Äîęóěĺíň âç˙ň čç ęýřŕ ďîčńęîâîé ěŕřčíű. Ŕäđĺń îđčăčíŕëüíîăî äîęóěĺíňŕ : http://star.arm.ac.uk/~ambn/hrts_sol.ps Äŕňŕ čçěĺíĺíč˙: Wed Jan 15 15:44:08 1997 Äŕňŕ číäĺęńčđîâŕíč˙: Mon Oct 1 22:03:02 2012 Ęîäčđîâęŕ: Ďîčńęîâűĺ ńëîâŕ: ion drive

Nonthermal Velocities in the Solar Transition Zone
and Corona
J.G. Doyle, E. O'Shea and R. Erd'elyi
Armagh Observatory, Armagh BT61 9DG, N. Ireland.
jgd@star.arm.ac.uk eos@star.arm.ac.uk rfs@star.arm.ac.uk
K.P. Dere and D.G. Socker
E. O. Hulburt Center for Space Research, U.S. Naval Research Laboratory,
Washington DC 20375­5000, USA.
kdere@solar.stanford.edu
F.P. Keenan
Department of Pure and Applied Physics, The Queen's University of Belfast,
Belfast BT7 1NN, N. Ireland.
F.Keenan@qub.ac.uk
(Received ..... ; Accepted in final form .....)
Abstract. Nonthermal velocities are presented for spectral lines covering the tem­
perature range 10 4 K -- 10 6 K, measured from high spectral resolution data for several
solar features observed at the limb by the High Resolution Telescope and Spectro­
graph (HRTS), including a coronal hole, `quiescent regions' and several small scale
active regions. These results are compared with predictions based on acoustic waves
and heating via Alfv'en waves. It is likely that more than one mechanism is operating
simultaneously, in particular, resonant Alfv'en wave heating, which is very sensitive
to background plasma motions.
1. Introduction
To heat the solar atmosphere by waves one requires an efficient dissi­
pation mechanism. There are three branches of possible theories that
have received the most attention: wave heating (e.g. acoustic waves,
Alfv'en waves), the resistive dissipation of D.C. electric currents and
finally the selective decay of a turbulent cascade of magnetic fields.
Based on transition region and coronal lines, Cheng et al. (1979) has
suggested that their observed non­thermal broadening favors Alfv'en
wave heating. This line broadening corresponds to random velocities in
the transition region and corona of 20 \Gamma 25 km s \Gamma1 (Dere and Mason,
1993; Hassler et al., 1990) under quiet Sun conditions, increasing slight­
ly in active regions and towards the solar limb (Mariska et al., 1979;
Hassler et al., 1990). However, Dere and Mason (1993) have shown
that the energy input from acoustic wave heating roughly equals the
radiative losses above 10 5 K, although they note that there seems to be
insufficient energy to account for the chromospheric losses.

2 J.G. Doyle et al.,
Figure 1. (a) The intensity of the O IV 1401 š A line (dashed line) and N IV 1486 š A
(solid lined) at the solar limb (1 scan line ¸ 0.43 arc sec). The position of the `quiet'
regions values tabulated in Table 1 are labeled Q1, Q2 and Q3 while the coronal
hole is labeled CH, (b) the electron density as determined from the O IV 1399/1401
line ratio
In looking again at the question of coronal heating, we analyse obser­
vational data taken during the fourth rocket flight of the High Reso­
lution Telescope and Spectrograph (HRTS) on the 7th March 1983.
These observations are of a coronal hole and a `quiescent' region at
the solar limb in the wavelength region 1200 -- 1600 š A, with a spectral
resolution of 0.1 š A (see x2 & 3 for further details). We compare the
data with the energy requirements assuming heating by both acoustic
and Alfv'en waves (x4:1) and with numerical results obtained in linear
dissipative (viscous) MHD extended by background equilibrium shear
flows (x4:2).
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HRTS data 3
Table I. HRTS nonthermal velocities in km s \Gamma1 for three `quiet' Sun positions
using the mean of three separate scans at Q1 == line scan 600, Q2 == line
scan 170 & Q3 == line scan 713, plus the coronal hole (CH = line scan 1200)
for ions in the temperature region 10 4 to 10 6 K
Ion – ( š A) Log T Three different quiet Sun regions Coronal Hole
Q1 Q2 Q3 CH
O I 1355 4.2 16.7\Sigma0.2 ... 18.0\Sigma0.7 ...
O I 1358 4.2 14.2\Sigma2.9 ... 22.0\Sigma0.9 ...
Fe II 1372 4.3 15.2\Sigma1.4 ... 14.8\Sigma1.1 ...
