Cyclotron damping by suprathermal fluxes of protons and electrons in the interplanetary medium will greatly attenuate high frequency Alfven waves and whistler waves within distances  1 AU of the sun. Electrons with energies between 50 eV to 2 KeV are heated as a result of damping interplanetary whistler waves with frequencies 2  omega meson/2 pion  30 Hz in the frame of the solar wind. This heating may account, in part, for the observed suprathermal tail of solar wind electrons. Protons with energies approximately  50 KeV damp Alfven waves with frequencies .001  omega meson/2 pion  .01 Hz. This damping mechanism may explain several features of a scatter free solar electron events and high intensity, anisotropic solar proton streams

Fisk, L. A.

Goldstein, M. L.

English

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DAMPING OF HIGH FREQUENCY
(NASA-TH-X-66254s) DAMPING OF HIGH
FFPEQUECnY WAVES IT THEr SOLR WTIND (NASA)
1-7 p HC $3.03 CSCL 03B
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G3/29 06226
M. L. GOLDSTEIN
L. -A. FISK
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JUNE 1973
, GODDARD SPACE FLIGHT CENTER - I
GREENBELT, MARYLAND
I
WAVES IN THE: SOLAR WIND
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https://ntrs.nasa.gov/search.jsp?R=19730017120 2020-03-11T19:12:22+00:00Z
DAMPING OF HIGH FREQUENCY WAVES IN THE SOLAR WIND
M. L. Goldstein*
Laboratory for Extraterrestrial Physics
and
L. A. Fisk
Laboratory for High Energy Astrophysics
June 1973
* NAS-NRC Postdoctoral Resident Research Associate
Goddard Space Flight Center
Greenbelt, Maryland
PRECEDING PAGVeBLANK NOT FILMED
DAMPING OF HIGH FREQUENCY WAVES IN THE SOLAR WIND
ABSTRACT
Cyclotron damping by suprathermal fluxes of protons and electrons in the
interplanetary medium will greatly attenuate high-frequency Alfven waves and
whistler waves within distances < 1 AU of the sun. Electrons with energies
between 50 eV to 2 KeV are heated as a result of damping interplanetary
whistler waves with frequencies 2 < cw/2 7T < 30 Hz in the frame of the solar wind.
This heating may account, in part, for the observed suprathermal tail of solar
wind electrons. Protons with energies < 50 KeV damp Alfv6n waves with fre-
quencies 10- 3 < w/2 r <10-2 Hz. This damping mechanism may explain several
features of a "scatter-free" solar electron events and high-intensity, anisotropic
solar-proton streams.
Preceding page blank
iii
I. INTRODUCTION
In this report, we consider the consequences of cyclotron (or magnetic-
Landau) damping of high-frequency waves in the interplanetary medium. The
frequency range of interest is from 10 -2 Qp < c <10-1 I|e in the plasma
(solar wind) frame, where Qp and Qe are the gyrofrequencies of protons and
electrons, respectively. In the frame of a spacecraft, the frequency range is
from about 10-2 to 102 Hz with a magnetic field, B, of 5y, and a solar wind
speed of Vsw = 300 km/s.
Waves in this frequency range have wavelengths comparable to the Larmor
radii of protons with energy < 500 KeV and electrons < several MeV, and thus
will resonantly scatter these low-energy particles. In fact, in current diffusion
theories (Jokipii 1966, 1967, 1968) the diffusion coefficient Kll, parallel to the
mean field is quite small (KI 10 20 cm 2 /s for protons, and Kll 10 21 cm 2/s
for electrons) provided that the spectral index of the power spectrum of mag-
netic field fluctuations is less than about two. This small value of Kll reflects
the efficiency with which high-frequency magnetic field fluctuations can reso-
nantly scatter low-energy particles. Thus a mechanism that attenuates these
waves, such as cyclotron damping, will produce observable consequences in the
spectra and anisotropies of low-energy particles.
