Ion Fractionation and Mixing Processes in the Turbosphere
and the Solar Wind Formation Region: Scaling Approach
I. S. Veselovsky
Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia
Abstract. The kinetic theory of the ion distribution function formation in the plasma is briefly considered.
Models are discussed of the individual particle and collective behavior. Governing dimensionless parameters
are presented for numerous collisional and collisionless regimes of the ion generation, separation and mixing
in the solar atmosphere. Global steady state and local non-stationary processes are shown to be interrelated in
a complicated manner possibly leading to the observed diversity in composition variations of the solar wind.
The strongest composition variations are produced at the shocks, but the mixing prevails in the sense that protons
remain the dominant component even under these extreme conditions. The dynamical balance between the mixing
and fractionation depends on many space-time scales simultaneously present in the plasma and electromagnetic
fields, which explains the lack of simple universal classifications and poor understanding of the laminar or
turbulent processes of the ion composition formation. For example, it is not clear if the persistent difference
between the composition of the quasi-stationary slow and fast solar wind streams could be related mainly to the
vertical stratification or to the horizontal inhomogeneity of the solar corona and dynamical processes in holes and
streamers.
INTRODUCTION
Variations of the solar wind ion composition and their
origins are poorly known and understood in many instances (see, e.g., reviews, [1, 2, 3, 4] and references
therein), though the thermal diffusion, electric fields and
Alfven waves are often involved in the separation processes.
The aim of this paper is to bring theoretical arguments
helpful in attempts of a qualitative understanding of the
observed diversity of the solar wind composition variations as a result of the multi-scale plasma evolution with
a competition of different physical processes of the ion
separation and mixing in the turbosphere and the solar
wind formation region. The term turbosphere is used here
to designate the region around the Sun where the velocity
of the regular radial outflow v is less than the velocities
Sv of irregular vertical and horizontal motions: v < Sv.
KINETIC REGIMES
In the deep regions of the solar atmosphere (chromosphere) where ionization and recombination processes
happen with other atomic and radiative transitions and
many-particle interactions are essential, one should consider correlation functions which could be in the equi-
librium state or not. The macroscopic diffusion approximation can be used only when collisions are sufficiently
frequent. Dimensionless parameters like ratios of the ionization, recombination, excitation and radiation times are
very important physical quantities which are needed for
the better understanding of the situation. In addition to
this, several dimensionless parameters are useful indicators of the importance (unimportance) of the local/nonlocal conditions (the so called Trieste numbers which are
ratios of internal and external energy, momentum and
mass fluxes in the coronal and heliospheric structures
under consideration). Trieste numbers characterize the
openness degree and vary from zero in the open system
to infinity in the isolated structures. They are of an order
of one for many morphological entities in the solar wind
formation region, which makes the analysis of the turbulent and "recuperating" systems very difficult and sensitive to the imposed initial and boundary conditions. It is
very important, but sometimes neglected without sufficient grounds, that the plasma in the solar wind formation region is radiative. The useful dimensionless parameter in this respect, Ve, is the ratio of the kinetic plasma
power to the emitted radiation power. The parameter Ve
is small for active regions and large for coronal holes as
a rule [5].
In the fully ionized coronal regions with the negligible
radiation role (coronal holes) the one-particle distribu-
CP598, Solar and Galactic Composition, edited by R. F. Wimmer-Schweingruber
© 2001 American Institute of Physics 0-7354-0042-3/017$ 18.00
305
tion functions F(r,v,r) obey the kinetic equations with
the Landau collision term
permagnetosonic. Because of this, the cold gas of noninteracting particles can be a valuable zero order approximation when considering the kinetic equations for the ion
distribution functions. In this approximation, the forcefree radial motion is used in the method of characteristics neglecting transversal thermal velocities, electromagnetic and gravity fields [8].
The gas-kinetic theory of weak and strong inhomogeneities in the collisionless solar wind was developed
using the small parameters M"1 and M^1, where M and
MA are the Mach and Mach-Alfven numbers for ions.
