The preferential acceleration and heating of solar wind heavy ions by the resonant cyclotron interaction were studied. It is concluded that this interaction is incapable of producing the observed differential speeds for reasonable solar wind parameters

Isenberg, P. A.

English

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ACCELERATION OF HEAVY IONS IN THE SOLAR WIND
Philip A. Isenberg
Space Science Center
University of New Hampshire
Durham, NH 03824
Abstract. Several recent studies of the preferential acceleration and heating
of solar wind heavy ions by the resonant cyclotron interaction are reviewed. It
is concluded that this interaction is incapable of producing the observed
differential speeds for reasonable solar wind parameters.
Heavy ions in the solar wind are often observed to have undergone preferen-
tial heating and acceleration relative to protons. At 1 AU, the heavy ions are
seen to flow faster than protons and to have comparable thermal speeds, the
temperatures tending to be proportional to the ion mass. These preferential
effects are positively correlated to the value of the solar wind speed. The
differential speeds of the ions, Av. -Iv. - ÷v L are also correlated with the
1 ' 1 I
local Alfven speed. Helios measurements of a_pha particles showed that, closer
to the sun, the differential speed was often equal to the Alfven speed, even
though the Alfven speed increased substantially in the inner solar system. At
these times, the kinetic energy flux of the alpha particles amounted to almost
one-third of the kinetic energy flux of the solar wind as a whole, and it is
important to understand how so much energy is concentrated into this so-called
"minor" component.
The close connection between the differential speeds and the local Alfven
speed suggests that the responsible mechanism is a wave-particle interaction.
One such mechanism which would favor the energization of low charge-per-mass
p_ti_i_ is Lh_ ±_un_nL uyclutzon i**te_dction. Si1_ce the Solar Wind Four
Conference in 1978, several workers have been investigating the quasi!inear
resonant cyclotron interaction between parallel-propagating waves and solar wind
ions, and this work will be reviewed here.
A parameter study of quasilinear effects of left-polarized (Alfven and
ion-cyclotron) waves on test populations of heavy ions was produced by Dusenbery
and Hollweg (1981). They found that the resonant interaction is capable of
accelerating heavy ions from speeds less than the proton speed to speeds greater
than the proton speed. They also found indications that the resonant interac-
tion could produce mass-proportional heating. However, they also found that the
dispersion relation for ion-cyclotron waves in an electron-proton plasma result-
ed in a critical value of Av. /V A above which the heavy ions were no longer in
resonance with the waves. T_s implies an upper limit to the differential speed
which is locally obtainable from the resonant interaction. For He , this
limiting value is only 0.2 V A. This point was also discussed by McKenzie and
Marsch (1982).
Dusenbery and Hollweg also estimated that approximately 20% of the total
wave energy near the sun would be required to accelerate the solar wind concen-
++
tration of He to the Alfven speed at 0.3 AU, as observed. The problem here is
that the bulk of the solar wind wave energy is seen at low frequencies and the
wave power declines with increasing frequency as P _ _-7 where typically 1.5 _ y
655
https://ntrs.nasa.gov/search.jsp?R=19840005050 2020-03-22T07:43:14+00:00Z
0.7
0.6
0.5
0.4_
0.3
0.2
0.1_
0.0
_ V,.,±/VA) z
(vp,,/v,,)_
_Ve H/VA )2
/UalVA
--
10-2 Re_ 102VA/_,p
Tp±/Tp._l.9
Ta±/Tall _, 1.8
-0.1
-0.2 ,
-0.3
20.00
J , I , L k , h
20.02 20.04 20.06 20.08 20.10
HELIOCENTRIC DISTANCE Z=R/R e
Figure i. Calculated proton and
alpha speeds relative to their cen-
ter of mass frame and squared ther-
mal speeds, all normalized to the
local Alfven speed (after Marsch
et al., 1982).
< 2.0. The enerqy in the high-frequency, resonant regime which is available to
the cyclotron interaction is several orders of magnitude too low.
This work, however, was a calculation of the local energization of test
heavy ions, and as such, it did not consider two important points. Firstly, a
spectrum of waves which can cyclotron resonate with heavy ions will also inter-
act with the solar wind protons. One must study this simultaneous interaction
and determine the extent to which the heavy ions are accelerated and heated in
excess of the protons. Secondly, results of this local calculation must still
be incorporated into a solar wind model to determine the evolution of the heavy
ion populations under the continuous action of the resonant process allowing for
the ongoing adjustment of the model parameters (such as the Alfven speed and the
wave power) in the expanding solar wind, as well as for the effects of other
participating forces. In particular, a local upper limit of _v. /V_ is not
l
necessarily significant when V A is decreasing: a process which _ou_d produce
Avi = VA/3 at 0.i AU might result in AVip = V A at 0.3 AU if the actual particle
spe_ds do not change.
