Heating in coronal funnels by ion cyclotron waves
Xing Li
Department of Physics, University of Wales, Aberystwyth, UK
Abstract. Plasma heating by ion cyclotron waves in rapidly expanding flow tubes in the transition region, referred as coronal
funnels, is investigated in a three-fluid plasma consisting of protons, electrons and alpha particles (α ’s). Ion cyclotron waves
are able to produce a transition region and a hot corona over a distance range of 104 km by directly heating alpha particles.
Although only alpha particles dissipate the waves, protons and electrons can also be heated to about 106 K due to Coulomb
coupling. It is found that alpha particles can be much hotter and faster than protons. Beyond 1.02Rs , the particles return to
thermal equilibrium when the electrons reach about 106 K which is canonically defined as the base of the corona. These
results lead to the following implications: (1) A transition region and corona may be energized by depositing energy to minor
ions only. (2) If spectral lines formed at Te 106 K are observed at different heights, the inferred outflow velocities may vary
by a factor of 5 to 6. (3) If minor ions are indeed much faster than protons and electrons at Te 106 K, one cannot reliably
determine the bulk outflow velocity of the solar wind in that region by using minor ion outflow velocities. However, when the
wave dissipation in the corona occurs much further away from the transition region, the loss of thermal equilibrium between
plasma species is much less pronounced, or a transition region and a hot corona cannot be energized by the waves at all.
INTRODUCTION
est cyclotron frequency will be heated first. Since the
Coulomb coupling is likely very strong in coronal funnels and the plasma beta value is small, those minor ions
and protons are not expected to be far away from equilibrium. However protons and electrons may be not in thermal equilibrium in coronal funnels as shown in [17, 6].
A more relevant question is: is Coulomb coupling strong
enough to transport energy from minor ions to the major species protons and electrons? More recently it was
found that some TR spectral lines can be fitted by two
Gaussians [18, 19]. One possible explanation is that the
broad lines may be formed by ions in coronal funnels
along open magnetic field lines [19] and the ions are preferentially heated by ion cyclotron waves.
In this paper, the heating of minor ions in coronal funnels along open magnetic field lines by high frequency
Alfvén waves is investigated in a fluid approach. These
waves are assumed to originate below the TR. It is shown
that it is possible for ions in the transition region not to
be in thermal equilibrium: they may have different temperature and outflow speed. Readers may also reference
recent semi-kinetic calculations of cyclotron heating in
coronal holes, oxygen ions are found much hotter (but
not faster) than protons and alpha particles at the base of
the corona [23, 24, 25].
Recent observations from the Ultraviolet Coronagraph
Spectrometer (UVCS) and Solar Ultraviolet Measurements of Emitted Radiation (SUMER) on board Solar
and Heliospheric Observatory (SOHO) have found that
(1): the temperature of minor ions decreases with increasing mass per charge at the base of the corona in
a southern coronal hole [22]; (2) ions not only have
greater than mass-proportional temperatures in the inner corona, they are much hotter than electrons and are
highly anisotropic as well [10, 15, 3]. These recent progresses strongly indicate that ion cyclotron resonance
may play an important role in the coronal heating.
Transition region (TR) models often do not treat electrons and protons as two different fluids [16]. More recent TR models treat electrons and protons as two separate fluids [17, 6]. Ion cyclotron resonance as a mechanism to heat the corona is adopted [17, 6]. In a two
fluid approach, a power law spectrum of ion cyclotron
waves originating from below the TR was adopted as a
mechanism to heat plasma in coronal funnels [17] and
the waves are able to generate a hot corona. Electrons
and protons start to lose thermal equilibrium when their
temperature reaches 10 5 K. The electron thermal pressure
in [17] is low. As a result, the plasma can reach a bulk
flow speed of 140 km/s at the top of the funnels.
