The Microscopic State of the Solar Wind
Eckart Marsch
Max-Planck-Institut für Aeronomie, 37191 Katlenburg-Lindau, Germany
Abstract. The microscopic state of the solar wind is reviewed, in particular the measurements and models of proton and
electron velocity distributions and kinetic features of heavy ions in the fast solar wind and coronal holes. Apparently, electron
distributions are largely determined by Coulomb collisions. Concerning the ions, there is mounting evidence that pitch-angle
diffusion in resonance with ion-cyclotron waves is the main process forming the shape of ion velocity distributions. Moreover,
the absorption of high-frequency waves seems to play a major role in the heating of the corona and solar wind. Dispersive
plasma waves and associated wave-particle interactions are the key to this problem. Plasma stability analyses and model
calculations, as well as observations adressing these subjects are briefly reviewed, while focussing on the critical issues.
an Alfvén wave, Mm. Concerning the microstructure or kinetic properties we have at 1 AU as the
important scales: Coulomb free path, AU;
ion inertial length, km; ion gyroradius,
km; and Debye length, m. For comparison, the spacecraft diameter m. Here is
the Alfvén speed and the proton gyrofrequency.
Kinetic processes in the solar corona and solar wind
are important, because the plasma is dilute, multicomponent and nonuniform. As a result of this and
macroscopic forces, significant deviations from local
thermal equilibrium arise, and complexity is caused in
phase space through strong distortions of the VDFs in
the thermal regime and by the occurence of suprathermal particles, e.g., the electron strahl or nonthermal ion
beams and ion differential streaming. Owing to the weak
collisionality there is a remaining influence of the global
boundary conditions (in the corona), which are still reflected locally in the microscopic state of the solar wind.
INTRODUCTION
Recent observational and theoretical results on the microscopic state of the solar wind are reviewed, in particular the proton velocity distributions (VDFs) and kinetic
features of heavy ions in the fast solar wind and coronal
holes. We summarize some of the well established observations of electrons, and discuss especially the Coulomb
collision effects on their VDFs. Whereas for the electrons collisions are sufficient to explain the main kinetic
features, for the ions wave-particle interactions are the
key processes shaping their VDFs.
There is mounting evidence that pitch-angle diffusion
(PAD) of thermal protons, in resonance with dispersive
ion-cyclotron waves, is essential. Plateau formation plays
a crucial role in the wave absorption or opacity for kinetic
waves near the ion cyclotron resonances [18]. PAD has
been included in recent kinetic models of the evolution
of proton VDFs in the solar corona and wind, in addition
to collisions. Finally, regulation mechanisms through microinstabilities, e.g for the ion temperature anisotropy,
are briefly discussed.
Ions
From [3] we reproduce here Figure 1, for reminding the reader of the proton VDFs. The most prominent features are: Proton beams (B, E, H), temperature anisotropy in the core, with (F, J, H),
and ion differential motion with (see, e.g.
[22]) not shown here, whereby all heavies appear to
drift together with the alpha particles ahead of the protons in fast streams and reveal the temperature scaling, , indicating preferential heating
by waves more than in mass proportion. The free energy contained in these nonthermal features can be converted by microinstabilities into plasma waves of vari-
VELOCITY DISTRIBUTIONS
This paper reviews selected results, returned from various mission, on particle VDFs in the solar wind. Comprehensive surveys are available in the reviews [15], [3]
and [20]. The solar corona and wind are inhomogeneous, and their plasma parameters show strong gradients with respect to distance from the sun. Concerning
the macrostructure or fluid properties we have as main
scales: heliocentric distance, AU ( Gm
); solar radius, km; wavelength of
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
399
tle temperature gradient, i.e. . The electron
energy spectra obtained from Ulysses have often been
fitted [13] by kappa functions (generalized Lorentzians
with ), which have much stronger tails than
a Maxwellian for the hot halo but the wrong curvature of the spectrum at suprathermal energies. Moreover,
the electron pitch-angle distributions are non-isotropic
and highly focused (strahl) in fast wind at higher energies (beyond about 100 eV) according to the still bestresolved measurements made by Helios [21]).
Therefore, that suprathermal electrons could drive solar wind solely through the electric field is not compatible with the new coronal and old in-situ observations.
The expanding corona is not an electron exosphere, but
a weakly collisional medium for which Liouville’s theorem cannot be exploited straightforwardly. Electrons do
not matter dynamically in the solar wind acceleration,
and velocity filtration is a negligable effect [9]. As a result, the fast solar wind ions have certainly to be driven
by their own pressure-gradient force (see the reviews
[19] and by Hollweg, in this volume).