The first work to deal with these questions was Marsch et al. (1982). They
considered a system containing bi-Maxwellian distributions of protons and alpha
particles and an initial power-law spectrum of parallel-propagating waves.
Starting the calculation at 20 solar radii (_ 0.I AU), they investigated the
self-consistent interaction, following the evolution of the wave spectra, as
well as the velocities and temperatures of the particles. They included the
effects of the shifting model parameters as a function of radius, but did not
consider any non-resonant forces except those produced by a radial magnetic
field on a plasma with a double-adiabatic equation of state.
Their first result was that, as expected from the work of Dusenbery and
Hollweg, the observed wave power levels in the resonant frequency range were too
656
small to produce any significant effects. To continue the investigation, the
wave levels were increased. Figure 1 shows an example of the normalized veloc-
ities and squared thermal speedswhich result. The strong preferential heating
and acceleration of alpha particles is obvious, and it is clear that the cyclo-
tron resonant interaction can accelerate alpha particles through the proton
speed. Note, however, that the total scale only extends over 0.i R (R = solars . s
radius). In this m_el, the wave effects proceed very quickly, energlzlng the
particles within i0 -R . The wave energy, self-consistently calculated, is
essentially depleted b_ this time and further energization takes place on the
muchslower convective scale as the expanding solar wind and decreasing field
magnitude cause a rescaling of the resonant parameters. It would be interesting
to follow this model system further to investigate this slower energization, but
this evolution is obscured by another aspect of this model.
0.6
0.5
0.4
05!
02
0.1
0.0
2O
TalITaii_093
(Val/VA)Z /]
f _ (va"/vA)z
(vp/vA) z ....
\(Vp./VA )2 Tpl/Tp. _ 1.6
I L _ _'__ I I ' r721 22 23 24 25 26 2 28
HELIOCENTRIC DISTANCE Z=R/Re
:5.2
:5.0
2.8
2.6
rY
uJ 2.4
m
_2.2
zD 2.0
_ L.8
_ 1.6
1.4
1.2
t.0
0.81
2O
MAGNETOSONIC
WAVES (__.r
ION CYCLOTRON
WAVES (LHP) /
/
J AV_p _ 1.2VA
.....
I 1 t I I I I
21 22 2:5 24 25 26 27 28
HELIOCENTRIC DISTANCE Z=R/P_
(_) (b)
Figure 2. As in Figure i, except that the speeds shown in (b) are shown with
respect to the inertial frame, M i E vi/V A (after Marsch et al., 1982).
The resonant interaction which produced the results in Figure 1 took place
with left-polarized ion-cyclotron waves. Marsch et al. have also included the
effects of a resonant cyclotron interaction with an equally intense spectrum of
right-polarized magnetosonic waves. This is motivated by the observation that
T > T in the fast solar wind, which could be a signature of fast-mode heating.
z_llis _also suggested that this interaction could explain why Av is limited to
the Alfven speed. According to this model, the energization bye_he ion-cyclo-
tron waves, along with the decreasing phase speed in the expanding solar wind,
allows the alpha particle distribution to move into resonance with the right-
polarized waves, and this interaction completes the acceleration to Av = V .
In the example shown in Figure 2, this has all occured within six R . _t isA
certainly plausible that, once the alpha particles have attained differential
speeds comparable to the Alfven speed, an interaction with magnetosonic waves
could produce parallel heating and prevent Av /V from increasing further as V
declines. However, it is hard to see how th_e_as_ mode is itself responsible A
for the acce]eration up to Av = V A. In order for an alpha particle to cyclo-
tron resonate with the right-p_larized mode, it must be travelling along the
657
field at more than twice the Alfven speed with respect to the bulk plasma. Thus,
only the high-velocity tail of the distribution will be able to i_teract with
these waves. For the case shown in Figure 2, less than 5.7 x I0-- of the alpha
particles have this necessary speed at 25 R , where the model energization by
the magnetosonic mode has already started, sit is difficult to see how this
enormous acceleration and heating of the entire distribution from an interaction
with a tiny fraction of the particles can be other than a numerical artifact of
the model.
Another model which deals with some of these questions from a different
point of view has been presented by Isenberg and Hollweg (1983). This work does
not include as detailed a picture of the microscopic interaction as Marsch et
al., but it incorporates a reasonable simulation of the interaction into a full
solar wind model which includes all other forces of interest. The central point
of this model is that the energy problem is dealt with by invoking a "saturation
and cascade" scenario. The authors assume that large-amplitude Alfven waves in
the solar wind saturate when <_B2>/B 2 = ½, where <_B2> is the total magnetic
o
variance (integrated over the entire spectrum). It _s then hypothesized that
the energy lost from the waves cascades to resonant frequencies where it is
picked up by the particles through the resonant cyclotron interaction. The
details of these nonlinear processes were not specified, as their physics is not
well understood, but it was pointed out that this scenario is plausible and
consistent with observations.