Distribution function corrections were analytically calculated in this way for ions in response to the nonstationary plasma density, bulk velocity and temperature
perturbations at the given boundary near the Sun. The
leading terms are of the order M"2 and M^2. The details of calculations were described earlier [8, 9, 10]. In
the fast streams from coronal holes 7} ~ w/ and VTI are
the thermal velocities of the ion species nearly equal to
each other, hence partial Mach numbers M/ ~ V/VTI are
also approximately equal to each other for different ion
species. As a result, the variations of the composition
should be canceled in the linear approximation for this
case. In the slow wind, Ta ~ Tp because of the sufficient
collisions and M2 ~ 4M2. Hence, composition fluctuations are mostly produced by the proton parameter variations which are four times stronger than alpha particle
variations in the linear approximation. The linear approximation breaks down when SM~2 ^ > 1, where S = ^p
is the Strouhal number, 8A/A being the relative amplitude of parameter variations (density, velocity, temperature). It is clear that short time composition fluctuations
are steepen and mixed at the corresponding distances because of the nonlinear evolution with the formation of
interpenetrating streams or they are preserved when the
nonlinearity is saturated. Saturation processes are difficult to evaluate a priory. For this purpose the a detailed
information is needed about the dynamics taking into account the kinetic plasma and electromagnetic field characteristics in the real geometry situation. These considerations probably explain the lack of the universal composition behavior and the presence of the observed diversity with temporary mixed and fractionated regions. Nevertheless, the nonlinear mixing processes are obviously
very important limitative factors during the ion separation. As a result, the fractionation is not sufficiently
strong to eliminate protons from the ion composition or
make them a secondary component in the known regions
of the heliosphere. It appears that protons remain dominant ions everywhere and every time in the corona and in
the solar wind in spite of many different ion separation
mechanisms operating in the rarefied plasma.
The ballistic motion of particles in the force-free space
and in the gravity field preserves the composition only
where
is the transport operator in the phase space, St is the Landau collision term [6]. The dimensionless Knudsen number Kn = A//, where A is the Coulomb mean free path
and / is the characteristic length, delimits collisional and
collisionless regimes, Kn <C 1 and Kn ^> 1, correspondingly.
Ion separation and mixing processes under frequent
collisions when \St ^> \D\ or Kn <C 1 can be described
in the Coulomb diffusion approximation. The diffusion
coefficient along the magnetic field is proportional to
£)|l ~ A/ ~ a"1, where A/ is the mean free ion path, a
is the Coulomb cross section in the fully ionized plasma.
The laminar diffusion across the field is suppressed accordingly: D± K, D||(r//A/) 2 ~ Aj"1, where r/ is the ion
Larmor radius, D± ~B~2. The laminar diffusion regimes
[7] are violated by the multi-scale convection and waves
in the turbosphere around the Sun where the plasma is
out of equilibrium. In this case, some effective turbulent
mixing length L should be used instead of A in the corresponding transport estimates. The mixing length L in
the chromosphere and in the corona is generally comparable to the standard heights. Morphologically, vertical
motions in both directions along the loops and open magnetic structures take place here together with horizontal
shuffling motions due to convective electric drifts, magnetic stresses and thermal pressure gradients in the intermingled low- and high-beta regions. Turbulent processes
could enhance or depress the local and global transport
of the bulk mass and its ion components depending on
the situation and the geometry of the problem. It is very
likely, that vertical and horizontal turbulent mixing efficiency is different in the sources of the slow and fast solar
wind streams because of different plasma parameters and
the magnetic field geometry. Nevertheless, the vertical
mixing apparently prevails over the gravitational settling
and other separation mechanisms. As a result, composition variations are observed to be rather moderate in the
solar wind as rule.
GAS-KINETIC APPROXIMATION FOR
THE SOLAR WIND IONS
The free energy of the solar wind plasma resides mainly
in the inhomogeneous radial motion of ions, which is su-
306
when interpenetrating streams and the corresponding
mixing processes are absent. Such laminar flows are locally and temporary possible under appropriate boundary
and initial conditions in the solar corona. Due to the nonlinear kinematic steepening of the velocity profiles interpenetrating streams could appear in the kinetic approximation. Otherwise, the ballistic approximation breaks
down and shock fronts develop. The mixed state with interpenetrating streams, once created, develop in different
ways depending on the ratio of relaxation and dissipation times and lengths to the transit times and scales of
the inhomogeneity given by the corresponding Reynolds
numbers (ordinary and magnetic). In the one-scale approximation, only two possibilities exist in this respect:
the homogenization of the mixture or its segregation after
the transit stage. But the large free energy (kinetic and/or
magnetic) of the interpenetrating ion streams could be
a source of ion instabilities and the generation of new
small-scale structures is possible [11]. Hence, the fragmentation, segregation and relaxation coexist with the
merging, mixing and increase of structures. Both tendencies, the decay and growth of the structures are observed in the solar corona and the heliosphere permanently. They are documented in the composition measurements of quasistationary and transient solar wind
flows, but the knowledge and understanding of the evolution is rather limited. Interpenetrating ion streams in the
solar wind are sometimes observed, but their role in the
overall plasma mixing is not clear and needs additional
studies.