FIGURE 1. Solar wind proton velocity distribution functions
as measured by Helios at different locations and for various
flow speeds. Contours correspond to 80, 60, 40, 20, 10, 1, 0.1
percent of the maximum.
KINETIC TRANSPORT EQUATIONS
ous kinds, most prominently ion cyclotron waves propagating quasi-parallel to the background magnetic field.
However, sometimes collisions are strong enough in slow
wind to establish local Maxwellians (A, D).
General remarks
The most challenging but complete description of the
expanding corona and the solar wind is by means of
the Boltzmann-Vlasov kinetic equations for the ions,
i.e. protons, alpha-particles (with typically in abundance), minor heavier ions, as well as for the electrons
(core, halo, and strahl). Such a kinetic description of
the microscopic state of the corona and wind has, in the
past and still today, been a numerically almost intractable
problem, because the solar wind is multi-component and
the plasma is nonuniform, rather dilute and turbulent on
the fluid as well as microscopic scales. Yet, serious attempts have been made to overcome these difficulties.
Electrons
Solar wind electrons (see, e.g., Figures 1-3 in [3])
are characterized by two main populations: The thermal core electrons (typically 96%), which are bound
to the Sun by being trapped in the electrostatic potential well associated with the interplanetary electric field,
, and the suprathermal halo electrons (about
4%), which can escape from the Sun and surmount the
potential barrier. In simplified terms, the core population is local, collisional, bound by the electrostatic potential ( eV between Sun and Earth) and
reflected by the magnetic mirror in the corona, whereas
the halo population is more global, almost collisionless,
and free to escape to the outer heliosphere. Yet, halo
electrons may scatter by magnetic inhomogeneities and
wave-particle intercations.
The electrons in slow wind (from the closed streamer
belt in the corona) are comparatively hot, but the electrons from coronal holes that are open magnetically are
much cooler [2]. The halo electrons seem to carry most
of the heat [14] and are hotter, with . Heat conduction in the solar wind does not obey Fourier’s law,
resulting from a small skewness associated with a gen-
Coulomb collisions of ions and electrons
What is theoretically well understood is collisional
transport in terms of the Fokker-Planck operator. The importance of Coulomb collisions varies strongly with distance from the solar surface. Table 1 gives typical numbers for the electron density, temperature and collisionfree path. Note the wide range of parameters involved.
Concerning collisional effects on solar wind ions, in
[11] a kinetic equation with a simple relaxation term was
integrated. Since in most types of wind , with being the number of collisions of a proton per transit time
through the scale height , Coulomb collisions require a
400
FIGURE 2. Solar wind electron velocity distribution functions as modeled by the kinetic Boltzmann equation including Coulomb
collisions via the Landau collision integral. Left: isocontours at 54.9 and 29.7 (bottom) with pitch-angle focussing. Right:
corresponding cuts along the field illustrating the distinct skewness on the anti-sunward and clear cutoff on the sunward side.
TABLE 1.
Parameter
/cm
/K
/km
Collisional free path and plasma parameters
Chromosphere
(1.003 )
10
Corona
(1.2 )
(1-2)
model VDFs compare well with the observed ones, qualitatively in the pitch-angle distributions and even quantitatively in the energy spectra. In [24] it has recently been
shown that the contribution of the halo to the total electric field (in terms of the halo partial pressure gradient)
is dynamically negligable.
In conclusion, the behaviour of the majority electrons
is essentially understood on the basis of collisions alone,
and with the exception of the strahl electrons any waverelated process is hardly needed. Electrons seem to play
a passive role [19] in the acceleration of the solar wind
(which is to say of protons and other ions).
Solar wind
(1 AU)
10
kinetic treatment. It can be shown, though, that even very
few collisions suffice to remove the otherwise extreme
exospheric anisotropies. In slow wind one has for
about 10% of the time, and for about (30-40)%.
Therefore, here collisions do certainly matter, a result
confirmed also by the Ulysses measurements at lower
heliographic latitudes (see the review [20]).
The most advanced model for the collisional evolution of solar wind electrons has been presented in [9], in
which the Fokker-Planck equation for electrons was integrated in a realistic background model solar wind. Some
of the results are illustrated in Figure 2. Coulomb collisions were found to maintain a fairly isotropic core to
large heliocentric distances. The VDF is determined primarily by the electric field and the expanding geometry,
whereby velocity filtration is a rather weak effect. The
Quasilinear diffusion of ions
However, in order to explain the detailed kinetics of
ion VDFs, wave-particle interactions are indispensible,
and the Boltzmann equation complemented by wave
terms must be used. If the wave amplitudes are small
and the spectra broad in Fourier space, which is the
case in the solar wind kinetic regime, quasilinear theory (QLT) should be adequate to describe the waveparticle couplings. The diffusion equation for any type
of waves, propagating obliquely to the field in a magne-
401
are: , the frequency of a linear wave mode in
the plasma frame, and the wave vector. Strong waveparticle intercation, and thus diffusion in the wave frame,
occurs whenever an ion at speed fulfils the resonance
condition:
(3)
with the gyrofrequency , speed of light,
, and and , the charge and mass of the ion.