In Section II we outline briefly some of the observed features of low-energy
particle behavior that may be the direct result of cyclotron damping. Section III
contains a quantitative discussion of the theory, and a calculation of the appropriate
damping lengths.
1
II. Observed Behavior of Low Energy Particles
Listed below are some of the observed features of the behavior of low-energy
particles that may be the result of cyclotron damping of high-frequency waves.
We provide this list principally to motivate the theory reviewed in Section III.
1) Recent observations have shown that the thermal distribution of solar-
wind electrons has an extended suprathermal tail in the energy range
50 eV to - 2 KeV, as can be seen in the composite electron spectrum
shown in Figure 1 (Montgomery, et al., 1968; Ogilvie et al., 1971;
Anderson, et al., 1972; Lin, et al., 1972; Scudder, et al., 1973). Such a
suprathermal tail can result from heating nonthermal electrons through
the damping of high frequency-waves in the solar wind. In the next
section we argue that sufficient energy is probably available in high-
frequency field fluctuations to supply the required energy to 50 eV-2 KeV
electrons. One should note that at energies below - 100 eV the supra-
thermal tail may be a consequence of the heat flux of the solar wind.
However, it is not well understood why the heat flux is carried by - 100
eV electrons, rather than by electrons with energies of only a few eV
(Montgomery, 1972).
2) High-intensity proton streams at low-energies ( > 0.1 protons/(cm2 s
ster KeV) at = 500 KeV) are observed to be highly anisotropic (cf.
Krimigis, et al., 1971). Such large proton fluxes could damp the magnetic
irregularities that normally scatter these low-energy particles, and thus
permit the large anisotropies.
2
3) At times, solar electrons in the energy range 50 KeV to 1 MeV are ob-
served to propagate from the sun to earth with little or no scattering
(Lin, 1970; Wang, et al., 1971). These "scatter-free" events can be
understood if interplanetary conditions change at the time of the events
and the waves that normally scatter these electrons are damped.
III. Theory
Cyclotron damping is a resonant interaction in which a particle moving
with velocity V interacts with a transverse wave whose Doppler-shifted frequency
in the frame moving with velocity Vil - V ·B/B is Qv (v = p, e for protons and
electron, respectively). Thorough treatments of cyclotron damping can be found
in Stix (1962), Scarf (1962), and Tidman and Jaggi (1962). We outline below only
those parts of the theory that pertain to the damping of waves by a nonthermal
distribution of particles (Tidman and Jaggi, 1962; Tidman, 1966). We also
consider only the case of parallel propagating waves A IIB). However, waves
propagating at angles less than cos- 1 | Q I / c (c is the speed of light) also ex-
perience cyclotron damping (Tidman, 1966). For the waves considered here
damping can be important within a fairly large cone of angles - 600 about the
mean field.
The damping decrement,-, of transverse waves propagating parallel to the
ambient magnetic field is found by expanding the dispersion relation (Montgomery
and Tidman, 1964) in powers of /A) and equating the real and imaginary parts
3
to zero. Defining the particle distribution function fv, so that ffv dv = 1, one
finds (Tidman and Jaggi, 1962) that the real part, w, of the frequency satisfies
(1)
c2 - cC22  7T P dv 11) = 0
V P 1&
v OD I 1 (
and the imaginary part is given by
=
77 W 2
2co - i AV
v
7; 2 I T ()w ± Q)
I-
l]
where w2 = 47rN, e2 /M,, N is the density of the ambient plasma, and P denotes
the Cauchy principal-part. Also,
(3)an 2 I
yP(n)(Vl )-=- d~vjj J() - dv vfdv-'- + (vI
dvi dvi I11 11- · ·- · · ·
where
(4)
(VII) -= d v ,2dv a -V
At particle speeds several times the thermal speed, one can show that
thermal effects are unimportant in the numerator of (2). Furthermore, because
4
(2)
the density of the thermal plasma is assumed to be much greater than that of the
energetic particles, one can ignore the contribution of the energetic component
in (1) and in the denominator of (2). If we define the phase velocity, Vph = co/,
(1) becomes
.- 1/2
(phCO±e) w(C
In the frequency range 0p < < I ne only Vph of the extraordinary mode ("+") is
less than c and can be damped by this mechanism. The phase velocity of the
ordinary mode ("-") is greater than c and will not be considered further. For
C < Qp, both modes can be damped and V h - VA = B/(477 NM ) 1/2 the Alfven speed.