Fe II 1392 4.3 20.9\Sigma0.4 ... 20.1\Sigma0.2 ...
Fe II 1405 4.3 17.7\Sigma0.1 ... 18.1\Sigma0.1 ...
Si IV 1393 4.8 38.9\Sigma0.4 41.1\Sigma0.4 46.0\Sigma0.5 40.9\Sigma0.5
Si IV 1402 4.8 40.4\Sigma0.6 49.4\Sigma0.4 45.2\Sigma0.3 43.6\Sigma0.3
S IV 1406 5.0 36.7\Sigma0.4 34.1\Sigma1.4 37.5\Sigma0.6 33.4\Sigma0.4
N IV 1486 5.2 27.6\Sigma0.2 26.3\Sigma0.3 28.9\Sigma2.3 29.5\Sigma1.4
O IV 1397 5.2 24.2\Sigma0.4 22.5\Sigma1.3 24.8\Sigma0.1 24.9\Sigma0.4
O IV 1399 5.2 32.0\Sigma0.1 32.2\Sigma0.5 31.9\Sigma0.4 33.8\Sigma0.1
O IV 1401 5.2 28.2\Sigma0.7 32.8\Upsilon0.6 29.8\Sigma0.1 31.0\Sigma0.4
O IV 1407 5.2 31.0\Sigma0.7 30.4\Sigma0.6 32.4\Sigma0.1 29.1\Sigma1.1
N V 1238 5.3 41.0\Sigma0.5 47.3\Sigma0.6 44.5\Sigma0.5 39.6\Sigma0.1
N V 1242 5.3 38.2\Sigma0.5 38.0\Sigma1.3 38.3\Sigma0.2 35.8\Sigma0.1
O V 1371 5.4 26.3\Sigma0.3 26.9\Sigma0.3 27.0\Sigma1.9 28.6\Sigma0.4
Si VIII 1445 5.9 19.3\Sigma0.5 16.8\Sigma0.9 14.2\Sigma1.6 8.2\Sigma5.4
Fe XI 1467 6.1 11.6\Sigma4.5 ... 19.2\Sigma3.2 ...
Fe XII 1242 6.2 30.2\Sigma0.6 11.9\Sigma0.6 29.5\Sigma2.3 ...
2. Observational Data
Data used in this work was obtained during the fourth rocket flight of
HRTS on the 7th March 1983 (referred to as HRTS­4). The HRTS­4
instrument consists of a high spectral and spatial resolution spectro­
graph (see Bartoe and Brueckner 1975 for further details) with a slit
length of 900 arc sec and a slit width of ¸1 arc sec on the solar surface,
corresponding to 0.1 š A resolution. The images of the Sun were focussed
onto the slit­jaws of the spectrograph by means of a 30­cm Cassegrain
telescope. In addition, HRTS­4 consists of a double grating, zero dis­
persion broadband spectroheliograph which images the spectrograph
slit jaw plate in a selected UV wavelength band with a bandpass of
150 š A (Brueckner and Bartoe, 1983). The spectroheliogram was used
to produce C IV images above the limb at 1550 š A (Cook et al., 1984).
H ff images were also photographed from the slit jaw plate image. The
HRTS­4 instrument was designed and built specially for observations
at the solar limb via the use of a curved slit. In addition, two faster
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4 J.G. Doyle et al.,
mechanically ruled gratings were used instead of the original holograph­
ic grating, covering 1200--1310 š A and 1340--1560 š A. The curved slit had
a radius of curvature greater than the solar radius. During the mission,
the slit was placed tangential to the limb and displaced 5 arc sec inside
the limb at the center, with the two ends rising to 15 arc sec above the
limb. This meant that a sample of all heights from --5 to +15 arc sec
above the limb could be obtained on slit spectra.
During the rocket flight the curved slit was placed at the solar limb
with its center at the western edge of the south polar coronal hole,
giving both coronal hole and quiet region coverage. The mission itself
is discussed in some detail in the paper by Cook et al. (1984). The
stigmatic spectrum was recorded using photographic film and the abso­
lute calibration determined using the same method as Dere and Mason
(1993). Exposure times of 17.2, 4.9, 1.4 and 0.4 seconds were used in
this work. A summed or composite spectrum was made from each expo­
sure by weighting each intensity value by the slope of the characteristic
curve for that density (see Dere and Mason for further details).
