Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/~ambn/291jgd.ps Äàòà èçìåíåíèÿ: Wed Jul 1 13:46:40 1998 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:05:32 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ion drive

A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
02.12.7; 02.18.7; 08.01.2; 08.03.3; 08.12.1; 09.07.1
ASTRONOMY
AND
ASTROPHYSICS
Broadening of Si viii lines observed in the Solar polar
coronal holes
D. Banerjee 1 , L. Teriaca 1 , J.G. Doyle 1 , and K. Wilhelm 2
1 Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
emails: dipu@star.arm.ac.uk, lte@star.arm.ac.uk, jgd@star.arm.ac.uk
2 Max­Planck­Institut f¨ur Aeronomie, D­37191 Katlenburg­Lindau, Germany
email: wilhelm@linmpi.mpg.de
Received date, accepted date
Abstract. We study the variation of the line width and
electron density as a function of height above two coro­
nal holes from forbidden spectral lines of Si viii. The
spectra were obtained with the Solar Ultraviolet Measure­
ments of Emitted Radiation spectrometer flown on the So­
lar and Heliospheric Observatory spacecraft. The observa­
tions concentrate on the dark regions outside the plumes,
which are believed to be the locations, where the fast solar
wind originates. The line width data show that the non­
thermal line­of­sight velocity increases from 27 km s \Gamma1 at
27 arc sec above the limb to 46 km s \Gamma1 some 250 arc sec
(i.e. ¸180,000 km) above the limb. The electron density
shows a decrease from 1:1 10 8 cm \Gamma3 to 1:6 10 7 cm \Gamma3 over
the same distance. This data implies that the non­thermal
velocity is inversely proportional to the quadratic root of
the electron density, in excellent agreement with that pre­
dicted for undamped radially propagating Alfv'en waves.
We show that the energy flux associated with these hydro­
magnetic waves is sufficient to drive the high speed solar
wind streams.
Key words: Sun: Corona ­Ultraviolet: SOHO--Sun: waves
1. Introduction
Doschek & Feldman (1977) measured line profiles of for­
bidden lines and inferred non­thermal velocities between
17 and 25 km s \Gamma1 at altitudes of 10--20 arc sec above
a coronal hole and several `quiet' Sun regions. Based on
SKYLAB data, Doschek et al. (1977), Nicolas et al. (1977)
and Mariska et al. (1978) have shown that the line width
increases with height above the limb. These earlier ob­
servations were restricted to temperatures in or below
the transition region and thus inferring details concerning
coronal heating was not possible. Hassler et al. (1990) ob­
tained line profile data for the coronal lines Mg x 609/625 š A
Send offprint requests to: D. Banerjee
up to ¸ 140; 000 km above the limb. Recently Doyle et al.
(1998) showed an increase in line width up to ¸ 70; 000 km
above the limb for 'quiet' Sun conditions at the equatorial
Western limb. Such data has important implications for
deciding the merits of the different coronal heating mech­
anisms. This paper provides observational evidence sug­
gesting that a significant energy flux may be transported
by Alfv'en waves in the solar polar coronal holes.
To isolate line broadening due to hydromagnetic waves,
it is necessary to observe the line widths of high temper­
ature coronal lines (¸ 10 6 K) above the limb. With the
launch of SOHO, and in particular with the, SUMER in­
strument (Wilhelm et al. 1995), line width measurements
of coronal lines as a function of position above the limb
are possible. Here, we report an observational sequence
involving the Si viii line pair (1445.75 š A & 1440.49 š A).
Observations of this line pair allows us to determine si­
multaneously the electron density and the line width.
Recently Doschek et al. (1997) have used some of the
same datasets, although they determined only the elec­
tron densities. We go one step further, measuring the line
widths along with the electron density. Wilhelm et al.
(1998) also have determined electron densities in coronal
holes, but their main interests were on the electron tem­
perature estimates in polar plumes and inter­plume lanes.
Ulysses observations have indicated that fast solar wind
originates from polar coronal holes. During the year 1996,
part of the minimum sunspot activity, the polar coronal
holes were well developed and relatively stable. This gives
us the ideal opportunity to study the fast solar wind ac­
celeration site, namely the polar coronal holes. Koutchmy
& Bocchialini (1997) and Wilhelm et al. (1998) have sug­
gested that the source of the fast solar wind lies in the
low density background plasma (and not the plumes). So
in this study we concentrate on the darkest region outside
plumes, namely the inter­plume lanes (see Fig. 1). Our
objective is to infer the cause of the line width variation,
plus the possible role of waves on coronal heating and the
driving of the solar wind.

