University of Warwick
Friday 19th November 2010
Periodic Spectral Line
Asymmetries In Solar Coronal
Structures From Slow
Magnetoacoustic Waves
Erwin Verwichte
In collaboration with Mike Marsh, Claire Foullon, Tom Van
Doorsselaere, Ineke de Moortel, Alan Hood & Valery Nakariakov.
Verwichte et al. ApJL 724 194 (2010)
Centre for Fusion, Space and Astrophysics
THE UNIVERSITY OF WARWICK
Propagating intensity perturbations...
Since SoHO we see intensity perturbations
propagating along large coronal structures
[Ofman 1997, De Forest & Gurman 1998, Berghmans & Clette 1999].
TRACE 21/04/2001
TRACE 14/07/1998
SoHO/EIT 21/04/2001
2
...are slow magnetoacoustic waves
These perturbations have been interpreted as slow magnetoacoustic
waves propagating along plume/loop structures.
[De Moortel et al. 2000, Nakariakov et al. 2000, Robbrecht et al. 2001, ..., ...].
Characteristics:
- Projected propagation ≤ sound speed (tube speed) of structure
Multi-bandpass observations [Robbrecht et al. 2001; King et al. 2003]
STEREO showed that Vph = Cs [Marsh et al. 2009]
- Periods of 3,5, 10, ... minutes
- Long duration of quasi-periodic behaviour
- Damped over a short distance (thermal conduction)
- In phase intensity and velocity perturbations [Wang et al. 2009]
SoHO/EIT 21/04/2001
3
Example: this week’s SDO
4
But are we all wrong? They are flows!
Recently a new ‘school of thought’ has appeared that claim that the
intensity perturbations are signatures of high-speed flows(~100 km/s), based
on measurements of periodic spectral line asymmetries.
[Schrijver et al. 1999, Sakao et al. 2007, hara et al. 2008, De Pontieu et al. 2009, McIntosh et al. 2010].
SoHO/EIT 15/09/2001
5
Their arguments
- Many of the observed propagating disturbances (especially in coronal
loops emanating from quiet Sun network and active region plage) do not
show evidence of significant quasi-periodic signals.
- The wave interpretation was compatible with the lack of strong Doppler
shifts.
- EIS measurements of intensity and velocity oscillations are accompanied
by oscillations in the line width and recurring asymmetries in line profiles
across a range of temperatures.
However, a periodic flow model does not yet exist. The whole alternative is
a superposition of a periodic upflow component in the blue wing on top
of a static background
,
λc (x, t) = λ0 1 −
asymmetries in
#
c
celeration
in
these
structures
as
well
as
for
the
seismologi0
sight, i.e. I(λ)= %(λ, x)dx. The quantities λc and ∆λ are the
der of 1-5% core
$
%
cal
exploitation
slow respectively.
waves (e.g. Robbrecht
al.broaden2001;
line centreusing
and width,
For thermal et
line
" 1/2
e suggestion that
T
King eting,
al. the
2003;
Wang
al. form
2009).
width
is ofetthe
∆λ = λc vth /c0 where c0 is the
∆λ(x, t) = ∆λ0 1 +
.
turbations could
−1
T
0
speed of light and vth is the ion thermal speed. We shall ilkm s ) upflows
lustrate
our 2.
findings
using
an emission spectral
line from
Can
slow
waves
explain
features?
09; McIntosh &
SLOWthroughout
WAVE
MODEL
an iron ion minority species at a temperature of 1 MK (vth =
The Doppler velocity shift involves the line-ofA
slow
magnetoacoustic
wave
in
a
coronal
structure
of
low
−1
12 km s , cs /vth = 12.5), which is convolved by the spectral
not mutually excomponent v" cos α where α is the angle between
plasma-β
is guided
to propagate
alongThe
theintensity
structureis parallel