3. Observational results
At low to moderate spatial resolution all emission lines formed in
the transition region are broadened in excess of their thermal width
(Boland et al., 1975; Athay and White, 1978; Cheng et al., 1979). This
excess width may be associated with velocity fields. When the scale
of these motions is small compared with the instrumental resolution,
they are often accounted for by adding a component (¸) to the Gaussian
width
oe 2 = – 2
2c 2
`
2kT
M
+ ¸ 2
'
+ oe 2
I (1)
where ¸ is the most probable nonthermal velocity. This, of course,
assumes that the nonthermal velocities also have a Gaussian distri­
bution. The term oe I is the Gaussian instrumental width. The non­
thermal velocity is also occasionally expressed in terms of the root­
mean­squared velocity, v rms , where the factor of 3 assumes isotropic
conditions (we will return to this later).
v rms = (3=2) 1=2 ¸ (2)
In this work and as defined by Dere et al. (1984), moments of the line
profiles were calculated. These are the total line intensity I, in ergs
cm \Gamma2 s \Gamma1 sr \Gamma1 ,
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HRTS data 5
Figure 2. Non­thermal velocities in Fe II, O IV, Si IV, N V, Si VIII & Fe XII
along the solar limb observing the `quiet Sun' up to line scan 1090 and the coronal
hole thereafter. Note that for multiplets or doublets, we plot the mean of the values
derived from the individual lines. The regions at either end of the slit are at ¸ 10 \Gamma 15
arc sec above the limb while the position in the center is at ¸ 5 arc sec inside the
limb
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6 J.G. Doyle et al.,
I =
Z
I – d– (3)
the net line shift \Delta– in š A
\Delta– = I \Gamma1
Z
(– \Gamma – r )I – d–; (4)
and the line width \Delta– 2 in š A 2 ,
\Delta– 2 = I \Gamma1
Z
(– \Gamma – obs ) 2 I – d– (5)
where I – is the specific intensity in the line in ergs cm \Gamma2 s \Gamma1 sr \Gamma1
š A \Gamma1 , – the wavelength in š A, – r the rest wavelength, and – obs =
– r + \Delta–. When Gaussian line profiles are analyzed in terms of wave­
length moments, the second moment ! \Delta– 2 ?= oe 2 . Therefore by
calculating the second wavelength moment it is possible to determine
the nonthermal velocity for a wide range of ions via Eqn. (1).
In Fig. 1a we show the intensity of the O IV 1401 š A along the length
of the slit (1 scan line ¸ 0.43 arc sec). In this figure, the coronal hole
region begins at scan line 1090 upwards and is noted as CH on the
graph. Using the O IV 1399 š A/1401 š A theoretical line ratio, the varia­
tion of the electron density in the transition region at a temperature of
10 5 K is given in Fig. 1b. The atomic data adopted in the calculation
of the O IV I(1399.8 š A)/I(1407.4 š A) intensity ratios are as discussed by
O'Shea et al. (1996), with the exception of Einstein A­coefficients for
the 2s 2 2p 2 P -- 2s2p 2 4 P intercombination lines. Brage et al. (1996) have
recently recalculated radiative rates for these transitions, and shown
that the data of Nussbaumer and Storey (1982) employed by O'Shea et
al. are in error. We have therefore used the Brage et al. results in our
line ratio calculations, although we note that these lead to theoretical
values within a few percent of those estimated by Cook et al. (1995).
This is not the case for line ratios involving the O IV 1404.81 š A transi­
tion, where the Brage et al. diagnostics are up to '20% different from
those of Cook et al. However we do not employ the 1404.81 š A line as a
diagnostic, as it is blended with a S IV transition at 1404.77 š A (Cook
et al.,).
The electron density in the `quiescent' Sun is variable, ranging from
2 \Theta 10 10 cm \Gamma3 to 10 11 cm \Gamma3 . On­the­other­hand, a lower electron density
approaching 10 10 cm \Gamma3 is derived for the data beginning at scan line
1090 (i.e. in the coronal hole). A low value for N e is also derived for the
data around scan line 600. These latter observations are from a position
of ¸ 5 arc sec inside the limb, while the higher N e 's are towards the
end of the slit which is at a position ¸ 10 \Gamma 15 arc sec above the limb.
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HRTS data 7
The non­thermal velocities measured in the coronal hole and `quiet'
Sun regions are shown graphically in Fig. 2 for ions ranging in tem­
perature from 10 4 to 10 6 K. The values derived from the Si IV lines
(formed around 7 \Theta 10 4 K) are ¸ 40 km s \Gamma1 in the `quiet Sun' and coro­
nal hole, although there are small regions around the limb which show
values as large as 45 \Gamma 48 km s \Gamma1 . The O IV lines formed at 1:6 \Theta 10 5 K
have values ¸ 10 \Gamma 12 km s \Gamma1 lower than Si IV. The low temperature
Fe II (¸ 2 \Theta 10 4 K) lines have non­thermal velocities of ¸ 19 km s \Gamma1 ,
while the coronal ion, Fe XII, has velocities of ¸ 27 km s \Gamma1 in the `quiet
Sun'. Neither of these ions are detectable in the coronal hole region.