SELF-SIMILAR ION FRACTIONATION
IN THE SOLAR WIND
The collisionless cold ion dynamics with the compensating electrons near the thermal equilibrium can be
described by multifluid equations taking into account
electric and magnetic fields [6]. The nonstationary selfsimilar variations in the ion composition of the solar
wind have been analyzed based on this approach [12].
The governing equations for protons (/ = 1) and alphaparticles (/ = 2) in the case of motions along the external
magnetic field are written as follows:
,
.drii
dui
x
du\
du2
-
kriiUi
„,.
^ _
1 dn
1 miZ2dn
+
=Q
307
aredimensionless ion velocities and the self-similar variable,
k = 0, 1,2 for the planar, cylindrical or spherical geometry. The electric potential (p in the corona is of an
order of 100 V.
There are two possible asymptotics at t ^> 1 :
3 m\Z2
-a,
where Ci}2 and n 10,20 arc arbitrary constants;
In the both cases (I and II) the condition n2 <C n\
is assumed. The formulae are applicable also for other
types of ions (/ = 2), and not only for alpha-particles.
It is seen from these formulae that enhanced (depleted)
n2/n\ regions could propagate faster or slower than unperturbed plasma flow depending on the boundary and
initial conditions (which are beyond the scope of the
self-similar assumptions). In the case (I) for weak perturbations, alpha-particles are moving faster than protons.
For strong perturbations (II), alpha-particles are moving
slower than protons. The potential electric fields and the
electron distributions are controlling factors. The velocity difference u2 — u\ is positive in the case (I) when
Ci w c2 and u2 — u\ ~ i~2 decreases with increasing distance from the Sun, in a qualitative accordance with the
Helios observations [13].
Theoretically, it is obvious that electric fields are involved in the ion dynamics not only along the magnetic
field, but also in the nonstationary and quasistationary
drifts across the magnetic field, which are strongly inhomogeneous and variable in the solar wind formation
region. As a consequence, the characteristic velocities
of the nonstationary ion mixing or separation along the
magnetic field could be determined by the ion sound
speed, and by the drift velocities across the magnetic
fields [14, 15]. The drift velocities are often scaled as
L/t, where L and t are characteristic lengths and times
of the magnetic field variations. Contrary to the prejudice that magnetic fields always suppress the transport
processes across them, observations in space and laboratory experiments, as well as theoretical arguments show
many examples of the enhanced energy and mass transports perpendicular to magnetic fields in the presence of
the inductive or potential electric fields due to the imposed initial or boundary conditions. Both external nonlocal fields and local fields generated in situ by the free
energy available in the open, closed and partially open
plasma systems of the solar corona and the heliosphere
could be and are the drivers of the electromagnetic "separators" or "mixers" operating there.
The dimensionless Faraday number F = -j™, where
j and p are the electric current and the electric charge
densities, could indicate on the dominance of the potential F <C 1 or inductive F > 1 electric fields. The case
F <C 1 was considered here.
for example due to the nonlinear acoustic wave damping. The opposite case r < 1 could mean the nonequilibrium Joule heating of electrons. The situation in the
high P regions is much more complicated because of the
anisotropy introduced by the magnetic field.
The dimensionless scaling approach is useful for a
better understanding of the different physical situations
in the binary cases (inertia and collisions, inertia and
electromagnetic forces, inertia and gravity etc.) as well as
the ternary and more complicated regimes when several
physical factors are essential, for example, combined
inertia, gravity and electromagnetic forces. In addition
to the one-scale approximation and classification, multiscale states and mechanisms represent very interesting
practical applications.
There are many tens of the physically different
regimes of the dissipative macroscopic plasma behavior
in the stratified solar atmosphere and the turbosphere
around the Sun [5]. Electromagnetic and gas-kinetic
drivers are known to be effective in the "mixers" and
"separators" of the ion composition. Subsonic thermal
pressure pulses in the low P regions, field aligned interpenetrating streams, secondary field-aligned flows
driven by the nonstationary electric drifts and the continuity conditions steepen and decay with the formation
of the 7} ~ rnt states. The same states can be obtained
as a result of the ion cyclotron In the situations far
from the thermal and mechanical equilibrium in the
solar corona and the heliosphere anomalous transport
processes by Alfven waves and convective motions
are ubiquitous. Collisions between particles tend to
equalize the temperature differences. Both mechanisms,
the convective mixing of different plasma parcels and
wave-particle interactions are also capable of producing
observed deviations from the relation 7} ~ m\. Stronger
or weaker dependences could result from different initial
or boundary conditions in the case of the mixing. Instead
of these nonlocal causes, locally steeper or shallower
wave spectra can also lead to same qualitative results.