to
resolution
of
the
EIS10
instrument.
furthermore
e and red shifts
propagation
the line-of-sight. Equation
Why
throw
away
years
of
slow
wave
work?
Canof
we
possibly and
explain
the magnetic
field.toHence,
the wave
almost completely
proportional
a function,
whichiscontains
information lonabout
on loops, and inscribe the
effects
of waves?
intensity variations due to t
periodic
large
line-width
with
slow
gitudinal
and line-asymmetries
one-dimensional.
Forand
simplicity,
we
model
ionisation
and depends on temperature
and (weakly)
on the
den-variations
of the order of
sity perturbation, Doppler shifts due to the wave
sity. as
Wea shall
assume, for thesmall
sake amplitude
of clarity inplane
what follows,
slow wave
one-dimensional,
sound
l Zanna 2008),
and thermal line broadening due to the wave tem
that
over
the
range
of
temperatures
that
the
slow
wave
covers,
wave
propagating
upwards
in
a
static
equilibrium
plasma
that
Downflows are
We model the spectral line in the presence of a simple
slowBecause
wave:the density and temperature
turbation.
this function
constant (see
De aMoortel
Bradshaw
re found in faint,
is uniform
along theismagnetic
field.e.g.
Such
wave is&described
are in phase with the velocity for a propagating
2008,
for (Landau
a study of&the
effect of
ionisation on slow wave diby the
solution
Lifshitz
1987)
wave
perturbations
the emission from the plasma is enhanced during
agnostics).
ow waves in the
E.
Verwichte
et
al.
" emitting
(blue-shift) propagating phase of the wave and
plasma
mod1 inTthe
n"
v"
iksen & Maltby
" The presence of the slow wave
vifies
= the
a csstrength,
cos(x −centre
cs t) , and width of =the emission
=
,line (1)
during the downwards (red-shift) propagating
as a
et al. 1978; Mc(γ
−
1)
T
n
c
0
0
s
3.
EMISSION
LINE MODIFIED
BY asymmetric
A SLOW WAVE
T)
on
board
Hinode
have
revealed
signatures
of
quasifunction of space and time as
wave.
This inherently
behaviour is
dsen et al. 2003).
"
"
"
intensity
perturbations
acerdic
thepropagating
new obser1. From
Eq. (2)
it can be
thata the
where
n (x, t),
T (x, t) andatvthe
(x, edge
t)$ areofthe
The emission ofFigure
a coronal
resonant
spectral
lineseen
from
% wave perturbations
"
egions
(temperature
around
1.1 MK),
travelling
speeds
f quasi-periodic
n
will
symmetric
in number
density,
temperature
and1atvelocity,
The
coronal volume
element
isamodelled
asGaussian profile at any giv
line be
emission
, respectively.
(3)local
+
n(x,
t)
=
n
−1
0
slow
0 kmwave
s , interandquantities
which arguably
were
interpreted
by
Sakao
a single plasma element in which a slow wave
with subscript ’0’ indicate nthe
0 equivalent constant
!
"
sequence
for
our
$
%
(2007) in terms
of continuous
flowsAlso,
(see γalso
Heratio
et" al.of specific heats, cs
2
order
for
the
line
to
be
asymmetric
either the em
equilibrium
quantities.
is the
(λ
−
λ
)
c
v cos α
2
on
and
wind
ac∼ over
n exp
−
). Also, a correlation
between sound
Doppler
.
(2)
,
(4) %(λ)
λc (x,
t)shifts
= λand
−
aged
a period
of oscillation
or an additiona
0 1and
is the equilibrium
speed
a is line
the
relative
wave am2
2(∆λ)
c
the
seismologidenings as well
as significant
blue wing
plasma source is taken along the line-of-sight.
plitude,
which isdeviations
assumed in
to the
be small,
i.e.0 %1/2
a # 1. The wave
$
recht
et
al.
2001;
"
ne profiles have
foundφ=kx-ωt,
in EIS observations
T frequency ω=cs k.
