Ion Heating Due to Plasma Microinstabilities in Coronal
Holes and the Fast Solar Wind
S. A. Markovskii and Joseph V. Hollweg
Space Science Center, University of New Hampshire, Durham NH 03824, USA
Abstract. There is growing evidence that the heating of ions in coronal holes and the fast solar wind is due to cyclotron
resonant damping of ion cyclotron waves. At the same time, the origin of these waves is much less understood. We suggest
that the source of the waves in the coronal holes is a heat flux coming from the Sun. The heat flux generates ion cyclotron
waves through plasma microinstability, and then the waves heat the ions. We use a new view according to which the heat flux
is launched intermittently by small-scale reconnection events (nanoflares) at the coronal base. This allows the heat flux to be
sporadically large enough to drive the instabilities, while at the same time to satisfy the time-averaged energy requirements
of the solar wind. Depending on the plasma parameters, the heat flux can excite shear Alfvén and electrostatic ion cyclotron
waves. We show that, for reasonable parameters, the heat flux is sufficient to drive the instability that results in significant
heating of protons and heavy ions in the inner corona.
INTRODUCTION
mal energy of electrons. The thermal energy density of
the electrons nT can be estimated as the energy density
B2 8π of the reconnecting magnetic field. Then, for reasonable parameters at the base of a coronal hole, B 12 G
and n 4 108 cm 3 the electron temperature T 108 K.
In a collisionless plasma, the electrons escaping from the
site of local heating can form a beam-like distribution
function, through velocity dispersion, at least at the initial stage of the evolution (e.g., [10]). The situation at
the coronal base is certainly more complicated. Nevertheless, it appears to be reasonable to use a two-component
model of the electron distributionfunction consisting of a
Maxwellian core and a heat-flux carrying component that
moves with respect to it (Figure 1).
It is now widely believed that the heating of the ions in solar coronal holes and the resulting generation of the fast
solar wind is due to cyclotron-resonant damping of ion
cyclotron waves; see, e.g., recent reviews by Hollweg and
Isenberg [1] and Cranmer [2]. Much less understood is
the origin of these waves in coronal holes. The mechanisms of the wave generation proposed so far are associated with a number of difficulties; see, e.g., Ref. [3] for a
review. We suggest that the waves in the proton cyclotron
frequency range are generated by a plasma microinstability due to a heat flux coming from the Sun. The electron
distribution functions observed in the solar wind often
indicate the presence of a nonzero heat flux in the solar
wind frame (e.g., [4–6]). Therefore, it is natural to assume
that the heat flux also exists in the inner corona. The heat
flux can then excite microinstabilities that heat ions. Similar ideas are widely used in ionospheric physics and they
were also discussed in connection with the solar corona
by Forslund [7], Toichi [8], and Coppi [9].
It is usually thought that the heat flux arises simply
from the gradients in a steady corona and solar wind. We
explore the consequences of a new view. We assume that
the heat flux is launched intermittently by small-scale reconnection events (nanoflares) at the coronal base. This
allows the heat flux to be sporadically large enough to
drive the instabilities, while at the same time to satisfy the
time-averaged energy requirements of the solar wind.
In the reconnection process, a significant part of the reconnected magnetic field energy is released into the ther-
LINEAR ANALYSIS
We investigate an electron heat-flux ion cyclotron instability. The instability is due to the fact that the maximum
of the electron distribution function is displaced with respect to that of the ion distribution function (Figure 1),
so that the destabilization results from the electron Landau resonance. In some sense, it is similar to the instability driven by a parallel current considered, for instance,
by Forslund et al. [11], except that we assume a zero net
current condition to be valid.
At large plasma beta, β 0 1 the primary growing
heat flux instability is a whistler mode (e.g., [12]). If the
plasma beta is between 0.1 and 0.01, then a shear Alfvén
heat flux instability is the most important one [13]. In
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
307
1
ω
0.8
ω/Ωp, γ/Ωp
reduced distribution function
1
protons
0.6
0.4
electrostatic
Alfven
0
100 x γ
-1
electrons
0.2
-2
0.04
0
-4
0
-2
2
4
6
0.1
0.12
0.14
FIGURE 2. Frequency and growth rate of shear Alfvén and
electrostatic instabilities as functions of the parallel wave number between the growth rate maxima at β p βc 0 002 and
uc 3 7VTc The plot is a cut in the k –k plane along a line
going though the points where the growth rate is maximum.