2 Banerjee et al.: Si viii line broadening in a coronal hole
Table 1. SUMER Polar coronal hole observations, N(S)PCH = north(south) polar coronal hole. Slit 1,2 = 4x300, 1x300 arc
sec 2 respectively.
Date Region Detector Slit Solar X­Y Co­Ord Starting time End Time Exp. Time (s) No. of spectra
4 Nov'96 NPCH B 2 0,+1107 04:54 10:36 320 63
10 Dec'96 NPCH B 1 0,+1229 03:30 09:11 320 68
5 May'96 SPCH A 2 0,\Gamma1100 03:46 04:18 320 6
2. Observations and Data Reduction
SUMER is a normal incidence spectrograph operating over
the wavelength range 400 š A to 1610 š A. The off­axis parabola
mirror is moveable in two dimensions around the focal
point allowing pointing of the instrument independent of
the spacecraft altitude. Four slits are available; 4 \Theta 300,
1\Theta300, 1\Theta120 and 0:3\Theta120 arcsec 2 . For the data obtained
here we used the 4 \Theta 300(slit 1) and 1 \Theta 300(slit 2) arc sec 2
slits. The detectors (see Siegmund et al. 1994) have 1024
spectral pixels and 360 spatial pixels, each. The central
area is coated with KBr which increases the quantum effi­
ciency by up to an order of magnitude in the range 900 š A
to 1500 š A. The dates of observations, locations, pointing
Fig. 1. The locations for slit 1 (on Dec 10) & slit 2 (on Nov 4)
datasets in the north coronal hole on an EIT image (courtesy of
the EIT consortium) of He II 304 š A obtained on 4th November,
1996 at 17:51 UT.
positions of the centre of the slits, slit sizes and exposure
times are given in Table 1. The slits were, for a nominal
spacecraft altitude, oriented in the north­south direction
parallel to the solar radius. The spatial resolution (1 arc
sec ¸ 715 km) along the slit provides the intensities as
a function of height outside the limb. For the observa­
tion in the north polar coronal hole (NPCH) we used two
sequences of temporal serial images (i.e.a series of spec­
tra taken in the same pointing position but at successive
times). All the images taken with the same slit and at the
same location were summed resulting in two final spectra
relating to the NPCH and one for the south polar coronal
hole (SPCH). Details on the total number of individual
Fig. 2. Density dependence of the Si viii line ratio
(1445.75 š A/1440.49 š A) for the atomic data within CHIANTI
(solid line) and for comparison the ratios given by Doschek et
al. (1997) (+).
spectra used is given in Table 1. The objective of the ob­
serving programme was to obtain high S/N spectra of the
Si viii 1440.49 š A and 1445.75 š A lines as a function of posi­
tion in the N(S)PCH. Fig. 1 shows an image of the North
polar region taken with EIT in He II 304 š A at 17:51 UT
on November 4, 1996 with the slit 1 & 2 locations super­
imposed. Note that the slits are positioned in the darkest
region of NPCH, in the inter­plume lanes. We have also
checked, with the EIT images on May 5, 1996, that for
the SPCH, the pointing of the slit was in the inter­plume
region.
The inference of the electron density in an ionized
plasma using the line ratio of forbidden lines of higher ions
like Si viii has been used by several authors, e.g. Feldman
et al. (1978), Doschek et al. (1997), Wilhelm et al. (1997)
and Doyle et al. (1998). Fig. 2 shows the variation of the
Si viii line intensity ratio as a function of electron density
obtained from the CHIANTI atomic data base (Dere et
al. 1997). The (+) symbols indicate the values used by
Doschek et al. (1997). This figure clearly shows that the
Si viii ratio is a useful density diagnostic for densities in
the range 10 7 cm \Gamma3 to several 10 9 cm \Gamma3 . For the SUMER
instrument, the process of data reduction involves three
main steps; destretching, radiometric calibration and slit
effects corrections, which were done using various IDL rou­

Banerjee et al.: Si viii line broadening in a coronal hole 3
Fig. 3. (a,b) Sample Si viii 1445.75 š A & (c,d) 1440.49 š A line fits to the data at 47(a,c) and 139(b,d) arc sec above the limb
respectively (solid line are the data and the dotted line is the fit). Intensities units are in erg cm \Gamma2 s \Gamma1 sr \Gamma1 š A \Gamma1
tines from within the SUMER software tree (see Doyle et
al. 1998 for details).
