On the acceleration and wave heating of the solar wind:
implications of the mean free path of solar energetic particles
T. Laitinen , R. Vainio 1 and H. Fichtner†
†
Space Research Laboratory, VISPA and Department of Physics, University of Turku, Finland
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik,Ruhr-Universität Bochum, Germany
Abstract. Wave damping and cascading processes have been found to be important for the heating and acceleration of the
solar wind. However, it remains a difficult task to extract details of these processes from observations of the thermal plasma
only. The wave power required for efficient heating and acceleration of the solar wind also affects the acceleration and transport
of solar energetic particles. Thus, their observation could provide valuable clues for the actual evolution of the wave power
close to the coronal base and, in turn, give constraints for solar wind modeling. Pursuing this idea, we have developed a
steady-state two-fluid model for the wave heating and acceleration of the solar wind. The dissipation frequency determining
the heating is obtained from a cyclotron damping rate that depends on the plasma beta and, thus, differs from the usual
assumption, a fixed fraction of the ion cyclotron frequency. We present first results obtained with the two-fluid code and, in
particular, discuss the implications of the corresponding mean free path of energetic particles.
INTRODUCTION
Tu and Marsch [2]. According to their results, the wave
power needed for heating was orders of magnitude too
high to explain the properties of energetic particles observations in equatorial plane.
The study made in [4] however lacked solar wind modeling and was in this sense not consistent. Thus, in order to improve the study, we have created a steady-state
two-fluid solar wind model using the self-consistent dissipation frequency, and re-iterated the energetic particle
study. The model itself is presented in more detail in
Laitinen et al. [6], this paper concentrates on the consequences for energetic particles.
Wave related phenomena are considered a strong candidate for heating solar wind on open field lines. Waves
transfer energy efficiently from the sun to corona, and exert pressure to drive the plasma. Cyclotron damping has
been suggested to strongly contribute to heating [1], and
a global model using dissipation at cyclotron resonance
for wave heating was presented by Tu and Marsch [2].
Subsequently the model has been extended to include
cascading that efficiently transfers wave power to dissipation [3]. These models compare well with observations
of coronal velocity, density and temperature profiles in
the fast solar wind.
However, information on the available wave power is
more difficult to obtain. Recently, Vainio and Laitinen
[4] suggested that solar energetic particles could provide an important observational constraint for the waves.
While the transient energetic particle events, related to
solar flares or coronal mass ejections (CMEs), do not
contribute significantly to the solar wind dynamics, the
waves isotropize the energetic particle pitch angle distributions very efficiently [e.g., 5]. Vainio and Laitinen [4]
estimated the energetic particle mean free path, an essential parameter for both particle acceleration and transport, resulting from the requirements for the model of
SOLAR WIND MODEL
The basic solar wind model is based on a set of hydrodynamical two-fluid equations for mass, momentum and
energy continuity. We include Coulomb collisions and
heat conduction for both electrons and protons, but neglect radiative losses. The proton heating is related to the
wave power by Tu and Marsch [2] as
Qp
u vA cos ψ
PL fH r d f H
4π
dr
(1)
where PL f r is the wave power of left-hand polarized
waves. The wave power spectrum evolves radially as
1
Present address: Department of Physical Sciences, University of
Helsinki, Finland
P f r
θ f T r f P0 f CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
303
M0 M0 12
M rM r 12
(2)
FIGURE 1. Examples of the dissipation and the spectral termination frequencies (dashed curve and filled circles, respectively) and contours of the relative damping decrement (solid
curves) at levels 0.01, 1, and 100.
FIGURE 2. Solar wind solution for typical slow solar wind
(solid lines) and coronal hole (dashed lines). On the top left
panel thick lines represent the wind velocity, thin lines Alfvén
velocity. On bottom left panel thick lines represent proton
temperature, thin lines electron temperature. The bottom right
panel shows the spectral power at f0 1 Hz.
where M uvA cos ψ is the Alfvén Mach number and
vA B 4π m p n, and f T r is the frequency at which
the spectrum terminates. The spectrum at the solar surface is a single power law P0 f P¬ f 1 . The magnetic field is described by a parametrization suggested
by Kopp and Holzer [7], which allows faster than radial
expansion close to the sun, with radial behaviour in the
interplanetary space, and Archimedean spiral with spiral
angle ψ . We also assume vanishing net magnetic helicity, i.e. PL PR , which results in approximately two times
stronger wave pressure gradient for momentum equation,
compared to [6] and earlier solar wind heating models.
In earlier approaches the dissipation in cyclotron heating models has been taken to occur at a fixed fraction
of the local ion cyclotron frequency. We employ a more
self-consistent treatment, suggested by Vainio and Laitinen [4], which includes the thermal effects and results in
dissipation frequency
fH r
Y r Ωp
2π
Y
β p ln lnY
ENERGETIC PARTICLE TRANSPORT
The standard quasilinear theory (SQLT) relates Alfvén
waves with energetic particle transport through the
Fokker-Planck pitch angle diffusion coefficient
Dµ µ
1 2
where Ω p
and β p 8π nT B2 the proton plasma beta.