Ion cyclotron waves propagating in the TR have important implications for minor ions: since minor ions
have lower cyclotron frequency, the species with the low-
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
283
MODEL ASSUMPTIONS
If the maximum frequency, f d , of the Alfvén waves
originating below the TR is still lower than the frequency
fH , ion cyclotron resonance does not occur. Hence, Q
0, and the waves will not heat the plasma. Then p w will
Êf
d P d f Following [17],
be related to P f r as 8π p w
f
Standard three-fluid plasma transport equations in an
isotropic, charge-free plasma including electrons, protons and α ’s are considered. The time dependent mass,
momentum and energy equations describing a onedimensional, three-fluid conductive plasma flow in coronal funnels are the same as those used in [14, 8]. To save
space, they will not be repeated here. In the following,
various assumptions are summarized.
1. It is assumed that plasmas in coronal funnels are
collision dominated and classical thermal conduction
parallel to the magnetic field is included for the electrons,
protons and α ’s [1, 12].
2. The radiative energy loss in the electron energy
equation is assumed to have the form given by [20] for
an optically thin medium. 3. A power law spectrum of
Alfvén waves originating below the TR is assumed to be
the only energy source of coronal heating. When the high
frequency Alfvén waves propagate along open coronal
magnetic field lines, ion cyclotron resonance may produce an upper limit on the frequency of Alfvén waves,
fH . This frequency is below the lowest gyrofrequency at
a given height. Following [21, 8], f H is taken as f H
α f fα c vph vA where f α c is the gyrofrequency of α ’s, v ph
is the phase speed of dispersionless Alfvén waves, v A is
the Alfvén speed, and α f is a constant less than unity,
taken to be 0.7 in this study. Previous studies have as02 [8, 12]. However, at such a low fresumed α f
quency, ion cyclotron waves are not in resonance with
alpha particles in a low beta plasma such as the one
treated here. At α f
07, linear Vlasov theory shows
that ion cyclotron waves already show some weak dispersion. This weak dispersion has been neglected. Ignoring nonlinear cascade processes, the evolution of Alfvén
wave spectrum in a multi-ion fluid can be written as [9]
2
∂P
v2A
∂ vph vph vcm Pa
∂ t vph vph vcm a ∂ r
v2A
P f ∝ f 1 is assumed.
It was found that while Q is determined by (2), the
total wave dissipation rate by resonant cyclotron interaction is vph vph vcm Q v2A [8]. The total dissipation rate
is divided between the acceleration and heating for different ion species, and is determined by the microphysics
of the resonant interaction. The method in [8] is used to
determine this energy distribution, cold plasma dispersion relation and Pk ∝ k 5 are assumed in the dissipation range, and only α ’s are heated by the waves.
L
NUMERICAL RESULTS
The same numerical method in [9, 14] is used in the paper. Model calculations start from a point in the lower
TR where the temperature is 6 10 4 K and extend to
15000km above it. The radial step dr gradually increases
from 35m at the lower boundary, to about 300km at
the top of the computational domain. The lower limit
on the temperature is chosen such that both hydrogen
and helium are almost fully ionized. The corresponding electron density at the boundary is 333 10 9cm 3 .
Thus yielding an electron thermal pressure of n e Te
2 1014 K cm 3 SUMER observations showed that ther142)
mal pressure is roughly a constant (log n e Te
in the transition region when the electron temperature
varies from 2 10 4K to 106 K in coronal holes [26]. The
total Alfvén wave amplitude at the base of the funnel is
δ V 27km/s. The magnetic field geometry and a are
taken from [11] and parameters are as follows: f m 13
is the expansion factor of the cross-section of a flux tube,
r1 1004RS is the heliocentric distance at which the major expansion takes place. The parameter σ 0005R S
represents the length over which the expansion occurs.
The magnetic field at the bottom of the funnel is 130 G.
This means that we assume that the coronal funnels considered here may originate from deep in the photosphere
[5]. The low- and high-frequency ends of the wave spectrum are fixed at f L 10 3 Hz and f d 30000 Hz.