3. Results
A Gaussian profile was fitted to each spectral line. Only
data from outside the solar limb could be fitted due to the
faintness of the Si viii lines on­disk and blending problems
(Doyle et al. 1998). In Fig. 3, we show sample fits to the
data at 47 arc sec above the limb for slit 2 and 139 arc
sec for slit 1. In all instances, both Si viii lines can be
fitted with a single Gaussian, although the 1440.39 š A line
is slightly affected by the presence of a nearby line.
Table 2 summarizes the results for the N(S)PCH. Figs.
4a & 5a shows the variation of electron density above the
limb, calculated from the CHIANTI data in Fig. 2. In the
case of the NPCH data, the densities are derived from line
profiles averaged over 10 arc sec (1 spatial detector pixel
corresponds to ¸1 arc sec) up to 190 arc sec, with the last
three data points being the average of 20, 20 and 30 arc
sec respectively. For the SPCH we have averaged over 20
arc sec throughout.
The error in deriving the electron density in Figs. 4a &
5a depends not only on the line strengths but also on the
atomic data. The relative measuring error in deriving N e
is estimated to be 12% to 15%, similar to the expected ab­
solute error (see Laming et al. 1997). Recently Doyle et al.
(1998) measured electron densities at the equatorial limb,
again using the Si viii line ratio. Comparing their Table 1
with the present values, the densities in the coronal holes
are significantly lower than the `quite' Sun, approximately
a factor of two as already noted by Doschek et al. (1997).
The spectral line profile of an optically thin line results
from the thermal broadening caused by the ion tempera­
ture T i and broadening caused by non­thermal motions.
Assuming a Gaussian distribution one obtains,
FWHM =
''
4ln2
`
–
c
' 2 `
2kBT i
M + ¸ 2
' # 1=2
; (1)
where T i is the ion temperature, and M the ion mass. ¸
is the non­thermal speed, related to the wave amplitude
by ¸ 2 = 1
2
! ffi v 2 ?, where the factor of 2 accounts for
the polarization and direction of propagation of a wave
relative to the line of sight. The measured line profiles
will also depend on the instrumental characteristics. The

4 Banerjee et al.: Si viii line broadening in a coronal hole
Table 2. A summary of the measured and corrected line width for Si viii 1445.75 š A, plus the derived non­thermal velocities
(¸) and the electron densities for N(S)PCH = north(south)polar coronal hole and slits 1,2 = 4x300, 1x300 arc sec 2 respectively,
using T i = 1 10 6 K.
Limb posn. Region Slit I(1445.75 š A) Ratio FWHM (mš A) ¸ log Ne/cm \Gamma3
(arc sec) (erg cm \Gamma2 s \Gamma1 sr \Gamma1 ) Measured Corrected (km s \Gamma1 )
27 NPCH 2 7.42 9.31 308 291.6 27.0 8.06
37 NPCH 2 6.87 10.31 326 310.5 30.1 8.13
47 NPCH 2 4.94 8.62 344 329.4 33.1 8.02
57 NPCH 2 3.45 8.10 348 333.6 33.7 7.98
67 NPCH 2 2.41 6.69 374 360.6 37.8 7.87
77 NPCH 2 1.78 6.24 372 358.5 37.5 7.83
87 NPCH 2 1.41 6.04 381 367.9 38.9 7.81
98 NPCH 2 1.02 5.36 382 368.9 39.0 7.74
108 NPCH 2 0.72 4.99 395 382.3 41.0 7.69
118 NPCH 2 0.57 4.82 382 368.9 39.0 7.67
119 NPCH 1 0.50 4.91 432 402.5 43.9 7.68
129 NPCH 1 0.45 4.21 445 416.5 45.8 7.59
139 NPCH 1 0.37 4.23 421 390.6 42.2 7.59
149 NPCH 1 0.27 3.88 444 415.4 45.7 7.54
159 NPCH 1 0.22 3.49 447 418.6 46.2 7.48
169 NPCH 1 0.19 3.61 442 413.3 45.4 7.50
179 NPCH 1 0.15 3.59 446 417.5 46.0 7.49
194 NPCH 1 0.11 2.80 453 425.0 47.1 7.33
219 NPCH 1 0.07 2.69 428 398.2 43.3 7.30
250 NPCH 1 0.04 2.39 449 420.8 46.5 7.22
17 SPCH 2 7.87 10.81 297 285.5 26.0 8.16
37 SPCH 2 4.03 7.94 305 293.8 27.4 7.97
57 SPCH 2 2.28 6.62 327 316.6 31.1 7.86
77 SPCH 2 1.32 5.41 360 350.6 36.3 7.74
spectrometer will introduce an instrumental width, which
is 77 mš A(FWHM ) for detector A and 95 mš A(FWHM )
for detector B at 1445 š A. In addition, the slit width has
to be considered. The latter effect can not be treated as
a Gaussian distribution and a de­convolution is required.