The only high temperature ion which can be detected in the coronal
hole is Si VIII (formed at 8 \Theta 10 5 K), which has non­thermal veloci­
ties of 19 km s \Gamma1 in the `quiet Sun' dropping to ¸ 11 km s \Gamma1 inside the
coronal hole.
Checking the line widths as derived via the moments method as
opposed to fitting Gaussians we find very good agreement for all the
stronger lines. For the Q1 region the two weak lines due to Fe II 1372 š A
and O I 1358 š A have line widths which are ¸ 20% smaller with the
moments method compared to Gaussian line fits. This would explain
the slightly smaller non­thermal velocities derived from these lines as
tabulated in Table 1.
Since we are observing at the limb, it is possible that some of the
stronger resonance lines are effected by opacity. For example, Shine et
al. (1976) noted that lines such as C IV 1548/1552 š A showed a non­
thermal velocity increase from ¸ 28 km s \Gamma1 on disk to ¸ 35 km s \Gamma1
off the limb. In the present data, the intersystem lines, such as the
different O IV lines, N IV 1498 š A and O V 1371 š A all formed in the
1 \Gamma 3 \Theta 10 5 K temperature range, show good internal agreement in the
calculated values of ¸. Similarly, for the strong resonance lines of Si
IV and N V, there is a similar good internal agreement in the derived
non­thermal velocities. However, the values derived from the resonance
line and those from the intersystem lines differ by 12:3 \Sigma 1:5 km s \Gamma1 ,
see Fig. 3. This we interpret entirely in terms of optical depth effects.
This is consistent with the line intensity ratios of the two Si IV lines
and the two N V lines being close to 1.5 compared to the optically thin
value of 2.0. Thus in the 10 5 K temperature range, it is unlikely that ¸
reaches values more than ¸ 30 km s \Gamma1 .
The temperatures used are the ionization equilibrium temperatures
of Arnaud and Rothenflug (1985) except for the ions of O I and Fe
II. Here the temperatures used were those that maximized the func­
tion T \Gamma1=2 exp(\Gamma\DeltaE=k T )AZI , where \DeltaE is the excitation energy of
the transition and AZI is the relative ion abundance, also taken from
Arnaud and Rothenflug.
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8 J.G. Doyle et al.,
Figure 3. Non­thermal velocities derived from the mean of the strong resonance due
to Si IV and N V compared to those derived from the intersystem lines of O IV, N
IV and O V
As noted in x1, many authors have used the excess line widths to
infer information on the coronal heating mechanism. In Fig. 4 we plot
the nonthermal velocities versus temperature for the three `quiet Sun`
positions as given in Table 1; also plotted on the graph are the off­limb
values of Mariska et al. (1978, 1979). These latter values are in good
agreement with the present estimates. Below we discuss these results
in terms of heating via acoustic and Alfv'en waves.
4. Discussion
Most theorists agree that the energy source of coronal heating lies in the
convective motion in and below the photosphere. We will not discuss
how these waves are generated and propagate up to coronal levels, but
instead discuss the merits of both heating via acoustic waves and Alfv'en
waves.
4.1. Acoustic and Alfv' en waves
The nature of the observed nonthermal broadening is still unclear. It is
uncertain whether the observed broadening should be considered as the
result of propagating waves, either acoustic or Alfv'en, or whether the
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HRTS data 9
Figure 4. Nonthermal velocities for Q1 (triangles), Q2 (plusses), Q3 (squares), the
off­limb data of Mariska et al. (1978) (crosses) and Mariska et al. (1979) (stars).