The Whistler Mode.
We consider the first the damping of the whistler mode (Qp < c < IQeI ). These
waves have wavelengths comparable to the gyroradii of suprathermal solar-wind
electrons (50 eV to - 2 KeV), and can provide an energy source for heating these
particles. We use an isotropic distribution function under the assumption that
the particles are scattered and thus isotropized while damping the waves.
However, the assumption of isotropy is not essential to our argument because
we show below that an anisotropic distribution function enhances the damping
of the whistler mode. After the waves are damped electron fluxes are expected
to be anisotropic because less high-frequency power will be available to produce
isotropy via pitch-angle scattering.
5
We let fv = fv + fv 2where 1 and 2 refer to thermal and suprathermal
particle distributions, respectively. We have defined fv 2 = M jJ/(N v2) where
-Fj,(v) = Av(1/2 M Iv2) v is the differential energy spectrum measured in units
of particles/cm 2 s ster KeV. In this frequency range the protons do not contribute
to the damping because the velocities involved are a few times the thermal
velocity and at quiet times the observed velocity distribution in this range is a
good fit to a Maxwellian (Ogilvie, private communication). Consequently,
7T2 V nMjph QMe e e co Ph (6)
Nre
We define a damping length L 1 = (V5, + Vph )/I | I. Substituting numbers
appropriate to 1 AU (B = 5 x 10-5 G, N = 10 cm- 3 , Vsw = 3 x 107 cm/s,
Ae = 10 s , r = 3) we find that Lll < 3 x 1012 cm for electrons between
50 eV and 2 KeV (20 fp < W < lieI /5. Thus, suprathermal electrons near
1 AU will absorb wave energy within a characteristic distance 0.2 AU.
In order to heat sufficiently the suprathermal electrons via this mechanism
one must be able to find a region of interplanetary space between the sun and
1AU in which the energy density in the magnetic field fluctuations is comparable
to the energy density in the particles. In the energy range 50 eV - 2 KeV most
of the particle energy is found in 50 eV electrons. However, these electrons
damp the highest frequency waves, which contain the least magnetic energy.
Thus, if there is a heliocentric distance less than 1 AU at which the energy
6
density in 50 eV electrons is comparable to that in the waves that they damp,
we will have shown that the suprathermal electrons between 50 eV and 3 KeV
can be, at least in part, produced by this wave damping mechanism. Of course,
an element of conjecture will remain in this discussion because the required
high-frequency fluctuations cannot be observed at 1 AU let alone at other
heliocentric distances.
We assume that the power spectrum of magnetic fluctuations near the sun
can be estimated by extrapolating to smaller heliocentric distances the power
spectra that are observed near 1 AU. We preserve the spectral shape in this
extrapolation, and therefore we assume that the power spectrum for fluctuations
along the mean field direction is approximately given by
PZZ() -A>A (7)
where A is wave number and &c = 277A with k the correlation length of the
field fluctuations. The z-component of the field strength has a mean square value
denoted by (B'z).
On assuming isotropic fluctuations, the energy density in field fluctuations
above k, is given by (Batchelor, 1956)
47r j ' d & / d A ) (8)
We choose k to be the wave number of fluctuations damped by - 50 eV electrons.