This has been done using the standard SUMER software
and the corrected values are tabulated in Table 2. The
magnitude of this correction is small, although it does
result in a correction in the derived non­thermal veloc­
ity of ¸2 km s \Gamma1 for slit 2 and ¸2.5 km s \Gamma1 for slit 1.
Note that the formation temperature of Si viii in ion­
ization equilibrium is T e = 8 10 5 K. Recently Wilhelm
et al. (1998) have estimated the electron temperatures in
the inter­plume lanes of coronal holes to be Ÿ 900; 000
K. They have also indicated that ions in a coronal hole
are hotter than the electrons. Furthermore, observations
at larger distances from the limb obtained by UVCS (An­
tonucci et al. 1997), demonstrate that the assumption of
collisional ionization equilibrium(CIE) and the common
notion that T i = T e in the coronal hole plasma are not
realistic. Tu et al. (1998) have argued that in a coronal
region a few tens arc seconds above the solar limb the
CIE assumption may not apply, and therefore the ion ki­
netic temperatures may not have anything to do with the
electron temperature or the formation temperature. To
explore further, the ion temperature versus electron tem­
perature question, we derive the non­thermal velocity (¸)
using Equation (1), assuming three different ion tempera­
tures, 8 10 5 K, 1 10 6 K and 2 10 6 K. From the comparison of
our observed dataset and theoretical implications (which
will be discussed in the next section) we find T i = 1 10 6 K
to be the best choice. This is consistent with the estimates
given by Tu et al. (1998), who have measured the effective
ion temperatures in a solar polar coronal hole observed by
SUMER. In Figs. 4b & 5b we show the variation of ¸
with height above the limb for T i = 1 10 6 K. For the non­
thermal velocity measurement, we have used the width
of the (stronger) Si viii 1445.75 š A line. The accuracy in
the ¸ measurement is around 0.5 km s \Gamma1 for slit 2 and
around 1 km s \Gamma1 for slit 1. The Si viii data indicates that
for the NPCH, the non­thermal velocity increases from
¸ 27 km s \Gamma1 at 25 arc sec above the limb to ¸ 46 km s \Gamma1
at 250 arc sec above the limb for plasma around 1 10 6
K. For the SPCH, the non­thermal velocity increase from
¸ 26 km s \Gamma1 at 17 arc sec above the limb to ¸ 36 km s \Gamma1
at 80 arc sec above the limb. In Figs. 4 & 5 the dashed
lines are second order polynomial fits to the observed data.
Note that for the last three points in Fig. 4b (beyond 200
arc sec) in the case NPCH, the uncertainty in measuring
¸ is ¸ 1.5 km s \Gamma1 . The (+) symbols are obtained from the

Banerjee et al.: Si viii line broadening in a coronal hole 5
Fig. 4. (a) The electron density as measured via the Si viii
1445 š A/1440 š A line ratio as a function of position above the
North Polar coronal hole, (b) the non­thermal velocity as de­
rived from Si viii using T i = 1 10 6 K. The dashed line is a sec­
ond order polynomial fits, while the (+) symbols correspond
to theoretical values (see text).
theoretical expression (Eq. [3]). The implications of these
plots will be discussed in the following section.