The dash (­ ­ ­) line, Pe corresponds to the nonthermal velocities assuming a con­
stant acoustic flux of 1:5 \Theta 10 6 erg cm \Gamma2 s \Gamma1 based on a constant electron pres­
sure of 3 \Theta 10 15 cm \Gamma3 K. The double­dot­dash (­ .. ­ .. ­) line, Ne , corresponds to
a constant acoustic flux of 1:5 \Theta 10 6 erg cm \Gamma2 s \Gamma1 based on a constant density of
2 \Theta 10 10 cm \Gamma3 . The solid line, PAlfven , corresponds to a constant Alf'ven flux of
1:5 \Theta 10 6 erg cm \Gamma2 s \Gamma1 , a pressure of 3 \Theta 10 15 cm \Gamma3 K and a magnetic field of 10
Gauss. Note that values of ¸ above ¸ 30 km s \Gamma1 are probably not correct due to
opacity effects.
nonthermal broadening just represents turbulent motions in the atmo­
sphere. Earlier papers by Boland et al. (1975), Cheng et al. (1979) and
Mariska et al. (1978, 1979) have assumed that the observed nonthermal
broadening is a result of a propagating wave mode. More recent papers
by Dere and Mason (1993) and McClements et al. (1991) support this
view. Under this assumption it becomes possible to estimate the energy
flux due to either acoustic or Alfv'en waves. Firstly however we must
make the further assumption that the observed nonthermal broaden­
ing is isotropic in nature, i.e. it has the same magnitude at disk center
and the limb. We feel that this assumption is justified as earlier line
widths measurements of quiet Sun conditions (Mariska et al., 1978) and
flares (Mariska, 1994) taken on the disk and at the limb are in good
agreement.
hrts—sol.tex; 9/01/1997; 10:28; no v.; p.9

10 J.G. Doyle et al.,
The mechanical energy density is given by
E = aev 2
rms (6)
where ae is the mass density and v rms is the rms velocity along the line
of sight. Using Eqn. (2) the above equation can be rewritten as follows
(assuming isotropic conditions),
E = 3
2 ae¸ 2 (7)
The energy flux passing through the transition zone is given by
OE = EC (8)
where C is the propagation velocity of the particular wave mode being
examined. For acoustic modes C is the sound speed and we have
OE s = 3
2 ae¸ 2
`
5
3
P
ae
' 1=2
(9)
where P is the pressure and we assume fl is equal to 5=3. For Alfv'en
waves C is the Alfv'en speed, and we have
OE A = 3
2 ae¸ 2 B
(4úae) 1=2
(10)
where B is the magnetic field strength. Using the perfect gas law to
eliminate the mass density from Eqns. (9) and (10) above we get the
following,
OE s = 3
2
`
5
3
Żm p
k
' 1=2
¸ 2 P
T 1=2
(11)
and
OE A = 3
2
`
1
4ú
Żm p
k
' 1=2
¸ 2
`
P
T
' 1=2
B (12)
where Ż is the mean molecular weight, m p is the mass of a proton, k is
Boltzmann's constant, T the electron temperature and P the electron
pressure. In Fig. 4 we plot ¸ as a function of temperature determined
from Eqn. (11) with OE s constant at 1:5 \Theta 10 6 ergs cm \Gamma2 s \Gamma1 . We have
taken a pressure of 0:83 dyne cm \Gamma2 and a value of Ż = 0:62 in calcu­
lating these values. These values of ¸ are marked on the graph (dashed
line) with the letter P e , which refers to a condition of constant pres­
sure. There have been a few suggestions, e.g. Doyle et al. (1985) in the
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HRTS data 11
analysis of sunspot data, that the electron density can be constant in
the upper atmosphere, i.e. from 10 5 K to 10 6 K.
With the same constant flux and a constant density of 2 \Theta 10 10 cm \Gamma3 ,
we get the line marked N e . The line marked P Alfven represents the val­
ues of ¸ calculated from Eqn. (12) with OE A constant at the same flux
with B=10 Gauss. In addition we plot on this graph our observation­
al results of non­thermal velocity for three different regions along the
slit. Also included are the nonthermal velocity values published in the
papers by Mariska et al. (1978, 1979).
Via an emission measure curve, Dere and Mason (1993) estimated
radiative losses above 10 5 K of 6 \Theta 10 5 erg cm \Gamma2 s \Gamma1 in the `quiet Sun',
this may be compared to losses of 3 \Theta 10 5 erg cm \Gamma2 s \Gamma1 if we use the
Raymond and Doyle (1981) `quiet Sun' emission measure curve and
the radiative loss function of Cook et al. (1989). Over the whole atmo­
sphere, i.e. chromosphere, transition zone and corona, the total losses
are ¸ 1:4 \Theta 10 6 erg cm \Gamma2 s \Gamma1 . These figures therefore suggests that
there is more than sufficient energy in the observed nonthermal veloc­
ities to heat the solar chromosphere, transition zone and corona via
acoustic waves if one assumes a constant electron pressure atmosphere
below 10 5 K and a constant electron density above this temperature.
