Cyclotron-resonant diffusion regulating the core and beam of
solar wind proton distributions
C.-Y. Tu£ , E. MarschÝ and L.-H. Wang£
Department of Geophysics, Peking University, 100871, Beijing, China
Max-Planck-Institut für Aeronomie, 37191 Katlenburg-Lindau, Germany
Abstract. Ion diffusion as predicted by quasi-linear theory has been compared with in-situ solar wind proton measurements. It
is found that the observed phase-space-density contours match very well those corresponding to the time-asymptotic plateau
generated by proton diffusion in cyclotron-wave resonance. Observations show that the perpendicular temperature of the
beam distribution is of the same order as its parallel one. A perpendicular heating mechanism is needed to balance the radial
tendency for adiabatic cooling. Outward and inward propagating cyclotron waves may together be able to control the thermal
anisotropy of the core distribution. However, there are hardly any cyclotron waves, which could resonate with a proton beam
having a drift velocity equal to or greater than the Alfvén speed. Therefore, we consider also outward-propagating waves,
with both left and right hand polarization, on a second dispersion branch existing in a cold plasma with electrons, protons
and alpha particles. These waves can resonate with the beam protons. The resulting diffusion can indeed explain the shape
of the beam distribution. A time-dependent kinetic code, in two-dimensional velocity space, has been developed to integrate
the quasi-linear diffusion equation. An initial shuttle-like distribution function is shown to develop into a distribution having
a core and a beam. The beam is found to drift at the Alfvén speed and be less anisotropic than the core. The radial evolution
of the beam density in the model is found to be consistent with the observations.
INTRODUCTION
RECENT DEVELOPMENTS
The velocity distribution of solar wind protons is known
to have usually two components, an anisotropic core and
a drifting beam. Such non-thermal distributions were
found first near 1 AU in the early days of space observations [5] and later on by Helios near 0.3 AU [16]. Recently, these non-thermal features were also identified in
the Ulysses [8] and Wind [1], [13] spacecraft observations.
Many theoretical papers were published to explain
these phenomena, which have nevertheless not been understood fully. Several papers, (such as [2], [9], [10],
[17], [24], [25], and [29]) tried to explain the perpendicular heating by means of the heating rates as calculated in the quasilinear theory of cyclotron resonance. It
was further assumed that the velocity distribution function (VDF) was a fixed drifting bi-Maxwellian. These
theoretical works yielded proton anisotropies somewhat
lower than the observed ones. However, especially when
considering the damping of the waves at frequencies
near the ion gyro-frequencies [24], such kind of models
failed in describing the large anisotropy of the O oxygen ions in the solar corona [14]. A common weakness
of the models is that resonant diffusion is not properly
described. In order to consider these wave-particle processes the rigid bi-Maxwellian VDF must be given up.
Here we will briefly review some recent work. The
thermal anisotropy instability was invoked ([7], [8]) to
explain the limits of the observed value of the proton thermal anisotropy, . Recently, the
anisotropy was analysed again, using data obtained in
low-speed wind by the WIND plasma instrument [13],
and found to have an upper bound close to the theoretical linear instability threshold. However, the majority of
the data points are far from this threshold and cannot be
explained as being limited by the instability. The theoretical instability threshold, [7], is somewhat lower than the observed values of , in particular
when the plasma beta, , is less than 0.1 as in high-speed
wind near 0.3 AU [26]. This failure to predict a proper
upper limit may be considered as a clear indication that
the core VDF in high-speed wind is not described adequately by a bi-Maxwellian VDF, which was used in
the instability calculation. Moreover, a theory involving
a shape-invariant VDF can also not explain how the thermal anisotropy is formed in the first place.
Concerning wave dissipation and generation, it should
be pointed out that the energy source required for the
primary wave heating of the protons is entirely different from the energy source of the secondary waves resulting from marginal stability. With regard to primor-
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
389
FIGURE 1. The dispersion relation of a cold plasma with alpha-particle abundance, , and normalized drift, .