4. Discussion
The measurement of line widths can provide information
concerning ion temperatures, sub­resolution turbulent mo­
tions and velocity fluctuations associated with magneto­
hydrodynamic (MHD) waves in the corona. Although a
number of mechanisms, including Alfv'en waves, have been
shown to be consistent with previous observations (Has­
sler et al. 1990; Doyle et al. 1998), none of these observa­
tions of the line broadening have been done for the coro­
nal hole regions. Doyle et al. (1998) have discussed, that
purely acoustic waves are not important for coronal heat­
ing (Hollweg, 1990) and the slow and fast magnetoacoustic
Fig. 5. (a) The electron density as measured via the Si viii
1445 š A/1440 š A line ratio as a function of position above the
South Polar coronal hole, (b) the non­thermal velocity as de­
rived from Si viii using T i = 1 10 6 K. The dashed line is a
second order polynomial fit, while the (+) symbols correspond
to theoretical values (see text).
waves are also not able to supply the coronal energy re­
quirements.
It is well known that the Alfv'en waves propagate vir­
tually undamped through the quasi­static corona and de­
posit their energy flux in the higher corona. Low frequency
Alfv'en waves are reflected in the chromosphere­corona
transition region, but the higher frequency Alfv'en waves
with a wavelength that is shorter than the Alfv'en speed
scale height, may play an important role in heating coronal
holes (Hollweg 1990). Axford & McKenzie (1992) first sug­
gested that the high speed wind originates directly from
the chromospheric network at the bottom of a coronal
hole, and that high frequency waves at 10­10 4 Hz are pos­
sibly created there, by small scale reconnections. Based
on these ideas Marsch & Tu (1997) have developed a two­
fluid model and demonstrated that Alfv'en waves are quite
appropriate for both heating the coronal funnels through

6 Banerjee et al.: Si viii line broadening in a coronal hole
Fig. 6. Variation of electron density with non­thermal velocity
for the N(S)PCH. The squared boxes represents the measured
values as given in Table 2 and the solid lines represent the
theoretical relation (Eq. [3]) for magnetic field strengths as
indicated.
cyclotron dissipation and accelerating the wind with the
help of wave pressure gradient. They have considered spa­
tial inhomogeneity in the form of rapid expanding flux
tubes, implying an increase of its cross sectional area and
a decrease of magnetic field strength with height. From a
parametric study they inferred that for acceleration of the
solar wind the wave amplitude required are of the order of
several 10 km s \Gamma1 , which is consistent with our observa­
tions. Thus our observations will provide initial conditions
for such model calculation.
Although Parker (1958) developed the theory of solar
wind acceleration, the mechanism responsible is not yet
fully understood. In particular, observations of the high
speed solar wind indicates the need for a substantial en­
ergy flux that is transported outward from the coronal
base by some process other than convection or classical
thermal conduction. Marsch & Tu (1997) have pointed out
that the most natural means of energy transport guided
by field lines is through the Alfv'en mode and its high fre­
quency extensions up to the local cyclotron frequency. The
energy flux density in the corona due to Alfv'en waves is
given by (Doyle et al. 1998),
F =
r
ae
4ú ! ffi v 2 ? B (2)
where ae is the plasma mass density (related to N e as
ae = m p N e , m p is the proton mass), ! ffi v 2 ? , the mean
square velocity is given previously, and B, is the magnetic
field strength. From our dataset at 120 arc sec above the
limb for the NPCH using N e = 4:8 10 7 cm \Gamma3 , ! ffi v 2 ?=
2 \Theta (43:9 km s \Gamma1 ) 2 (see Table. 2), we find F = 4:9 10 5 erg
cm \Gamma2 s \Gamma1 for B = 5 G, which is only sightly lower than
the requirements for a coronal hole with a high speed solar
wind flow (Withbroe & Noyes 1977), which is estimated to
be 8 10 5 erg cm \Gamma2 s \Gamma1 . The average field strength in coro­
nal holes is estimated to be 5­10 G (Hollweg, 1990). One
should note that we have used the WKB approximation.
Hollweg (1990) have shown that a non­WKB approach
enhances the energy flux somewhat. Furthermore, non­
linear effects could also be important. Recently Torkels­
son et al. (1998) have studied numerically, the propagation
of nonlinear spherical Alfv'en waves in a radial magnetic
field. They show that the wave damps by forming current
sheets, in which the Poynting flux is lost to Ohmic heating
and the acceleration of an upflow. This process could be
important for the fast solar wind.