As shown in Fig. 1 (dashed line), this form of dissipation frequency is not necessarily a monotonously decreasing function of heliocentric distance. If the spectral
termination frequency f T fH , there are no waves available for heating. Thus, if the waves are generated only at
solar surface, we may write f T as
fT r
min fH r¼ r¼ r µ 2
kres I kres B2
(4)
where the resonant waves are described by the wavenumber spectrum I k, and the particles scatter at a single
Ωvµ .
resonant wavenumber, k res
In the case of a spectral cut-off, the used form of
the resonance condition will cause a resonance gap on
pitch angles µ ΩvkT , where there are no waves
to scatter particles. The problem of such resonance gap
has been addressed by several authors, with modeling of
thermal damping [8], turbulent spectral evolution [9] or
medium scale fluctuations [10], which act in widening
the resonance, or with a power law dissipation region
spectrum with waves propagating in both directions [11].
Here we assume that these effects close the resonance
gap efficiently and use the SQLT model of the pitch angle
diffusion coefficient.
If we further assume vanishing net magnetic helicity, we may derive the wave number spectrum from the
frequency power spectrum by k I k 2 f PL f 2,
u vA k 2π . The mean free path λ is
where f
then, for power law spectrum with spectral index of 1,
given by
1
π Ω r r p
¬
u vA cos ψ
eB m p c is the proton cyclotron frequency
u
vA cos ψ
π
Ω 1
2
λ
(3)
1 µ2
3v 1
dµ
8 1
Dµ µ
2
B2
2 v
,
π Ω 2P¬L Rr
where Rr is the radial dependence in Eq. (2).
304
(5)
FIGURE 3. Energetic particle SQLT mean free path for 10
MeV protons, for slow wind (solid curve) and a coronal hole
model (dashed curve).
FIGURE 4. An energetic particle event (solid curve) compared with transport simulation in slow (dashed curve) and fast
(dash-dotted curve) solar wind solution for energy 12-20 MeV,
with wave power reduced to 10 %.
RESULTS AND DISCUSSION
fixed to an X-ray flare maximum. The slow wind model,
representative to equatorial conditions, on the other hand
has to t f 15, clearly contradicting the observations.
The coronal hole fast wind, on the other hand, provides
very fast convection and, thus, the onset occurs earlier,
with to t f 7. The three events during ULYSSES’s first
south polar pass reported by Bothmer et al. [15] give the
ratios of around 10 during quiet sun conditions. Dalla
et al. [16] analyzed the events during the second pass,
during active sun conditions, and found the transport to
be much faster, with t o t f 1 – 3 in fast wind events.
This discrepancy would suggest that fast wind streams
are significantly different during quiet and active sun.
However, the low statistics make more accurate analysis
difficult, as e.g. bad magnetic connection to acceleration
site may cause large delays. In addition, as the spacecraft
was over 3 AU from the sun for the first pass events
and around 2 AU for the second, onset time comparisons
between the two and with 1 AU simulations may be
unreliable and difficult to interpret.
Waves contribute also to particle acceleration in
corona and interplanetary space, as the estimated mean
free path provides favorable conditions for particle
acceleration by CME driven shocks. To demonstrate
this, we simulate shock acceleration with a Monte Carlo
code [13], using the slow wind profiles and a shock with
speed us 1000 km/s. Particles of spectrum ∝ p 5 , from
50 keV to 50 MeV, are injected as an outward directed
beam in front of the shock as the shock is launched at
the sun. The intensities observed at 0.2 AU are shown in
Fig. 5, as a function of time and energy.
The acceleration is limited by adiabatic focusing in
the diverging magnetic field. Particles that are focused
more efficiently than diffused back towards the shock,
escape the shock and are not accelerated further. The
We have used the solar wind model to generate two distinct cases of solar wind to represent a typical slow solar wind and a coronal hole with fast wind. The velocity, density, temperature and spectral power profiles are
shown in Fig. 2. The magnetic field geometry for the
coronal hole wind is super-radial, with the magnetic field
6 times higher than for the slow wind model in the low
corona. Both models require coronal base wave power of
P¬L 5 10 4G2 .
The SQLT energetic particle mean free paths derived
for these winds, for 10 MeV protons, are presented in
Fig. 3. In the corona, the stronger magnetic field of the
fast wind model results in a longer mean free path, as
suggested by Eq. (5). The interplanetary mean free path
for both models remains very short, and the resulting
energetic particle transport is dominated by convection,
as the diffusion time scale, t d r2 2vλ , is an order of
magnitude longer than the convection time scale.
We compare the resulting transport with an energetic
particle event observed by ERNE instrument onboard
SOHO spacecraft on 28 December 1999, analysed in
more detail by Torsti et al. [12]. We use Monte Carlo
simulations [see, e.g., 13] of 12 – 52 MeV protons with
a spectrum ∝ p 2 , where p denotes particle momentum.
We increase the mean free path by an order of magnitude
from the SQLT values, in order to ensure that it isn’t
underestimated due to the assumptions of the SQLT.