The left panel shows the dispersion relation of LHP waves. The dashed lines delineate the cyclotron-resonance condition with
and , respectively. The thick curves show the dispersion relation used in this numerical simulation.
.
The right panel shows the dispersion relation for RHP waves. The dashed line shows the resonance condition with dial proton heating, pre-existing high-frequency waves
are damped while being in resonance with the particles.
The energy may be replenished from waves in the lowfrequency range, perhaps by cascading [2], [20], [22] or
the sweeping mechanism [21], [23]. In contrast, in the
case of an instability the wave energy comes from the
kinetic thermal energy of the ions, and the threshold is
determined by assuming a small growth rate. It is clear
that these mechanisms cannot work simultaneously at the
same frequency, although they both predict asymptotic
states with small absolute values of the growth rate.
The proton VDF is expected to be isotropic near the
base of the corona as the result of collisions. Due to
the mirror force, the proton thermal anisotropy, , will
initially decrease with the expanding solar wind and thus
tend to have negative values. Consequently, instability
theory does not explain how the proton core anisotropy
can attain values between 3 and 4, as observed near 0.3
AU in high-speed streams.
There are only few papers in which one tried to explain the original formation of the proton beam distribution. Tam and Chang [19] presented a hybrid solar
wind model showing the formation of a proton beam.
However, the differential speed of the beam is about
three times the Alfvén velocity (see their Figure 3). This
value is too large when campared with observations. This
large differential speed may come from the resonance of
the protons with RHP waves. Clack et al. [1] compared
the proton beam differential velocity observed by WIND
with the limit set by the Alfvén instability [4] and found
that most data points lie considerably below the instability threshold, especially in cases where the core plasma
beta is low. They concluded that how exactly the protonproton relative velocity is constrained remains unclear.
It is obvious that non-thermal features of solar wind
protons should be explained by resonant diffusion. Recently, quasilinear diffusion theory has been used to develop kinetic solar wind models [3], [6], [9], [11], [12],
[28]. Marsch and Tu [18] provided from Helios observations the first direct evidence of proton velocityspace diffusion driven by cyclotron resonance. Tu and
Marsch [26] showed that the observed proton core distribution (and its anisotropy) could be explained by quasilinear diffusion driven by resonance of protons with LHP
waves, having an inward as well as outward sense of
propagation. Marsch [15] presented a brief review of this
topic in these proceedings. Tu et al. [27] showed that the
proton beam might be understood as the result of the resonance of protons with outward propagating waves (both
LHP and RHP), which follow the dispersion of the second branch that exists in a plasma with alpha particles
drifting along the field at about the Alfvén speed.
FORMATION OF THE PROTON BEAM
DISTRIBUTION
For a given wave power spectrum density, the quasilinear diffusion equation [2] is solved by means of a twodimensional code based on the Lax-Wendroff integration
scheme [27]. In Figure 1 the used dispersion relation is
shown, which pertains to a plasma consisting of protons,
alpha particles and electrons. The relative abundance of
the alpha particles is taken to be 0.05, and their drift
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.
FIGURE 2. Formation of a proton beam VDF. The upper
panel shows the initially assumed bi-Maxwellian distribution
with parameters: km/s, km/s, and km/s. The bottom panel shows the simulation results after
50000 time steps in the numerical run. The magnetic field is
nT and the density, cm
FIGURE 3. Radial evolution of a proton beam VDF. The
upper panel shows the final stable VDF resulting at 0.3 AU,
which is simulated from an initial distribution function that is
the same as the lower distribution in Figure 2. The middle panel
shows the final stable distribution at 0.6 AU. The bottom panel
shows the final stable VDF at 0.9 AU.
velocity relative to the protons is . The solid lines
show the dispersion relation and the dashed lines show
the resonance conditions.