case is relevant for spectral raster scans where te
hasbeen
a phase
wavenumberbyk Hara
and
∆λ(x,
t)
=
∆λ
1
+
.
(5)
The
observed
intensity
total for
emission
the line-of0
(2008). Moreover,
observationsstratification,
of asymmetries
traded
spatialalong
resolution,
whilst the
EffectsEIS
of gravitational
dissipation
(e.g. thermal # lutionisisthe
T 0in
sight,
i.e. the
I(λ)= %(λ,
x)dx.forThe
quantities
λc and ∆λ are the
profiles, with faint
blue-wing
in order
of 1-5%
relevant
spectral
slit measurements.
conduction)
andexcess
variations
in loop
crosscore
section will
cause
local spectral
signatures
line centre
and width, respectively. For thermal line broadensity, were used
to support
thebealternative
suggestion
that
The a
Doppler
shift
the line-of-sight
velocity
amplitude
to
avelocity
function
of involves
height with
increasing
amplistructure of lowpropagating
ing,
width
∆λ = λc vth /c0 where c0 is the
quasi-periodic
intensity
component
v" cos αperturbations
where
α is the could
angle
between
thethe
direction
tude
due
to
stratification
and decreasing
amplitude
due
to
dis-timeis of the form
ucture
parallel
to
Integrate
over
Integrate
over line-of-sight
−1
Asymmetry
of aWe
time-averaged
l
speed
of has
light
ion thermal
speed.
shall ilxplained as periodic
high-speed
(50-100
kmline-of-sight.
s ) upflows
of and
propagation
and the
Equations
(3)-(5)
de- and vth is the 3.1.
sipation
cross
section
divergence.
These
phenomena
completely lonour
findingsTo
throughout
using
from win
scribe
the
intensity
variations
tolustrate
the or
wave
denad
of model
slow waves
Pontieu
et regimes
al.of2009;
& due
been (De
studied
in effects
the
ofMcIntosh
wavelengths
shorter
simiunderstand
howananemission
averageline
blue-red
y, we
the
sitytypical
perturbation,
Dopplerlength-scales
shifts due to the
wave
anvelocity
iron et
ionfield
species atby
a temperature
of we
1 MK
(vth =the li
ontieu
2009;
McIntosh
et
al. 2010).
lar to the
longitudinal
(Nakariakov
al.minority
is produced
a slow wave,
expand
ude plane
sound
−1
and
thermal
due to the
temperature
12
km s ,percs /vth =relative
12.5), to
which
is convolvedusing
by the
aves
and flows
are inVerwichte
general
ofetline
course
notDe
mutually
ex2000;
al.broadening
2001;
Moortel
&wave
Hood
2004).
the equilibrium,
s =spectral
(λ − λ0 )/∆λ
rium plasma
that
turbation.
Because
the
density
and
temperature
perturbations
resolution of the EIS instrument. The intensity is furthermore
ve phenomena.
Doppler blue and red shifts
wave
is described Persistent
are
in
phase
with
the
velocity
for
a
propagating
slow wave to
, a function, which contains information about
proportional
been observed, using EIS, in active region loops, and inthe emission from the plasma is enhanced during the upwards
Slow waves have inherent spectral asymmetry
Because velocity and density perturbations are in-phase, upwards
propagating waves have a stronger blue wing than red wing!
¯ "=! " which
¯2 ! Marel
asymmetry
is characterised using ,quantities B and
"2 ! & f¯m
d"2 F(s!1993),
fder
spectral lines2  (Van
Franx
canThe! line
(γ − 1)(3γ
+which
1)theblue
+integrated
∗f)2"
the
wing
small
amplitudes
and
wing
bias
for
largema
even
the
wave
T "∗ ) + v" cos
α
T  , n"4 (8)
"
R,
which
are
intensity
1 and 3though
line red
!¯ (s) ≈ n0  f¯0 −T " 3F(s
through
the
slow for
isbetween
propagating.
This
is modm
m
m
2
theslow
contributio
2centre
vwave

