the one considered in the previous section. However, even for such low wave intensity, oxygen
magnetosphere is considered. It is shown that in the primary auroral electron-beam region of the boundary plasma sheet, lower hybrid waves could be an efficient mechanism for the transverse heating of ions of ionospheric origin (H and 0~) although for oxygen ions to be energized by such a ±
wave—particle interaction mechanism, some sort of pre-heating process would be required. Lower hybrid waves can also be the agent responsible for the generation and energization of the so-called “counterstreaming electrons” along auroral field lines. In the central plasma sheet, where intense low frequency electromagnetic turbulence are detected, it is suggested that oxygen ions can be conveniently energized to the observed energy levels via electromagnetic ion cyclotron resonance provided the wave spectra contained a small fraction of left-hand polarized component. Monte Carlo calculations provided the first successful comparison of an observed conic with a realistic theoretical model. Asymptotic properties of this sort of heating allowed the search for a self-similar solution yielding a more efficient method for calculating the detailed shape of the ion distribution function as well as some additional insight into the physics of the heating process.
Acknowledgements The authors are indebted to M. André, B. Basu, J. Burch, B. Coppi, D. Gurnett, J. Jasperse, N. Hershkowitz, M. Hudson, P. Kintner, D. Kiumpar, H. Koskinen, R. Lysak, M. Mellott, W. Peter-
74
T. Chang et aL
/ Electromagnetic tornadoes in space
son, I. Roth, E. Shelley, M. Temerin, D. Vvedensky and J.D. Winningham for collaborative research and/or useful interactions. This research is partially supported by the US Air Force Office of Scientific Research under Contract No. F49620 86-C-0128, and the Air Force Geophysics Laboratory under Contract Nos. F19628-86-K-0005 and FY7121-84-0-0006. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.
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