The LHP first-branch waves with outward sense of
propagation are in resonance with inward moving protons. The LHP first branch waves with inward propagation sense (see the thick solid line) are in resonance
with outward moving protons, having a parallel velocity less than a critical velocity, , say of 50 km/s. The
protons with velocity greater than 50 km/s, but less than
km/s are in resonance with the second-branch
LHP waves with outward sense of propagation. Protons
with velocities greater than are in resonance with
second-branch RHP waves with outward sense of propagation.
The wave power spectrum, , is assumed to have a
power-law form: , where cm , and G cm, which
is consistent with a power density of 1 nT /Hz near the
proton gyrofrequency according to the Helios observations at 0.3 AU. The power densities of the four wave
modes involved are all given. The initial velocity distribution is assumed to be elongated along the magnetic
field direction and shown in the upper panel of Figure 2. After integrating the diffusion equation for 50000
steps in time, corresponding to 15 seconds (9 ,
where rad/s), we obtain a rather stable distribution, which is shown in the lower panel of Figure 2
and reveals two components, core and beam. The resulting anisotropies of the two components are 1.8 and 1.1,
respectively. The velocity difference between beam and
core is 96 km/s and about equal to . In conclusion, the
numerical results describe the observations well.
EVOLUTION OF THE DENSITY RATIO
BETWEEN CORE AND BEAM PROTONS
With this simulation code we can also calculate the radial evolution of the ratio of the beam density to the
core density. Assuming an initial distribution function at
AU and typical plasma parameters there, i.e.
km/s, we ran the numerical program and obtained finally a stable distribution (still for AU).
Then we used this final distribution function as the initial input distribution at AU and typical plasma
parameters at that distance, where km/s, and
found another final distribution. We further applied the
same procedure at AU and thus got finally the
stable VDF at AU. We also assumed that the ratio, is constant. The radial evolution of the distribution function is shown in Figure 3.
The corresponding radial evolution of is shown
in Figure 4, where is the beam density and the total
proton density. The crosses show the Helios observations
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ACKNOWLEDGMENTS
Tu’s and Wang’s work was supported by the National
Natural Science Foundation of China under projects with
contract numbers 40174045 and 49990452, and by the
Foundation of Major Projects of National Basic Research
under contract number G-2000078405.
FIGURE 4. Radial evolution of the relative beam density
( ). The crosses are the Helios observations (taken from
[16]), and the triangles are our simulation results with the
model density and magnetic field from [22].
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in high-speed solar wind [17]. Both model results and
observations exhibit the same radial trend.
However, the possible weakness of this comparison
should be pointed out. The observational data shown do
not have error bars. The discrimination of the beam from
the core protons in not easy and may lead to a somewhat ambiguous separation in the measured data. Moreover, in the model calculations we did not consider the
self-consistent evolution of the wave dispersion relation.
Also, the effects of the magnetic field direction changing
between 0.3 AU and 0.9 AU were not considered.
CONCLUSION AND DISCUSSION
The proton core temperature anisotropy and the differential speed of the proton beam have been explained by
invoking cyclotron resonance diffusion. This scenario is
supported by observations as well as numerical simulations. The core protons can be in resonance with firstbranch LHP waves with outward sense and first-branch
LHP waves with inward sense of propagation. The beam
protons can be in resonance with second-branch LHP and
RHP waves having an outward propagation direction.
So far, it has been impossible to identify the secondbranch waves from magnetic field observations made
on single spacecraft. To detect and observe the secondbranch waves and to identify the possible different wave
modes remains a task for future solar wind missions. One
should also try to find a physical mechanism to create
the second-branch waves in the solar wind plasma. For
example, a highly anisotropic alpha-particle beam may
be able to generate the second-branch waves.
The dispersion relation in our model calculations was
simply assumed. This dispersion relation may not be
consistent with the evolution of the model distribution
in the numerical run. A self-consistent treatment of both,
the velocity distribution function and the dispersion relation, is required and should be considered within the
rationale of this model as the next reasonable step.
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