Subphotospheric convective motions generate electric
currents and magnetic fields seen in the photospheric
and chromospheric network elements, granules, spicules,
loops and other nonstationary solar activity manifestations. These nonstationary magnetic fields produce inductive electric fields according to the equation [V x
E] = — ^^. As a result, nonstationary electric drifts
DIMENSIONLESS MULTIFLUID MHD
SCALING
The demarkation of physically different regimes is possible with the use of the following dimensionless parameters and their combinations:
Kn =fki/l is the Knudsen number,
S = l/vt is the Strouhal number,
M = v/cs is the Mach number,
MA = V/VA is the Mach-Alfven number,
Fr = v2R(MQGl)~l is the Froude number,
where R is the heliospheric distance, M0 is the mass
of the Sun, G being the gravitation constant, and VA,
the Alfven speed. Additionally, the group of dissipative
parameters appears:
Re = vl/v is the (viscous) Reynolds number,
Rem = alv/c2 is the magnetic Reynolds number,
Pr = v/% is the Prandtl number,
where v is the kinematic viscosity, % is the temperature conductivity and a is the electric conductivity. All
dissipative coefficients are tensor quantities in the strong
magnetic field [7].
In the multifluid approximation, relative abundances
of the species oc/J = nt/tij are important dimensionless parameters where n\ and HJ represent corresponding number densities. All parameters above are assumed
to be partial quantities referred to the given type of ions
(indexes i,j). For example, nonstationary S > 1, quasistationary S < 1 and intermediate S ~ 1 regimes are delimited by the corresponding Strouhal numbers S = ^.
The high (low) ion-electron temperature ratios r =
Ti/Te could indicate on the dominant ion (electron) heating. The situation is rather simple in the low P regions:
r > 1 usually means the viscous mechanical heating,
V(r,t) = c ^i arise. Small-scale plasma elements are
mixed across the magnetic field under the action of fluctuating transversal electric field.
The corresponding mixing lengths L for the mentioned
motions could be less, comparable, or greater than the
standard height //,- (0.1-100 Mm) for ions in the solar
atmosphere at different levels with characteristic times
102 -105 s and velocities of an order of ~ 0.1 -100 km/s,
308
but usually they are comparable to standard heights ///.
Local vertical up- and down motions are nearly sonic
or trans-sonic in spicules, jets, loops. Horizontal motions
are sometimes faster, but sometimes slower than vertical
ones. Convective flows and wave branches are involved
in the motions in different proportions.
The turbulent mixing length L could also be less,
comparable or greater than the Coulomb mean free path
for ions A,/.
Accordingly, different diffusion and mixing regimes
are locally possible: molecular or turbulent ones under
essential or negligible gravitational sedimentation convective or wave-like, with subsonic or supersonic motions in a magnetically dominated or unmagnetized conditions.
DISCUSSION
Ion-atom separation mechanisms in the partially ionized
chromospheric plasma lead to the fractionation of elements according to their first ionization potentials, see
e.g. the review of models [16].
Solar wind charge states are used in attempts of obtaining rough estimates of temperature profiles in coronal holes, when comparing calculated expansion times
and ionization-recombination times assuming local collisional equilibrium with electrons. The situation in the
slow wind and coronal mass ejections is much more uncertain, variable and complicated.
First ionization potential (FIP) fractionation takes
place in the solar wind: elements Fe, Mg, Si with the low
FIP (< 10 eV) corresponding to the Lyman-alpha radiation contribution are more abundant than high FIP elements like inert gases. In the models of this phenomenon
(VonSteiger and Geiss, 1989) the ratio of the ion trapping time in the magnetic field to the ionization time by
UV radiation is important controlling factor and different
types of the solar wind flows exist according to the role of
the ion-atomic separation processes in the partially ionized solar atmosphere.
The difference between elemental compositions and
charge states of the fast and slow solar wind streams was
confirmed and investigated in detail in the recent study
of the Ulysses data [17]. The charge state distributions
are consistent with a single frozen-in temperature for
each element only in the fast streams. The first ionization
potential fractionation is clearly seen in the slow streams,
it is much more weaker in fast streams.