FIGURE 1. Schematic of the reduced distribution function,
i.e., integrated over velocities perpendicular to the background
magnetic field (arbitrary units), for protons and electrons.
coronal holes, the plasma beta is presumably between
0.01 and 0.001. In this case, an electrostatic heat flux instability is competitive with the shear Alfvén instability
[14]. The reason is that the growth rate of the electrostatic instability increases much faster than that of the
shear Alfvén instability with increasing heat flux above
the threshold. Therefore, the electrostatic instability can
dominate even though its threshold is somewhat greater.
We solve the full electromagnetic linear dispersion
equation for a quasi-neutral electron-proton plasma. For
simplicity we use an isotropic Maxwellian distribution
function for the protons. The electron distribution function consists of two isotropic Maxwellian components, a
cold and dense core denoted by the subscript “c” and a hot
and tenuous heat-flux carrying halo (“h”). The two electron components move with respect to each other along
the background magnetic field, so that a zero-current condition is satisfied.
The results of our calculations are displayed in Figures
2–6. Both instabilities are proton cyclotron resonant. As
can be seen from Figure 2, at β 0 002, the resonance
factor ω Ω p k VT p for the fastest growing electrostatic waves is 1 36 Therefore, the electrostatic waves
are strongly proton-resonant. The proton resonance with
Alfvén waves is weaker but still not negligible: the resonance factor is 2 56
Because the instabilities are driven by the Landau resonance with the core electrons, the waves propagate upstream in the solar wind. Similar instabilities on a qualitative level were considered for the first time by Forslund
[7]. It should be emphasized that if the waves are in a region of the solar wind where they can actually propagate
toward the Sun in the inertial frame, they will be able to
reach locations where they are resonant with He and
other heavy ions, even though they were only proton resonant at the generation site.
At greater plasma beta, the Alfvén instability takes
over. Figures 2-4 show the growth rate and frequency at
uc VTc 0 37 for different values of beta. At β 0 01
the electrostatic instability merges into the Alfvén instability. It is interesting to note that the Alfvén waves can
then have significant growth rate at a frequency equal
to the proton gyrofrequency. As a result, they become
strongly proton-resonant. The fact the ion cyclotron
waves can be excited at ω
Ω p is also suggested by
the solution obtained by Voitenko and Goossens [15] for
an ion beam driven instability. However, their approximate solution of the dispersion equation is not valid at
ω Ω p in contrast with our exact solution. Furthermore,
we show in Figures 2-4 that the physical reason why the
waves are excited at ω Ω p is coupling between shear
Alfvén and electrostatic ion cyclotron waves.
ION HEATING
0.08
k || R B p
v || / V T c , v || / V T p
0.06
8
Let us now compare the heat flux q that excites the instability with the energy flux at the coronal base q 0 needed
to drive the fast solar wind. Taking a typical density and
temperature in the vicinity of the coronal base n e 4
108 cm 3 and Tc 106 K and using the parameters nh
0 1n p Th
10Tc and uc
0 3VTc we obtain q 2
108 erg cm 2 s 1 The energy flux needed to drive the fast
solar wind can be estimated assuming that the expansion
factor is 10 and taking into account the fact that at 1 AU
the kinetic energy dominates over other forms of energy.
Then, it can be shown that q0 106 erg cm 2 s 1 for n p
308
1.25
1
Alfven
electrostatic
Alfven
2
ω/Ωp, γ/Ωp
ω/Ωp, γ/Ωp
2.5
ω
0.75
0.5
100 x γ
1.5
100 x γ
ω
1
0.5
0.25
electrostatic
0
0
0.03
0.06
0.09
0.12
0
0.15
0.03
0.06
FIGURE 3. Same as Figure 3 at β p
0.09
0.12
0.15
k || R B p
k || R B p
βc
FIGURE 4. Same as Figure 3 at β p βc 0 01 Electrostatic
wave merges into the Alfvén wave.