Figure 1c shows the gyrofrequency of α ’s f α c and
07 f α c as a function of height h, which is defined by
h r r0 . The r0 is the heliospheric distance at the lower
boundary (assumed to be R S ). In this study, flow velocities are much smaller than the Alfvén speed and v ph is
nearly identical to v A . As a result, f H 07 f α c , 07 f α c is
roughly the frequency at which cyclotron resonance dissipates the waves. If f d fH , no wave heating will occur.
Since the goal of the study is to investigate coronal fun-
0 (1)
where vcm is the center of mass speed, P f r is
the power spectrum density, related to the magnetic
field variance δ B2 and the Alfvén wave pressure p w
Êf
H Pd f . It is also assumed that the
byδ B2 8π pw
fL
Alfvén wave frequencies have a lower limit f L . Integrating (1) with respect to f over the range ( f L , fH ), and
ignoring nonlinear cascade processes leads to
2
v2A
∂ 2pw ∂ 2vph vph vcm pw a
∂t
vph vph vcm a ∂ r
v2A
Q
vph P fH r d fH
(2)
4π
dr
where Q is a measure of energy transfer from waves to
plasmas, a is the flow tube cross section area.
Q
284
FIGURE 1. A coronal funnel model in which the plasma is
heated by ion cyclotron waves. (a) speed of protons (solid line)
and alpha particles (dashed line); (b) temperatures of electrons
(dotted line), protons (solid line) and alpha particles (dashed
line); (c) Gyrofrequency of alpha particles in a coronal funnel;
(d) alpha particle heating rate. Here fd 30000 Hz.
FIGURE 2. The same as Figure 1 but here fd
12000 Hz.
105 K[7], but the outflow velocity of Ne 7 was 14km/s
along open magnetic field lines from SUMER data [27].
To assume that other minor ions behave similarly to the
α ’s, their velocity will vary in a similar manner. Hence,
if spectral lines formed at Te 106 K are observed
at different heights, the inferred outflow velocities may
vary accordingly.
The fact that minor ions can flow much faster than the
bulk plasma flow in the upper TR may have implications
for the analysis of TR spectral line observations. Traditional atomic physics calculations on which the diagnostics of both temperatures and densities depend often assume equal ion and electron temperatures. From Doppler
shift measurements of TR spectral lines, the mass flux in
the transition region was found more than enough to account for in situ measured solar wind mass flux [4]. If
minor ions are indeed much faster than protons and electrons at Te 106 K, one cannot reliably determine the
bulk outflow velocity of plasmas in that region by using
minor ion outflow velocities.
Figure 1d shows the alpha particle heating rate Q α
in the computation domain. This plasma heating rate is
similar to those given by [17, 6]. In those studies, heat is
directly deposited to protons. At the lower boundary of
the computational domain, f H fd , no wave heating is
possible. At 1080km, the maximum frequency f d finally
reaches f H and a strong heating is found. In Figure 1a
when h 10km, waves do not heat plasmas. The α ’s
are slightly faster than protons due to a larger pressure
gradient force. The electrons are little hotter than ions
due to a much a greater heat flux.
When the maximum wave frequency f d is smaller,
wave dissipation will occur at a higher height. Shown
in Figure 2 is a model in which the same parameters
as those in Figure 1 except f d
12000 Hz are used.
In this case, the initial plasma heating rate in Figure 2c
is much smaller than that in Figure 1d. This is because
nels in coronal holes where fast solar wind is believed to
originate, the abundance of α ’s is chosen at 6% [14, 8].
A steady state solution of the model calculations is
shown in Figures 1a and 1b. Ion cyclotron waves indeed produce a very rapid temperature increase and a hot
corona. When temperatures are still low, α ’s and protons are strongly coupled, so T p Tα . At a height of
600km, α ’s start to become much faster and hotter than
protons. The proton velocity at the lower boundary is
v p 291km/s, a normalized proton flux density at 1 AU
would be n p v p 2152 fm 65 22 10 8 cm 2 s 1 ,
a typical observed value in the fast wind. The α ’s reach
a maximum velocity of 26 km/s at h 1080km, totally
due to the introduction of ion cyclotron resonance. Below this point, the wave frequency at the high end of the
spectrum is smaller than the frequency where the α ’s can
resonate with the waves, no wave dissipation occurs. Figure 1 shows that at the top of the TR or at the base of the
corona, α ’s and protons (and electrons) can be significantly away from thermal equilibrium. This is the first
time shown in model calculations that minor ions flow
much faster and are much hotter than protons in the TR.