These results differ from previous analyses due to the constant density
assumption, which can be confirmed with the CDS and SUMER instru­
ments on board SOHO. However, these simple energy type analysis do
not address the problem of how acoustic waves could actually reach the
corona before being damped. It is therefore likely that more than one
heating mechanism is operating simultaneously, in particular Alfv'en
waves as these can easily reproduce the observed energy losses, even,
assuming a field strength of only a few Gauss. Furthermore because
of the inhomogeneous nature of the solar atmosphere, as can be seen
from the variations across the limb shown in Fig. 1, any conclusions
based on these rather simple expressions should be treated with cau­
tion. Below, we look more closely at Alfv'en waves and in particular an
efficient dissipation mechanism, i.e. resonant absorption.
4.2. Resonant absorption of Alfv' en waves
The solar atmosphere is a highly non­uniform plasma which is a nat­
ural medium for MHD waves. MHD waves might play an important
role in explaining the observed high temperatures in the solar corona.
These waves can be generated by the granular motions (e.g. foot­point
motion) and a part of their kinetic energy can be transformed into heat
in the magnetic loops. When the background equilibrium is character­
ized by steep variations in narrow layers it is common to represent the
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12 J.G. Doyle et al.,
Figure 5. Absorption coefficient ff versus the equilibrium flow, v flow
v A
for model I
(upper panel) and model II (lower panel) where vflow is the mass upflow/downflow
velocity and vA is the Alfv'en velocity. Note that the observational data suggest
v flow
v A
is in the range :01 to 0:05 depending upon the electron density and magnetic
field strength
background by one or more discontinuities, with uniform plasma on
both sides of the discontinuity. Such a medium can support waves that
propagate along the discontinuity but are strongly evanescent perpen­
dicular to the discontinuity. In nature it is hard to find true discontinu­
ities, instead the physical quantities vary continuously across an inter­
face in­between the different layers of plasma with different equilibrium
parameters. When the plasma is non­uniform a continuous spectrum
of Alfv'en and slow waves may exist in ideal MHD. There exists also
a third type of MHD waves, the fast magneto­acoustic waves. These
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HRTS data 13
waves have only discrete frequencies in the eigenspectrum which limits
their viability for coronal heating.
The physical basis of resonant absorption can be partly understood
in the context of linear ideal MHD. Let us shortly recall the concept of
resonant absorption and continuous spectra of Alfv'en and slow waves
in ideal MHD as given in a recent review by Goossens and Ruderman
(1996). We consider the stationary state of externally driven linear
MHD oscillations superimposed on a straight 1­dimensional cylindrical
plasma column, where all perturbed quantities oscillate with the fre­
quency of the external driver, !. If an external driver is turned on at
some instant, both free (at frequency !C for slow --cusp-- waves) and
forced oscillations (at frequency !A for Alfv'en waves) are excited in
the inhomogeneous system. After a transient time the free oscillations
decay due to phase mixing, and only the forced oscillations remain.
At the position where the resonant condition ! 2 = ! 2
A (r A ) is fulfilled
the global wave motion will be locally in resonance on a particular
magnetic surface. In ideal MHD the resonantly accumulated energy
will be infinite at the resonant position. This is of course unphysically
and tells us that part of the basic physics has been left out of the
mathematical description. This basic physics is dissipation which is
important close to the resonant point. What we learn from ideal MHD
is that the wave solutions are characterized by steep gradients and small
length scales (and that is exactly what is needed for efficient dissipation
of kinetic energy).
Since the first suggestion of resonant absorption as a dissipation
process (Ionson, 1978), much work has been carried out to justify the
resonant absorption mechanism. In the more recent theoretical models
of resonant absorption (Erd'elyi and Goossens, 1996), mass flows were
shown to be very important. Unfortunately, since we are observing at
the limb we have no information on the vertical motions, although such
information is available from previous data. For example, observations
from the SMM satellite (Gebbie et al., 1981) and HRTS (Achour et al.,
1995) show that there is a net background downflow of the order of a
few km s \Gamma1 . In the quiet Sun, the largest downflow reach 8 km s \Gamma1 at
10 5 K, while an active region has a value closer to 15 km s \Gamma1 . At coronal
temperatures, the net downflow velocity is ¸ 4 km s \Gamma1 in an active
region (the coronal lines are to weak in the quiet Sun disk spectrum).
Taking as typical of the corona a magnetic field strength of B = 10G,
a temperature T = 1:5 10 6 K, an electron density of N e = 2 10 9 cm \Gamma3
(i.e. we assume a constant pressure from the transition region to the
corona), would imply v A ¸ 440 km s \Gamma1 . Thus the observed plasma flow
is at most a few percent of the characteristic Alfv'en speed, i.e. v flow
v A
¸
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14 J.G. Doyle et al.,
0:01 to 0:02, although increasing N e by an order of magnitude could
take this value close to 0:05.