Note that the well-known quasilinear plateau in the
VDF implies a vanishing pitch-angle gradient, i.e.
. The wave growth rate or absorption coefficient is essentially proportional to it. If the growth
rate remains small, the slowly varying part of the ion
VDF is controlled by diffusion. Then the time evolution
of (1) will, if the wave power is large enough, lead to a
time-asymptotic state of the VDF given by:
FIGURE 3. Comparison of some measured proton velocity distributions obtained by Helios 2 with the theoretical
cyclotron-resonance plateaus as predicted by quasilinear theory. The thick dotted lines on the left-hand sides are the circular
arcs delineating the plateau, whereby the related centers of the
circles are marked on the axis by full dark dots (with the
same numbers attached to the contours). The dots indicate the
locations of the effective phase speed of the waves.
(1)
where the pitch-angle gradient in the wave frame was
introduced. It is given by the velocity derivative
¼
Plateau formation in proton VDF
(4)
where is the initial value of the parallel speed ,
which satisfies the resonance condition (3). In the case of
plateau formation, the particles conserve their energy in
the frame of reference propagating at the effective phase
speed, .
tized plasma, has originally been derived by [8].
The quasilinear diffusion equation describes the evolution of the velocity distribution function, ,
of any particle species , e.g., in the solar inertial frame
of reference, in which the particles and waves are supposed to propagate. With the nomenclature used in [16],
the diffusion equation can be generally written as
Observational evidence from Helios plasma data has
been obtained for the occurrence of pitch-angle diffusion
of solar wind protons [17]. Their VDFs show plateaus
defined by vanishing pitch-angle gradients (implying
marginal plasma stability). Parts of the isodensity contours in velocity space shown in Figure 3 are outlined
well by a sequence of segments of circles centered at the
phase speed (dots indicate its location), which is assumed to vary slightly and to be due to dispersion smaller
than the local Alfvén speed. For the contours between
0.2 and 0.4 of the maximum density, the plateau can be
as wide as 70 degrees in pitch angle.
(2)
KINETIC MODELS FOR THE IONS
The sums extend over the Bessel function index, ,
and wave mode number, . The magnetic field fluctuation spectrum is , which is normalized to the
background-field energy density. It turns out to be physically meaningful to introduce the ion-wave relaxation
or collision rate denoted by . For its definition see [16]. The quantities and are the the
plasma-frame velocity components parallel and perpendicular to the magnetic field, . Other symbols used
The diffusion in velocity space of plasma ions being in
resonance with waves is an old plasma physics subject
[8] and has been studied in the literature in very much
detail. Only recently [5], [7], [1], [29], [30], [23] has
QLT been applied also to the solar corona and wind. The
principle feature QLT predicts is that ions in resonance
with waves undergo merely PAD, while conserving their
kinetic energy in the frame moving at speed .
402
wave-diffusion operator and Coulomb collisions, and
then follows the proton VDF in the expanding wind. The
model can account for the bulk acceleration, produces
preferential resonant heating of alpha particles and occasionally a double proton beam. However, the shape of
the proton VDF does not correspond in detail to the observed one, and the model lacks selfconsistency concerning wave absorption or instability and the transport of the
wave energy.
In another recent hybrid model [25], [26] the VDFs
were fixed as bi-Maxwellians, but the wave spectrum
evolution was allowed to evolve selfconsistently. From
the results the following conclusions were drawn: It
is problematic to use a spectrum with a fixed spectral
slope near the cyclotron resonance, when one calculates
the partition of wave energy among the different ionic
species. This assumption neglects the important effects
of wave absorption in the dissipation domain, and thus
renders energy supply possible at extremely low amplitudes of the waves. But if the spectrum is allowed to
evolve through wave damping, the high perpendicular
temperature anisotropy as observed by SOHO does not
occur in the model. In conclusion, serious efforts must
be made to calculate the wave opacity selfconsistently
without making restrictive assumptions about the VDF.