In the nearby region below 600km, α ’s are significantly
slower than protons. This is due to the higher temperature
gradient of α ’s, and a smaller pressure gradient force for
α ’s. Since ion cyclotron waves only heat alpha particles,
the electrons are heated by Coulomb collisional coupling
with protons and the α ’s, and by the electron heat flux.
Observations of TR spectral lines often find various
ion velocities. These velocities can vary from a few to
about twenty kilometers per second. For instance, it was
found that in network boundaries an average outflow
velocity is several kilometers per second when T 63 285
the rapid decline of the magnetic field with height leads
to a rapidly declining Alfvén speed and FH . Hence Q
will decline very rapidly with height as well. The plasma
species are roughly in thermal equilibrium (Figures 2a
and 2b), plasma species have the same flow speed. The
α ’s are only slightly hotter than protons at the location where wave dissipation starts. SUMER observations
have shown multi-million kinetic temperatures of minor
ions at the base of the corona [22]. It seems that the result
shown in Figure 1 may fit SUMER observations better
than that in Figure 2. Note both models produce almost
identical mass fluxes, the proton velocity is 2.85 km/s at
the lower boundary in Figure 2 (compared to 2.91 km/s
in Figure 1).
solely through minor ions. In reality, the coronal heating physics may be far more complicated. Since there are
many ions heavier than the alpha particles in the TR and
their gyrofrequencies are smaller than that of α ’s, those
heavy ions will be heated first if the waves considered
here are indeed responsible for the coronal heating [2].
Due to their small abundances, it remains an open question whether the Coulomb coupling is able to transfer the
energy from these minor ions to the major species.
ACKNOWLEDGMENTS
This work was supported by a PPARC rolling grant to
UWA. Part of the work was supported by grant NAG510873 to Smithsonian Astrophysical Observatory (SAO).
DISCUSSION
The examples given above have some implications for
the coronal heating by a spectrum of ion cyclotron waves.
If the high frequency limit of the spectrum is much
higher than 30000Hz (Figure 1), more energy will be
released closer to the lower boundary, the temperatures
of species will increase more rapidly. It is expected that
alpha particles will become much hotter and faster than
protons in the region of initial ion cyclotron resonance.
However, a steeper temperature gradient will also lead to
a larger electron heat flux and a hotter electron temperature in the region close to the low boundary (as shown in
the Figure 1). This will violate our assumption of equal
species temperatures at the lower boundary. In such a
case, it is necessary to develop a model to include chromosphere. A full treatment of hydrogen and helium ionization processes will have to be included.
On the other hand, if the high frequency limit is lower
than 12000 Hz, it will lead to a smaller electron temperature gradient in the transition region. Eventually, a
smaller electron heat flux will not be able to balance the
radiation loss, and a hot corona cannot be created.
This study is limited to the case that the frequency of
ion cyclotron waves is smaller than the gyrofrequency
of α ’s. If waves with frequency between α and proton
gyrofrequency also exist, protons will certainly become
resonant with them. However these waves are difficult to
excite according to linear Vlasov theory.
In summary, plasma heating in coronal funnels along
open magnetic field lines by parallel propagating ion cyclotron waves is investigated. By heating alpha particles alone, the Coulomb coupling between α ’s and protons/electrons is strong enough to produce a TR and a
hot corona. Minor ions and protons may lose thermal
equilibrium in the transition region: they have different temperature and outflow speed. This study suggests
that the corona and the solar wind may be energized
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