Biirgi [18] have considered the steady-state three-fluid
model with the cusp geometry and found the depleted alpha to proton flux ratio in the coronal streamer. The useful dimensionless parameter here is the geometry expansion factor / = j1- for the corresponding flux tube cross
309
sections with a possible focusing (defocusing) of flows
and wave energy fluxes in the low P solar wind formation region [19].
Bochsler [20] have integrated numerically the energy
balance equations for a buoyant expanding plasmoid taking into account its heating and radiative losses together
with the ionization state changes which were shown to
be a sensitive indicator of the physical processes in the
inner corona. Useful dimensionless parameters here are
represented among others by the oc/-ionization degrees,
Ve - the ratio of the plasma kinetic power to the electromagnetic radiation (absorption) power [5].
The behavior of minor ions in the models is very complicated and depends on many factors: masses, charges,
gravity, thermal pressure, magnetic stresses, thermal diffusion, expansion factors etc. [21]. The Coulomb friction
plays different roles in the rapidly expanding magnetic
structures near coronal streamers and in polar coronal
holes. The solar wind composition and the composition
of the source regions in the solar atmosphere needs additional comparative studies. In this respect, recent SOHO
and ACE measurements of heavy ion composition variations can be used as a sensitive tool for the classification
of the solar wind acceleration conditions in the transition
region and in the solar corona [22, 23].
Ion composition measurements of coronal mass ejections bring some features which might be interpreted as
signatures of non-equilibrium plasma mixtures of different temperature components [24]. Turbulent mixing processes could be much more effective than molecular or
Coulomb diffusions in producing such mixtures.
CONCLUSIONS
Individual particle degrees of freedom (one-particle
molecular kinetic approximation), few or many particle correlations (ionization and recombination processes,
atomic and radiation phenomena) as well as collective
(macroscopic radiative MHD description) behavior are
possibly important ingredients to be taken into account
when considering fractionation and mixing processes in
the solar atmosphere.
Turbulent mixing makes the fractionation mechanisms
less effective in many instances. Sufficiently strong quasistationary magnetic fields are sometimes stopping the
transversal mixing and fractionation processes and could
preserve the memory of the original composition formation in the solar wind.
The scaling approach is useful for a better qualitative
understanding of the ion separation and mixing processes
in the solar wind.
ACKNOWLEDGMENTS
Res., 105, 27217-27238 (2000).
18. Biirgi, A., "Dynamics of Alpha Particles in Coronal
Streamer Type Geometry", in Solar Wind Seven,
Pergamon Press, Oxford, 1991, pp. 333-336.
19. Veselovsky, I. S., Geomagnetism and Aeronomy, 36, 1-7
(in Russian, also translated in English) (1996).
20. Bochsler, P., "Minor Ions - Tracers for Physical Processes
in the Heliosphere", in Solar Wind Seven, Pergamon
Press, Oxford, 1991, pp. 323-332.
21. Bodmer, R., and Bochsler, P., /. Geophys. Res., 105,
47-63 (2000).
22. von Steiger, R., and Zurbuchen, T. H., "Solar Wind
Composition as a Diagnostic Tool", in Solar and Galactic
Composition, Conference Booklet, Joint SOHO-ACE
Workshop 2001, March 6-9, 2001, Bern, Switzerland,
2001, p. 45.
23. Ko, Y.-K., Zurbuchen, T., Strachan, L., Riley, P., and
Raymond, J. C., "A Solar Wind Coronal Origin Study
from SOHO/UVCS and ACE/SWICS Joint Analysis", in
Solar and Galactic Composition, Conference Booklet,
Joint SOHO-ACE Workshop 2001, March 6-9, 2001,
Bern, Switzerland, 2001, p. 52.
24. Bochsler, P., "Mixed Solar Wind Originating From
Coronal Regions of Different Temperature", in Solar
Wind Five, NASA CP 2280, 1983, pp. 613-622.
The work is partially supported by the INTAS-ESA
grant 99-00727, RFBR grants 00-15-96623, 01-0216579, Federal Program "Astronomy" project 1.5.6.2
and the State Program "Universities of Russia" grant
99-0600. The author is grateful to the Organizing
Committee and the Swiss National Science Foundation
for the financial support enabling his attendance and
the presentation of the paper at the First SOHO/ACE
Workshop: "Solar and Galactic Composition".
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