To incorporate this scenario into a solar wind model, one needs information
on the propagation and dissipation properties of the waves in a multi-ion dif-
ferentially flowing plasma. Unfortunately, at present, the only available
theory of this type is restricted to Alfven waves in a thermally isotropic
plasma (Isenberg and Hollweg, 1982). For this reason, the model particle dis-
tributions were taken to be isotropic and the waves were taken to be nondisper-
sire, as well as parallel-propagating and left-polarized. The wave spectrum was
taken to be a power law with constant spectral index _. To simulate the inter-
action with ion-cyclotron waves, the spectrum was cut off at _ = _ in the
proton reference frame. P
The energization calculation proceeded as follows: The total plasma
heating rate was obtained from the divergence of the flux of wave action of the
saturated waves. This total was then distributed to the various ion species by
taking the heating rate for each species proportional to the appropriate quasi-
linear resonant cyclotron heating term. The wave acceleration of each species
was then the sum of the non-dissipative Alfven wave pressure and a term propor-
tional to the dissipative heating. This procedure allowed energy to be supplied
to the resonant particles on a convective time scale so that high resonant power
levels were no longer required. These wave effects were incorporated into the
spherically-symmetric, three-fluid, corotating solar wind equations which were
integrated from i0 R out to 1 AU.
s
A typical result is shown in Figure 3. For this case, the proton and wave
parameters were taken from the wave-driven high speed stream model of Hollweg
(1978). The alpha particles, 4% by number, had their velocity and temperature
set equal to the protons', and the spectral index was _ = 2. This result shows
definite preferential acceleration and heating when the waves saturate at 19 R
s'
but the differential speed falls far short of the observed Alfven speed value.
658
"_ 6OO
E
400
800 ' ' ' ' I ' ' ' ' I I ' ' ' I ' ' ' ' I ' ' ' w
'\Vp ÷VA
200 , , , , l , a , , I , , , i I , , , , I ....
0 50 I00 150 200 250
r(R s )
o
t-
i0 "t
1 ' , , , , ,,I ,
I06[-.. _ He++H+
105 - e-
:,..,j
Io4 ,
I0 ...... I'02 ....... I03
r(Rs)
(a) (b)
Figure 3. Ion radial speeds (a) and temperatures (b) from the model of Isenberg
and Hollweg (1983) for y = 2. Also shown in (a) is the radial component of the
proton plus Alfven speed, and in (b) is the electron temperature.
Furthermore, the temperature ratio at 1AU is only T /Tp = 3.6, where we would
hope to produce T /Tp _ 4.
The preferential effects shown here proved essentially insensitive to
changes in the initial particle velocities and temperatures. Increasing the
initial wave intensity generated faster, hotter populations, but the values of
Avid and T /T changed very little. The only parameter left to vary was the
__i i_de_ and _t _- easy to see w,i_-_--y is important. The quasilinear _eso-
nant interaction is proportional to the wave power at the resonant frequencies.
When the protons and heavy ions are not moving with respect to one another, they
resonate with waves near their respective gyrofrequencies. In the simplest
picture, then, the power available to the ions is a factor of (_./_)Y larger
• 1
than that for the protons where _. = q.B/(m.c). Clearly, uslng a s_eeper wave
• l . i . .
spectrum will _ncrease the preferentma_ energmzatmon of the heavy ions.
The magnitude of the interaction is also proportional to the (charge/mass) 2
of the particle, so one m_ght expect the total resonant acceleration to be
proportional to (A_/Z_) 7- . This simple picture would imply that a heavy ion
could not be prefe_en£ially accelerated by this process unless y > 2. In
reality, this picture is complicated by the thermal speeds, which smear out the
resonance; by the differential speed, which Doppler-shifts the resonances to
different relative points; and by the fall-off in wave power at _ . It turns
out to be the power cutoff that is most important. If the cutoffPis not in-
cluded and the model wave spectrum is continued with the same slope to infinite
frequency, the simple picture is verified by the numerical calculations: there
is no preferential acceleration of heavy ions for y < 2. The cutoff in the
spectrum reduces the power in the proton resonant range and allows helium to be
preferentially accelerated for Y > 1.5. To produce stronger preferential
effects, y must be increased to rarely observed values > 2, and it takes a Y =
4.7 to produce alpha particles with AVap = V A at 60 Rs as shown in Figure 4.
659
800 .... r .... I .... I ' , , , I ....
_ Vp + VA
x
600 ....................