2
2 (−1)
f2 =
+ d F(s)/ds
+
+ 2m F(s)H
,(s) where
th,0red
be seen by identifying
=
2
m!
ds
m
c
cos
α
a
a
widths
from
the
line
in
the
and
blue
wings
of
the
amplitudes
where
the
average
line
forms
a
heavier
red
wing.
elled fors small amplitudes using Eq. (6) by
replacingintensity
the term and Doppler
T0 2 T0
vth,0
T 0 n0
resulting
emission
in the2
f¯m=3
(γ
−
1)(3γ
+
1)
+
,
Hm (s) is the Hermite
polynomial
of
order
m
(Abramowitz
&
line,
respectively.
Hence,
(R
−
B)/(R
+
B)
gives
a
measure
!
"
2
2
2" =
2
Figure
2
shows
that
both
measures
show
similarly
aorder
growing
of
O(1)
in
f
by
1
+
I
/I
,
where
I
is
the
background
and
"
"3a
0
bg 03a
bg
c
cos
α
T
v
cos
α
T
the
contributions
of
O(a
s
4
2
v
of
the
asymmetry
in
the
wings
of
the
line
profile
with
negawhere
2
th,0
Stegun 1965).
f3 = 3
, f4 =f¯ 3=
simulated spec
I1)
the structure’s
plasma
emission.
Figure
1 ilblue-wing
bias
as ainfunction
of .(De
wave
amplitude.
, f¯equilibrium
=excess
(γ − tive
(γ
− 1)
(9)
0 isvalues
3 "2T 0 ,(7)
4an
representing
the
blue
wing
PonT 0 vth,0 ! 2
emission
theofblue wing. Fig
2forAntwo
Sincef¯the
perturbations
are all' proportional
2
lustrates
that
oscillation
phases,
φ=0,
π, the in
effect
2cosφ,αwhen
vth,0
2measure
a3a
cs cos α to acscos
a2 & 2
tieu
et al.
2009).
alternative
of line
asymmetry
3a
2
to the
line
as a
2
,
,
2
¯
¯3 −= only
¯4slow
8 −/2).
3(γThe
− 1)
=F(s)=exp(−s
1+
, is re- and
f0 − andover
3
equilibrium
thermal
speed
simulated
spectral
line
signatu
averaged
an
oscillation
period,
the
equilibrium
theskewness,
wave
on −
the
combined
line
is
variations
in
intensity,
Time-averaged
spectral
line
f
,
f
=
(γ
−
1)
(γ
1)
.
(9)
is
defined
as
((s
−
s
)/σ)
I(s)ds/
I(s)ds
where
0
2 to the16equilibrium line width as
42
vth,0
3.2.
Asymmetry
ofofathe
multi-component
lineof 5 minut
riod
lated
vth,0 /c
∆λ0 /λ0 v
. th,0
Ex0 =contributions.
2the line
Doppler
shift,
and line
asymmetry,
theline
strength
s0 and σ are
meanwidth
anddefined
standard
deviation
line.
It
quadratic
perturbation
terms
have
non-zero
We
The
argument
of
the
Gaussian
profile
is
as
to the
as aof
function
of tim
2 (6) is similar to the Gaussian-Hermite expansion of
pansion
is
consistent
with
the
R-B
measure
inintensity
showing
aand
bias
towards
Periodic
spectral
line
asymmetries
from
slow
magnetoacoustic
waves
c
cos
α
a
which
depends
on
I
/I
.
The
Doppler
velocity
s
with
I
=2I
First,
we
consider
the
profile
of
an
emission
line
which
conbg
0
denote time-averaged
quantities
by
a
bar.
The
average
intenbg
0.
spectral
lines
(Van der Marel
1993), which can
of
5Therefore,
minutes and a=0.15,
+The
3)
, &m Franx
f¯1 = (γ
the
blue
wingare
for reduced
small
amplitudes
and red1/(1
wing+riod
bias
for
large
,
argument
of
the
Gaussian
profile
is
defined
as
variations
by
a
factor
I
/I
).
2π/ω
m
m
bg
0
stitutes
from
plasma
components
in the
line-of- an
4 by identifying
vth,0
d F(s)/ds
= (−1)
F(s)H
where
m (s)
the
intensity
¯ be=seen
sv
−
vline
s +averf¯1 though
amplitudes
where
theemission
average
formstwo
areceives
heavier
red wing.
sity
I(s)
(ω/2π)
I(s,
t)dt
may
be
with
the
th,0
D
When
we
average
over
anwritten
oscillation
period,
the
blue
wing
more
with
I
=2I
Oscillations
bgand
0 . the
even
the
slow
wave
may
have
a
large
amplitude,
!
"
0
Hm (s) is2 the Hermite
polynomial
of
order
m
(Abramowitz
&
2
2
sight,
(1)
a
quasi-static
‘background’
(2)
the
plasma
struc-ofto
Figure 2=shows
that bothameasures
show
similarly
a growing
.
(10)
s
=
2
.
∗
heavy
ion
with
c
&
v
,
the
non-the
to O(aage
)
accuracy:
c
cos
α
a
a
the
tendency
s
th,0
s
¯1
emission
!(s)
as
the+sum
of a Gaussian
F(s∗ ) sv
and
resulting
intensity
and Doppler
velocity
be
small.
Stegun
theslow
intensity
and‘background’
Doppler velo
s + profile
emission
wing.
blue-wing
of wave
amplitude.may
th,0
D assupporting
¯2 − vbias
f¯ = 1965).
(γthan
−
1)(3γ the
1)are
+red
, fcos
ture
a propagating
wave.Again,
The
2a function
1
+
f
2)2
perturbations
all2sproportional
to
a
φ,
when
v
+
(∆v
.
(10)
=
=
the
contributions
of
order
O(a
)
in
Eq.
(6)
introduce
stronger
third and2Since
fourth
derivatives
of
a
Gaussian:
4 theorder
v
.
NTapproximately
∗  th,0
'and
th,0refers
the
tendency
totude
aapproxi
half-pe
ing
isFigure
toshow
∆vofNT
≈ acs
here
to another
plasma
inequal
theofsame
line-of-sight
2  2emission plasma
" ¯
averaged over
an oscillation
period, only&the equilibrium