7
We further assume that B 2  - 1/4 B 2, B = Bo/R 2 , kc = Co/R, and
I= A, /R 2. The subscript "o" denotes quantities evaluated at 1 AU, and R
is heliocentric distance in AU. We take B o = 5 AU, Aco = 3 x 10 - lI cm-', and
1A0 = 10 - 5 cm-'. The energy density in magnetic field fluctuations that is
available to heat 50 eV electrons is then given by
0.6R-7/ 2 eV/cm 3  (9)
The energy density in electrons above E = 50 eV is
47'je(E) E 2
4- e(E) E2  20 R-2eV/cm3  (10)
R2v(F e - 1/2) (Fe - 3/2)
where we have taken Fe = 3, je (> 50 eV) = 109 (cm 2 s ster keV) - 1 and
v= 4 x 108 cm s- 1. Therefore at R - 0.09 AU - 8 E', as required.
From Eqns. (5) and (6) it is easy to show that Lll R 2 , so that 50 eV electrons
damp waves even more effectively at the sun than at 1 AU.
The Alfven Mode
We consider next the damping of high frequency Alfven waves (co < Qp).
One can show that electrons alone cannot damp these waves within any
reasonable heliocentric distance because the required electrons are of quite
high energy (> several hundred KeV) and are few in number. Thus the damp-
ing must be done by suprathermal protons above -100 eV. Summing (2) over
polarization, the relevant damping decrement is (cf. Tidman, 1966)
- 2 o 2  Mp7T( 2 (p V ) i,( . (11)
Inel
8
The quasi-steady interplanetary proton flux is variable and at times can
reach intensities > 102 protons/cm3 s ster MeV at 1 MeV. At these times,
protons with energies up to - 1 MeV can effectively damp fluctuations and
consequently propagate in a highly anisotropic flux, in agreement with the
observations of Krimigis, et al. (1971).
The damping of high-frequency Alfven waves by low energy protons may
also affect electrons that resonate with these fluctuations. During most times
substantial proton fluxes, (- 0.5 protons/cm 2 s ster KeV) have been observed
at 60 KeV (Lin, private communication). If we assume here that jp = 105 E - 3
(protons/cm 2 s ster KeV) consistent with Lin's average observations (as opposed
to the quietest periods observed), then Alfv6n waves in the frequency range
10-2 Qp < co 10 - Qp will be damped within 1 AU. This damping could affect
the scattering of electrons with energies between -100 KeV and several MeV.
With normal interplanetary conditions, cyclotron damping cannot attenuate
waves that scatter electrons below - 100 KeV. This damping, then, cannot
account for electrons observed in scatter-free electron events (Lin, 1970;
Wang et al., 1971). However, it can be shown that the damping will be effective
to lower energies during periods of low Alfven speed. For example, if VA is
lowered by a factor of order two to - 20 km/s, the damping is effective for
electrons with energy down to -25 KeV. Thus, low VA together with particularly
effective cyclotron damping may be necessary in order that scatter-free events
occur.
9
Anisotropic Distribution Functions. In the above calculations, we used
isotropic distribution functions; however, we can readily include particle
anisotropies. Let F = fv2 (v) (1 + 5 cos a) be the suprathermal component,
where C < 1 is a measure of the anisotropy, and a is the particle pitch angle
defined so that a = 0 is directed away from the sun. Then one can show that the
denominator of (2) becomes
wv M, j( (± 9l+ 2F ( (12)
V ANU A rV 2rV + 1 (4Fr2 - l) kVph
For electrons that are damping whistlers,
w _+ , ne
-- V <0
ph
so that the anisotropy enhances the damping. However, an anisotropic flux of
protons will damp only left-hand circularly polarized Alfv6n waves (p/&A < 0),
while allowing the righthand mode to grow, provided that
1 + 2 < 2f__ (13)
r 2r +1 (4F2 - 1)
For Qp/w < 70 (13) is satisfied when C > 0.1. However, even in this case, the
right-hand mode cannot grow indefinitely. The presence of helium in the solar
wind will cause a mode coupling, changing the polarization from the right to the
left hand; the latter would then be damped (Smith and Brice, 1964; Scarf and
Fredricks, 1968). The problem is highly nonlinear and to our knowledge the
equilibrium state has not been calculated. Independent of this mode coupling
10
mechanism, the right hand mode could only grow until it becomes nonlinear. It
would then be damped via a variety of mechanisms (Hollweg, 1971; Lee and
Volk, 1973). In fact, this right hand mode together with the undamped, higher-
frequency, ion-cyclotron wave can produce at least part of the high frequency
power sometimes observed up to several Hz (Holzer, et. al., 1966; Fairfield,
private communication).