Mean free paths obtained from the energetic particle
observations can be more than an order of magnitude
longer than what the SQLT suggests, as phenomena such
as less scatter-effective wave modes are ignored [14].
As shown in Fig. 4, the observed event onset, t o =1.6 h,
is approximately twice the the scatter free arrival time
(t f 07 h for 20 MeV protons to 1 AU), with injection
305
characteristic velocity v c of the escaping particles can
be obtained from v c λ vc r3u1 r2 where u1
us u vA is the scattering center velocity in shock
frame [13]. This is shown in Fig. 5 as a function of
t r r¬ us .
As can be seen, the high-energy particles escape
the shock immediately, without re-acceleration. Particles
with lower energies, on the other hand, are either trapped
to the acceleration region or escape downstream. As a
result, lower energy particles are not observed until the
shock gets closer to the observer, contradicting observations.
Thus, both particle transport and acceleration observations suggest that the scattering wave power must decrease in the interplanetary space. This can be achieved
by including spectral transfer, as in [3]. Also the effect of
minor ions on the wave power should be considered, as
noted by Tu and Marsch [17], as well as the effect of 2D
fluctuations on heating.
Energetic particle observations may provide valuable
information also for wave power reduction mechanisms.
An increase in the mean free path may result in a gradually decreasing characteristic energy, which would subsequently be observed as a delayed release of lower energy particles. The observed intensities will thus contain information of the radial evolution of the wave
power. This information may also be obtained from
analysing interplanetary transport of flare accelerated
particles. Both studies would benefit from observations
made closer to the sun, as long interplanetary transport
brings uncertainty to the analysis.
FIGURE 5. Energetic particles accelerated by a shock with
us 1000 km/s as observed at 0.2 AU. Contours are set for
each half order of magnitude. The solid line describes the
characteristic energy for the shock acceleration
tant contribution to understanding solar wind heating.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
CONCLUSION
9.
We have studied an important constraint for solar wind
modeling provided by energetic particle observations.
We generated a typical slow and fast solar wind using
a new, self-consistent treatment for the wave dissipation
frequency, suggested by [4], which includes the thermal
dependence of the cyclotron damping process.
We find that the slow wind produces interplanetary
energetic particle transport conditions inconsistent with
energetic particle observations. However, in the corona,
the short mean free path provides favorable conditions
for particle acceleration by e.g. CME driven shocks. The
fast wind transport may fit better to the model results,
but as the observations are sparse and made at larger
distances, no definite conclusions can be made.
The phenomena reducing wave power must thus be included in solar wind modeling, to the extent that energetic particle observations may be explained. Energetic
particle observations made closer to the sun with the
planned Solar Orbiter spacecraft will provide an impor-
10.
11.
12.
13.
14.
15.
16.
17.
306
Marsch, E., Goertz, C. K., and Richter, K.,
J. Geophys. Res., 87, 5030 (1982).
Tu, C.-Y., and Marsch, E., Sol. Phys., 171, 363 (1997).
Hu, Y. Q., Habbal, S. R., and Li, X., J. Geophys. Res.,
104, 24819 (1999).
Vainio, R., and Laitinen, T., A&A, 371, 738 (2001).
Kallenrode, M., J. Geophys. Res., 98, 19037 (1993).
Laitinen, T., Fichtner, H., and Vainio, R., J. Geophys. Res.
(2002), manuscript no. 2002JA009479. Accepted.
Kopp, R. A., and Holzer, T. E., Sol. Phys., 49, 43 (1976).
Schlickeiser, R., and Achatz, U., J. Plasma Phys., 49, 63
(1993).
Bieber, J. W., Matthaeus, W. H., Smith, C. W., et al., ApJ,
420, 294 (1994).
Ng, C. K., and Reames, D. V., ApJ, 453, 890 (1995).
Vainio, R., ApJS, 131, 519 (2000).
Torsti, J., Kocharov, L., Laivola, J., Pohjolainen, S.,
Plunkett, S. P., Thompson, B. J., Kaiser, M. L., and
Reiner, M. J., Sol. Phys., 205, 123–147 (2002).
Vainio, R., Kocharov, L., and Laitinen, T., ApJ, 528, 1015
(2000).
Bieber, J. W., Wanner, W., and Matthaeus, W. H.,
J. Geophys. Res., 101, 2511 (1996).
Bothmer, V., Marsden, R. G., Sanderson, T. R., Trattner,
K. J., Wenzel, K.-P., Balogh, A., Forsyth, R. J., and
Goldstein, B. E., Geophys. Res. Lett., 22, 3369 (1995).
Dalla, S., Balogh, A., Krucker, S., Posner, A., MüllerMellin, R., Anglin, J. D., Hofer, M. Y., Marsden, R. G.,
Sanderson, T. R., Heber, B., Zhang, M., and McKibben,
R. B., Annales Geophysicae (2002), Submitted.
Tu, C.-Y., and Marsch, E., J. Geophys. Res., 106, 8233
(2001).