The hybrid model in [5] considered outward waves
only, but the cyclotron-resonant wave-particle interactions were calculated selfconsistently on the basis of a
scale separation between the small turbulent wave field
and large nonuniform background magnetic field. In contrast, the recently proposed kinetic-shell model [6], [7]
assumes again a rigid shape of the VDF, which is arranged in terms of shells centered according to equation
(4) at . Yet, fixing the VDF is not selfconsistent,
and the kinetic-shell approach applies only to a subset
of the observed VDFs (see Figure 1). However, the inclusion of outward waves allows energy transfer across
the boundary, and thus enables dissipation of outward waves and generation of inward waves. In [6] these
processes were not calculated locally, since by the shell
assumption the absorption is implied to be zero, but globally by means of the overall energy conservation laws.
FIGURE 4. Two-dimensional gyrotropic VDF of the heavy
coronal ion O . Note the contours with a perpendicular temperature anisotropy and skewness along the magnetic field.
Semi-kinetic model for coronal ions
A semi-kinetic model has been developed by [29],
[30] for the plasma dynamics of ions in the lower solar corona. This model consists of a closed set of reduced (with respect to the perpendicular velocity component) quasilinear diffusion equations. They involve
one-dimensional “reduced VDFs”, as they occur also in
the wave dispersion relations. This numerical model includes wave-particle interactions within the framework
of QLT and Coulomb collisions calculated by using the
Landau collision integral. Coupled Vlasov/Boltzmann
equations for these reduced VDFs were derived. The
semi-kinetic diffusion equations were solved for a coronal funnel and hole [30].
The results obtained for heavy ions in a coronal funnel show good agreement with SOHO observations and
yield preferential heating of the heavy ions, such as oxygen. This is illustrated in Figure 4. It was also found
that sizable temperature anisotropies and heat fluxes develop. The reduced VDFs of the heavy ions exhibit pronounced deviations from a Maxwellian, which cannot
be described adequately by higher-order polynomial expansions of the type that have been used [12], [10] to
model fast wind VDFs. The non-Maxwellian characteristics tend to increase further with height due to the ever
decreasing efficiency of Coulomb collisions. The wave
damping/growth rate shows that the VDFs can reach
marginal stability over a wide range of resonance speeds,
where wave absorption ceases. Such effects could never
be obtained if rigid VDFs were assumed at the outset.
Temperature anisotropy regulation
The observed velocity distributions such as in Figure 1
are often at the margin of stability. Wave-particle interactions are a key to understand ion kinetics in the solar
corona and wind, however the non-linear evolution has
not been fully investigated. In Figure 5 we show the proton temperature anisotropy as observed [27] and obtained
from various regulation models [4]. Apparently, the core
anisotropy is constrained and tuned by plateau formation.
Kinetic models for solar wind ions
The kinetic effects of wave-particle interactions have
been investigated [23] using a global hybrid model,
which takes the wave spectra as given, includes the QLT
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1.0
0.1
0.01
0.10
1.00
Beta
FIGURE 5. Proton core-temperature anisotropy as a function of the core-temperature plasma beta. The crosses show the
proton data taken from various high-speed VDFs. The solid line
shows the anisotropies as determined by the diffusion plateau,
calculated by means of the cold plasma dispersion relation. The
thin middle line is determined empirically by the pitch-angle
plateau. The line with full dots shows a numerical threshold
yielding: according to [4].
Similar investigations of the proton beam (see Figure 1)
indicate that the beam speed is also regulated by pitchangle diffusion [28] at the margin of resonantly driven
proton beam instabilities, with a modified dispersion in
the presence of alpha particles. Limited space does not
permit to fully discuss here such kinetic instabilies.
CONCLUSIONS
This short review of the microscopic state of the solar wind emphasised recent work on modelling the evolution of electrons by collisons and ions by cyclotronwave-induced diffusion. It seems clear that pure fluid and
hybrid-kinetic models cannot grasp the effects of waveparticle interactions. To describe them adequately, especially in the context of high-frequency plasma waves
heating the corona, requires kinetic physics. Theoretical
models using reduced or shape-invariant particle VDFs
and fixed wave energy spectral densities (ESDs) have
provided valuable first insights into the kinetics of the
solar wind but appear insufficient. Qualitatively new numerical results, which seem to approximate to the observations, were obtained in the past years.
However, the model assumptions of fixed VDFs or
rigid ESDs have to be given up. QLT when applied
selfconsistently in the case of weak turbulence seems to
capture the wave-particle kinetic processes in the solar
corona and wind, also for oblique wave propagation.
The problem of wave-energy generation, transport and
cascading (dissipation) in the dispersive domain remains
to be solved. As the next meaningful and feasible step,
selfconsistent kinetic calculations of the particle VDFs
together with the wave ESDs and opacities are suggested.
404