Assuming that the linear analysis is correct and gives a
fairly reasonable estimate of the energy flux, expression (2)
shows that the rms wave velocity amplitude and density
are related by,
! ffi v 2 ? 1=2 / ae \Gamma1=4 (3)
In Figs. 4b & 5b, the (+) represents the velocities cal­
culated on the basis of Equation (3), using the measured
N e . In both cases, we find excellent agreement, however
in order to explore the consistency further, we show addi­
tional plots in Fig. 6. The solid lines are the theoretically
predicted functional form of the variation of electron den­
sity with non­thermal velocity (Eq.[3]) and the squared
boxes are the values derived from the line ratios (see Ta­
ble 2). The proportionality constant have been chosen to
match the calculated energy flux. For the SPCH data we
have used B=7.5 G. Once again the agreement is very
good. Note that the first data point in Fig. 6a corresponds
to values close to the limb, thus it is possible that they
are contaminated to some degree by foreground and back­
ground `quiet' Sun emission. The other possibility is that
these are affected by compressional magnetic waves which
are believed to be present close to the solar limb. Lou
& Rosner (1994) showed that the behaviour of the mag­
netic waves are very similar to that of Alfv'en waves, with
compressible effects falling off much more rapidly with in­
creasing height, as compared to the Alfv'en waves. Lou
(1996) assumes a rough equipartition of overall wave en­
ergies associated with magnetic waves and Alfv'en waves

Banerjee et al.: Si viii line broadening in a coronal hole 7
at the coronal base and estimated the velocity fluctuations
to be ¸ 18 \Gamma 20 km s \Gamma1 . If that is the case then the first
data­point is well explained.
Using SUMER data Wilhelm et al. (1998) have ob­
tained the electron temperatures to be less than 800,000
K in a plume in the range from r = 1.03 R fi to 1.6 R fi de­
creasing with height to ¸ 330,000 K. Near an inter­plume
lane, they found electron temperatures between 750,000
K to 880,000 K in the same height interval, while Tu et
al. (1998) have found that the ion temperature remains
roughly constant. We show that for Si viii we require an
ion temperature of 1 10 6 K in order to obtain an excel­
lent fit to the observed data and the theoretically cal­
culated values (see Figs. 4b & 5b). This is only slightly
larger than the Si viii formation temperature assuming
ionization equilibrium, i.e. T e ¸ 8 10 5 . On the other hand,
T i = 2 10 6 is not consistent with the theory and our ob­
servations.
Applying the conservation of energy flux we can also
derive a relation between the non­thermal velocity and
magnetic field strength as,
¸ / B \Gamma1=2 (4)
Using equation (4) we find that a non­thermal velocity,
¸ = 30 km s \Gamma1 at 37 arc sec above the limb in a 5 G coronal
hole corresponds to 1.5 km s \Gamma1 in a 2 KG photospheric
field. This is comparable to the solar granulation velocity.
Thus, this procedure also allows us to estimate roughly
the magnetic field strength at different heights.
From a study of low latitude coronal streamers, Seely
et al. (1997) did not find any conclusive evidence for any
observable turbulent velocity at altitudes of 109 arc sec
and 209 arc sec above the equatorial limb. They argued
that their result casts doubt on the heating of the corona
and acceleration of the solar wind by hydromagnetic waves
in the streamers. Since only the low speed wind originates
from the coronal streamers, this problem is not relevant
for fast wind emanating from the polar inter­plume re­
gions. In the present study our results can consistently
be explained by hydromagnetic waves in coronal holes.
The observed non­thermal velocities at different heights
are also consistent with upward propagating Alfv'en waves.
To verify the consistency further, we have looked at the
line broadening of other coronal lines. Our preliminary re­
sults, which we hope to present in a subsequent paper,
shows a similar trend. Furthermore our observations im­
poses limits on the assumed wave amplitude for the model
calculation of Marsch & Tu (1997). Observed amplitudes
presented here for polar coronal holes are still adequate
for the requirements of coronal heating by the dissipation
of high frequency Alfv'en waves.
Acknowledgements. Research at Armagh Observatory is grant­
aided by the Dept. of Education for N. Ireland while partial
support for software and hardware is provided by the STAR­
LINK Project which is funded by the UK PPARC. This work
was supported by PPARC grant GR/K43315. We would like
to thank the SUMER and EIT teams at Goddard Space Flight
Center for their help in obtaining the data. The SUMER project
is financially supported by DLR, CNES, NASA, and PRODEX.
SUMER is part of SOHO, the Solar and Heliospheric Obser­
vatory of ESA and NASA.
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