_ 4oo
200 ......... ' .... _L .... _ ....
0 50 I00 150 200 25(
r(Rs)
108
107
A
106
t--
105
i04
I0
' ' ' ' ' '1
(_He ++
I I I I I I II
I02 i03
r(R s)
(a) (b)
Figure 4. As in Figure 3 for y = 4.7.
Once the alpha particles have been accelerated to Av > V resonant interac-
% A
tions with right-polarized magnetosonic waves can bec_me important. Beyond 80
R , in the case shown in Figure 4, almost 20% of the alpha population has suffi-
cient parallel speed to resonate with the fast mode. However, this number is
still small (% 3% at 40 R ) before the Alfven speed is reached and the right-
hand interaction cannot p_ay a significant role in the acceleration to V A.
Heavier ions, with lower gyrofrequencies, are able to resonate with more
intense waves and should be accelerated to higher speeds. However, this _s only
true if y > 2 since the dissipative processes are proportional to (A/Z) T-z.
When y = 2, the determining factor is still the reduced proton resonant power
due to the spectral cutoff, and the heavy ion velocities are the same as the
alpha particle curves in Figure 3. When Y is increased it is found that oxygen
ions, with A/Z near that of helium, are only slightly more accelerated. Acceler-
ation of iron ions, on the other hand, is significantly enhanced over alpha
+u
particles, especially in the lower ionization states, such as Fe , whlch
reaches V at 50 R when Y = 3 Calculations by Marsch (1983) generate similar
A
results for the interaction with left-polarized waves.
In summary, the Isenberg and Hollweg model of the resonant cyclotron
interaction cannot explain the observed preferential acceleration and heating of
heavy ions in the solar wind. To produce differential speeds on the order of
those observed requires extremely steep wave spectra. Although there have been
no direct observations of spectral slopes inside 60 R , all indications are that
. .S.
the spectra should be flatter there and spectral indlcles of y = 4 - 5 for m <
seem unlikely. The calculated temperature ratios, Ti/T P, at 1 AU are also
c_nsistently lower than the mass-proportional values.
Several questions remain, for instance: What is the effect of allowing
anisotropic particle distributions or including a spectrum of dispersive waves?
These modifications should be investigated, but I suspect they will not provide
660
substantially more preferential acceleration of heavy ions. The resonant
interaction tends to energize particles in the perpendicular direction, so the
requirement of isotropy amounts to an effective transfer of perpendicular energy
into parallel energy and it does not seemthat relaxing this assumption will
produce greater acceleration. Furthermore, dispersive ion-cyclotron waves would
interact with a smaller portion of the ion distribution than the assumed non-
dispersive spectrum does, and would presumably be less effective in energizing
the ions. It may also be important to consider the effect of including the
solar wind alpha particles in the dispersion relation of the waves. Preliminary
work (Isenberg, 1982) has shown that there may actually be no waves to acceler-
ate heavy ions to speeds faster than the proton speed by the resonant interac-
tion.
In conclusion, it appears that the resonant cyclotron interaction cannot be
responsible for the observed preferential acceleration and heating, and we must
look elsewhere to find a mechanism which produces the observed effects.
Acknowledgements. I am grateful for valuable conversations with J.V. Hollweg
and E. Marsch. This work was supported in part by NASA Grant NAG-5-130, and by
the NASA Solar-Terrestrial Theory Program under grant NAGW-76.
References
Dusenbery, P.B. and J.V. Hollweg, Ion-cyclotron heating and acceleration
of solar wind minor ions, J. Geo_hys. Res., 86, 153, 1981.
Hollweg, J.V., Some physical processes in the solar wind, Rev. Geophys.
Space Phys., 16, 689, 1978.
Isenberg, P. A., The cyclotron resonance condition in a proton-alpha
solar wind, EOS Trans. AGU, 63, 1087, 1982.
Isenberg, P. A. and J.V. Hollweg, Finite-amplitude Alfven waves in a
multi-ion plasma: Propagation, acceleration, and heating,
J. ueophys. Res., 87, 5023, 1982.
Isenberg, P. A. and J.V. Hollweg, On the preferential acceleration and
heating of solar wind heavy ions, J. Geophys. Res., in press, 1983.
Marsch, E., Resonant wave heating and acceleration of heavy ions in the
solar wind, J. Geo_hys. Res., submitted, 1983.
Marsch, E., C. K. Goertz, and K. Richter, Wave heating and acceleration
of solar wind ions by cyclotron resonance, J. Geo_hys. Res., 87,
5030, 1982.
McKenzie, J.F., and E. Marsch, Resonant wave acceleration of
minor ions in the solar wind, Astro_hys. Space Sci., 81, 295, 1982.
661


Acceleration of heavy ions in the solar wind

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