In Fig. 5a we plot as a function v flow
v A
, the computational results of
Erd'elyi and Goossens (1996) for Models I of Poedts et al. (1989). The
range of v flow
v A
is broader than the characteristic values for the observed
Doppler­shifts, i.e. v flow
v A
¸ 0:01 to 0:05. This figure shows that the flow
has a determinant effect on the resonant absorption, and even a very
small background mass­flow can drastically influence the absorption of
the incident waves.
By `decreasing' the equilibrium flow (from v flow
v A
= 0) in the parallel
direction (i.e. v flow
v A
? 0), the behaviour of the absorption rate will be
totally different from the one obtained by simulating flow in the anti­
parallel direction. The decrease of v flow
v A
can enhance the absorption rate
very strongly, even up to 100% at v flow
v A
¸ 0:05 (e.g. taking B = 10G
and N e = 2 10 9 cm \Gamma3 corresponds to ¸ 22 km s \Gamma1 downflow which is
slightly larger than the observational data would indicate. However,
assuming a larger coronal electron density, e.g. 2 \Theta 10 10 cm \Gamma3 , the max­
imum absorption rate would imply a mass downflow of 7 km s \Gamma1 in
excellent agreement with observations. This means a total resonant
absorption of the incoming driving wave, hence the inclusion of an
equilibrium flow opposite to the magnetic field lines can enhance con­
siderably the efficiency of the resonant absorption mechanism.
Fig 5b. shows the absorption rate versus the flow strength for Poedts
et al. Model II. Model II is characterized by the same type of equilibri­
um profile as in Model I, except that the profile of the current density
is more peaked than in Model I and the density profile is parabolic. The
aspect ratio is the same as in Model I (i.e. ffl = 20) and the local Alfv'en
frequency is increasing from the axis towards the boundary of the loop.
Here we see that for a broad range of flow strength parameters, the
coupling between the driver and the flux tube is very efficient. The
equilibrium does not really effect the absorption rate as dramatically
as in the previous case. The 100% absorption at v flow
v A
= 0:19 reflects
a discrete eigenvalue of the system.
Hence one can see that resonant absorption is strongly influenced by
equilibrium flows in coronal loops with constant density, in particular in
the case of an equilibrium plasma motion anti­parallel to the stationary
magnetic field lines with a value of ¸ 5% of the local Alfv'en speed. An
unfortunate limitation of these linear computations is that we cannot
give any estimate for the absorbed energy. Since in linear MHD the
amplitude of the driving Alfv'enic waves scales the whole solution.
hrts—sol.tex; 9/01/1997; 10:28; no v.; p.14

HRTS data 15
5. Conclusions
The different dissipation theories have different observational constraints
but neither the existing observational data nor the models are sophis­
ticated enough to give a definitive answer. Despite previous claims,
acoustic waves may be able to provide the necessary energy input to
the corona provided that a constant electron pressure assumption holds
below 10 5 K, while above this temperature a constant electron density
assumption is valid. This can be confirmed with the CDS and SUMER
instruments onboard SOHO. For example, SUMER has sufficient spec­
tral resolution and wavelength coverage to allow the observation of line
widths over a temperature range from the chromosphere to the coro­
na, while CDS will allow an evaluation of the electron density using a
series of different line diagnostics covering the above temperature range.
Recent model calculations by Laing and Edwin (1995) show that acous­
tic waves in a structured atmosphere can provide an energy input of
¸ 5 \Theta 10 7 erg cm \Gamma2 s \Gamma1 assuming nonthermal velocities of 50 km s \Gamma1
and a electron density of 10 11 cm \Gamma3 for wave periods in the 15--60 secs.
Decreasing the electron density by a factor of 10 reduces the energy
input by an order of magnitude, while the valid wave periods increase
to the 1--150 sec range.
As regards a distinction between the various heating mechanisms,
e.g. acoustic, Alfv'en wave heating, electric current dissipation and MHD
turbulent models, this might best be done in terms of the time­scales for
the various mechanisms. The dissipation of Alfv'en waves have a time­
scale comparable to the wave periods themselves. On­the­other­hand,
current sheets are supported by external driving forces, e.g. photospher­
ic motions, for intervals long enough compared to the dissipation. The
selective decay of MHD turbulence might involve A.C. electric currents.
Dere (1989) in the analysis of velocity fluctuations as measured via the
observed Doppler shifts in the C IV 1550 š A lines suggested that most
of the power was at higher frequencies than currently observable.