"
in
the
blue
wing.
3
shows
an
example
the
2
v
cos
α
2 !!
2
1
+
f
3.2.
Asymmetry
of
a
multi-component
line
−1


"
2
2
v
+
(∆v
)
4
3a
c
cos
α
3a
"


of
the
wave
co
quadratic
have
non-zero
contributions.
We th,0
s
+
NT
m 2
distinct
from
the
‘background
equilibrium’
plasma
structure
tude
of
approximately
1
km
s
+

− 1) snf¯terms
¯

fore,
mainly
through
its
velocity
perturbat
¯3 = 2perturbation
¯

simulated
spectral
line
signatures
from
a
single
Gaussian
fit
v
f
d
F(s
)
fdenote
,
f
=
(γ
(γ
−
1)
.
(9)
th,0 ∗inten 
m average
2
4by 
First,
we consider
the profile
of
an emission
line wing
which
contime-averaged
quantities
a bar.
asymmetries
-magnetoacoustic
Doppler
shift
to
the
blue
The
line
imposed
by
the
slow
wave
cause,

F(s
n020 1 f¯+
,slow
(8)
2n
vth,0
2 The&deformations
through
which
the
slow
wave
is
propagating.
This
mod,
−
!(s, x,!¯ (s)
t) ∼¯ ≈Periodic
'
exp
∗ ) +

0 ,−
of
theisisstatic
of
the
wave
linelin
to
to
the
line
as
a
function
of
time
for
a
slow
wave
with acontributed
pe
2π/ω
spectral
line
from
waves
3that

m
stitutes
emission
from
two
plasma
components
in
the
line-ofproduces
a
line
broadening
propor
"
m!
ds
sity I(s) = (ω/2π) 0n2 I(s, t)dt may
be written
with
the
aver-
T


¯
elled
for
small
amplitudes
using
Eq.
(6)
by
replacing
the
term
m=3a2Doppler
0
The
line
deformations
slowby
wave
cause,
firstly,
shiftby
in the
vamplitude.
=−
f1 and
vand
to
the
blue
riod
of
minutes
and
a static
plasma
component
sight,
(1)5a line
quasi-static
‘background’
the plasma
strucDa=0.15,
th,0(2)
of
the
static
line.
Theobservat
spectral
recent
The argument
of the Gaussian
defined
as1 +imposed
age emission
!(s) as theprofile
sum of aisGaussian
profile
F(s
T∗0) and
The
Doppler
velocity
and
lin
ture
supporting
a
propagating
slow
wave.
The
‘background’
¯
2
of
O(1)
in
f
by
1
+
I
/I
,
where
I
is
the
background
and
with
I
=2I
.
Oscillations
of
reduced
amplitude
are
seen
in
1/2
0
bg
0
bg
bg
0
firstly,
a
Doppler
shift
in
the
line
by
v
=−
f
v
to
the
blue
order derivatives
of
a
Gaussian:
Non-thermal
broadening
¯
a
heavy
ion
with
c
&
v
,
the
non-thermal
line
broadeno O(a ) accuracy:
D
1
th,0