IV. Conclusions
In this report, we have argued that cyclotron damping of high-frequency waves
by suprathermal particles in the solar wind can account for several features of
the behavior of low-energy protons and electrons. Electrons in the energy range
50 eV-2 KeV damp whistler waves, and as a result may be heated, contributing
to the observed suprathermal tail of the solar wind (Montgomery, et al., 1968;
Anderson, et al., 1972; Ogilvie, et al., 1971). Low energy protons damp high-
frequency Alfven waves. In quasi-steady, high-intensity proton streams, the
damping attenuates waves that scatter 1 MeV protons and may account for
the large anisotropies that are seen (cf., Krimigis, et. al., 1971). During more
average time periods, protons with energies in the frame of the solar wind
between 100 eV and 50 KeV attenuate the fluctuations that scatter electrons
with energies between -100 KeV to several MeV. During periods of low Alfven
speed, this damping attenuates waves that scatter electrons with energies down
to 425 KeV and, provided that the damping is particularly effective, may
account for some features of scatter-free electron events.
11
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Holzer, R. E., M. G. McLeod, E. J. Smith, Preliminary results from the OGO-1
search coil magnetometer; boundary positions and magnetic noise spectra,
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12
Lin, R. P., Observations of scatter-free propagation of - 40 KeV solar electrons
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from 18 KeV to 1.8 MeV during solar quiet times, Phys. Rev. Letters, 29,
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Ogilvie, K. W., J. D. Scudder, and M. Sugiura, Electron energy flux in the
solar-wind, J. Geophys. Res., 76, 8165, 1971.
Scarf, F. L., Landau damping and the attenuation of whistlers, Phys. Fluids, 5, 6,
1962.
Scarf, F. L., and R. W. Fredricks, Ion cyclotron whistlers in the solar wind,
J. Geophys. Res., 73, 1747, 1968.
Scudder, J. D., D. L. Lind, and K. W. Ogilvie, Electron observations in the
solar-wind and magnetosheath, submitted to J. Geophys. Res., 1973.
Smith, R. L., and N. Brice, Propagation in multicomponent plasmas, J. Geophys.
Res., 69, 5029, 1964.
Stix, T. H., The theory of plasma waves, pp. 159-162, McGraw-Hill, Inc.,
New York, New York, 1962.
13
Tidman, D. A., Damping of interstellar plasma waves by the cosmic-ray gas,
Astrophys. J., 144, 615, 1966,
Tidman, D..A., and R. K. Jaggi, Landau damping of transverse waves in the
exosphere by fast-particle fluxes, J. Geophys. Res., 67, 2215, 1962.
Wang, J. R., L. A. Fisk, and R. P. Lin, Observations of the Scatter-free solar-
flare electrons in the energy range 20-1000 KeV, Proc. 12th International
Conference on Cosmic Rays, Hobart, 2, 438, (1971).
14
10 10
, I I
0 \\
0 MAXWELLIANE MAXWELLIAN INTERPLANETAR
e INTENSITY \ * ELECTRONS
u)
z (KT ' 40eV)
u
* QUIET- TIME
J
,. OBSERVATIONS
EXCEPT FOR I
z I0 6w
z
I -
104
leV I KeV
ENERGY
F Figure 1. A compilation of quiet-time interplanetary electron data
(after. Lin, et al., 1972)
15


Damping of high frequency waves in the solar wind

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