Numerical work on resonant absorption of Alfv'en waves have shown
that a background velocity shear flow can significantly influence the
absorption of the incoming driving waves. The presence of the equilibri­
um flow might therefore be determinant for the power loss of the Alfv'en
waves. It is shown that even for rather small steady flows, the absorp­
tion rate can become zero (i.e. total reflection). For other values of the
equilibrium flow we find that the resonant absorption can be strongly
enhanced, even up to total absorption of the resonant wave. This total
absorption might reflect radiating eigenmodes (see e.g. Goossens and
Hollweg, 1993).
hrts—sol.tex; 9/01/1997; 10:28; no v.; p.15

16 J.G. Doyle et al.,
A possible method for testing the resonant absorption of Alfv'en
waves is to measure the polarity of the non­thermal motion. Erd'elyi et
al. (1995) have shown that the dominant non­thermal motion (in the
case of resonant absorption of Alfv'en waves in dissipative MHD with
stationary background flow) has a polarity perpendicular to the mag­
netic field lines. The perpendicular component of the perturbations in
the magnetic surfaces has a 1=x singularity, which is proportional to
the cubic root of the Reynolds number, which should be very large in
coronal regions. The other component on the magnetic surface (e.g. the
parallel displacement) is continuous. The third component, perpendicu­
lar to the magnetic surfaces, has a log(x) singularity. Thus observations
that can provide information concerning the polarity along the mag­
netic field lines in coronal loops (or in other hot structures in the solar
atmosphere) are fundamental for Alfv'en wave heating.
It should also be noted that in solar flux tubes (like sunspots, coronal
loops) the equilibrium flow is probably very complicated. The accuracy
and resolution of present instruments are not yet sufficient in order to
observe the detailed structure of the background plasma flows. In simu­
lations it is mostly supposed that equilibrium flows follow the magnetic
field lines, and the amplitudes of these flows are proportional to the
local Alfv'en speed. But this is just an approximation, maybe heuristic.
Although the present dataset allows an accurate determination of the
non­thermal velocities from 10 4 K to 10 6 K, we are limited by the lack
of density diagnostics in the coronal region. New observations, such as
those that will be obtained by the high resolution CDS and SUMER
spectrometers onboard SOHO, should give us a more accurate picture
concerning the Doppler­shift in different magnetic structures, but also
a determination of the electron density as a function of temperature.
The electron density at 10 5 K in the `quiescent' Sun is variable, rang­
ing from 2\Theta10 10 cm \Gamma3 to 10 11 cm \Gamma3 , with the coronal hole giving a value
of ¸ 10 10 cm \Gamma3 . A low value for N e is also derived for the data which
is at a position of ¸ 5 arc sec inside the limb, while the higher N e 's is
for a position ¸ 10 \Gamma 15 arc sec above the limb.
The derived non­thermal velocities from the intersystem lines, such
as the different O IV lines, N IV 1498 š A and O V 1371 š A all formed
in the 1 \Gamma 3 \Theta 10 5 K temperature range, show good internal agreement
with ¸ ¸ 28 km s \Gamma1 in both the `quiet Sun' and coronal hole. Simi­
larly, the strong resonance lines of Si IV and N V have good internal
agreement in the derived non­thermal velocities. However, the values
derived from the resonance line are 12:3 \Sigma 1:5 km s \Gamma1 larger. This dif­
ference can be interpreted entirely in terms of optical depth effects.
The low temperature Fe II (¸ 2 \Theta 10 4 K) lines have non­thermal veloc­
ities of ¸ 19 km s \Gamma1 , while the coronal ion, Fe XII, has velocities of
hrts—sol.tex; 9/01/1997; 10:28; no v.; p.16

HRTS data 17
¸ 27 km s \Gamma1 in the `quiet Sun'. Neither of these ions are detectable in
the coronal hole region. The only high temperature ion which can be
detected in the coronal hole is Si VIII (formed at 8 \Theta 10 5 K), which
has non­thermal velocities of 19 km s \Gamma1 in the `quiet Sun' dropping to
¸ 11 km s \Gamma1 inside the coronal hole.
Acknowledgements
Research at Armagh Observatory is grant­aided by the Dept. of Edu­
cation for N. Ireland while partial support for software and hardware
is provided by the STARLINK Project which is funded by the UK
PPARC. EOS is supported via a studentship from Armagh Observa­
tory while RE is supported via PPARC grant GR/K43315. RE is also
grateful to the Research Council of the K.U. Leuven where some of
these calculations were performed. FPK acknowledges support from
the Royal Society. The NRL HRTS rocket program is supported by a
grant from the NASA Space Physics Division.
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