0 005 4 cm 3 and VSW 750 km/s at 1 AU. Because q q0 the
heat flux must be launched intermittently. This means that
a sporadic heat flux, high enough to excite the instability,
exists for a short period of time followed by a long period
of a much lower flux, so that the time-averaged flux gives
the right energy of the solar wind.
It is important that the waves driven by the timeaveraged heat flux can provide effective proton heating
in coronal holes, even though the time-averaged flux
is much smaller than the sporadic one. To show this,
we calculate the proton heating rate by waves from
quasilinear theory as
n dT dt γW
The threshold of the instabilities depends on the
plasma beta and the ratio of the proton to core temperature. The threshold of the Alfvén instability u c is plotted
in Figure 6 as a function of the heliocentric distance
for typical coronal hole parameters. We used the simple
approximate formula
uc
1 3
Tp Tc
where W is the energy density of the turbulent fluctuations excited by the instability. The heating rate in the
vicinity of the coronal base can be estimated by order
of magnitude as 10 6 erg cm 3 s 1 from kinetic [16] and
fluid [17] models that give the right parameters of the solar wind.
Taking the growth rate of the instability γ 10 2 Ω p
105 rad/s, we obtain the following relative
with Ω p
wave energy density
(3)
to plot the curve. The beta-dependence of uc has the same
origin as that in the case of a parallel-current instability in
a low-beta plasma [11] and the numerical factor is calculated at β p 0 01 and Tp Tc This approximate formula
gives uc 0 16VTc at β 0 04 and Tp Tc 2 while the
exact value of uc is 0 15VTc
In the region of increased threshold the heating is less
effective or even stops, if the actual core–proton velocity
uc nh nc uh is constant. However, this is not necessarily the case. If the velocity of the halo uh is initially
almost constant, the halo density decreases with distance
faster than the density of the background electron population, which plays the role of the current-neutralizing
core. Therefore, the ratio nh nc can increase with the distance together with uc so that uc is always well above uc
At later times, the intermittently launched halo expands
in the radial direction. As a result, its density decreases
faster and the ratio nh nc does not become too high. In
any case, at large enough distances, the threshold of the
instability decreases and nearly all of the launched heat
flux is transformed to the ion kinetic energy flux.
(1)
0 18VTc β p 0 01 W nT
10
7
(2)
In (1) and (2) we imply that the heating rate and the energy density correspond to a time-averaged heat flux. The
sporadic wave energy density is thus a factor of q q 0 200 greater. Nonetheless, this wave energy density is consistent with the one produced by a current-driven instability at a moderately supercritical drift in the quasilinear limit [18]. Note that the Alfvén waves at large propagation angles are almost electrostatic, because their electric field perturbation is almost parallel to the wavevector. Therefore, the quasilinear theory well-developed for
the electrostatic instability gives qualitatively right results for the Alfvén instability.
ACKNOWLEDGMENTS
We are grateful to Terry Forbes, Philip Isenberg, Martin
Lee, and Bernard Vasquez for useful discussions.
309
0.5
1
0.4
ω
uc/VTc, β
ω/Ωp, γ/Ωp
1.25
0.75
0.5
100 x γ
0.25
0
0.18
10 x β p
u *c
0.3
0.2
0.1
10 x β c
0
0.2
0.22
0.24
0.26
0.28
0
0.3
2
4
6
8
10
R / R sun
uc/VTc
FIGURE 5. Growth rate of the Alfvén waves at β p βc 0 01 maximized over the wavenumbers and propagation directions and the corresponding frequency as functions of the core–
proton velocity uc FIGURE 6. Threshold core–proton drift velocity and core
and proton betas for typical coronal hole parameters as functions of the heliocentric distance.
This work is supported by NASA Sun-Earth Connection Theory program under grants NAG5-8228 and
NAG5-11797, Solar and Heliospheric Physics SR&T
program under grant NAG5-10988, and Living With a
Star program under grant NAG5-10835 to the University
of New Hampshire.
10.
11.
12.
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