recent
observations
by
De
Pont
s
th,0
where third and fourth
wing,v4Dsecondly
non-thermal
broadening
∆v
= line-of-sight
f2theplasma
vline
for the
plasma
refers
here
to Doppler
another
plasma
in
theNT
same
th,0
√
the
intensity
velocity.
Also,
width
has
strongly
correlated.
For
instance,
ahalf-per
Dopple
 am
s + f¯1!
Iline
isand
the
structure’s
equilibrium
emission.
Figure
1 il1/2
0 the
" svth,0 −+
+

4
¯
m
distinct
from
‘background
equilibrium’
plasma
structure


!
"


¯
¯


&
'
.
(10)
s∗¯ = 2-
=
.
2.
Thereing
is
approximately
equal
to
∆v
≈
ac
cos
α/


wing,
secondly
a
non-thermal
line
broadening
∆v
=
f
v
for
the
half-period
oscillation
f
(x,
t)
d
F(s)
f
d
F(s
)
f
2
2
NT
s
−1
NT
th,0


∗
m
2


m
"
the
tendency
to
show
a
half-period
oscillation
with
an
ampliand,
thirdly,
an
asymmetry
through
the
term
involving
the
addition
of
a mi
2
2
2
lustrates
that
for
oscillation
φ=0, π, the line
effectwidt
of
2has
!
"22 !¯n(s) ≈1 f+na0(x,
α2 ∗ )+ '
 , through
 , α (8)
¯ &+
f¯0 t)F(s)
−v cosvF(s
through
which
the slow
wave
is two
propagating.
This isphases,
mods
an
associated
non-thermal
a2 dsmcs cos
f
" f≈
−1

s
+
(6)
2
2
+
(∆v
)

0
¯


fore,
mainly
its
velocity
perturbation,
the
slow
wave
NT
2
m!
tude
of ssmall
approximately
1 can
km
s=20
and
isreplacing
caused
byterm
an
excursion
an derivative
asymmetry
term
involving
the
addition
of
modest
steady upfl
−vth,0
3(γth,0
− 1)
−
, Using
f n− = 01 +
 and,8thirdly,
ac
=5km/s
∆v
km/s
[Hara
et
ala2008]
 the
m
NTEq.
elled
for
(6)
by
the
theamplitudes
slow
wave
on
the
combined
line
is
variations
intensity,
−1
third
ofthrough
F.
Eq.
(9)
itusing
be
seen
that
for
the in
excursions
m!
ds
!(s, x, t) ∼ n20 1 +0 2 exp −16 &
'  ,m=3
20
km
s
,
consistent
with
observations
rep
4
v
of
the
wave
contributed
line
towards
both
blue
and
red
wing
th,0
produces
a
line
broadening
that
is
proportional
to
the
wave
"
of
O(1)
in
f
by
1
+
I
/I
,
where
I
is
the
background
and
m=1
0 be seen
bg
0 that
bg
T
third
derivative
of
F.
Using
Eq.
(9)
it
can
for
the
excursions
into
the
red
wing


Doppler
shift,
line
width
and
line
asymmetry,
the
strength
of
n
where
0
2 1 + T 0by the slow wave cause,
The line deformations
imposed
Iof
theThe
structure’s
plasma
emission.
1 ilstatic
line.equilibrium
The
spectral
signatures
are
consistent
2
0-isthe
Line
asymmetry
Doppler
velocity
and
lineFigure
broadening
are
(2008).
Furthemore,
forand
anwith
iron emission
!
"2 amplitude.
c
cos
α
a
which
depends
on
I
/I
.
The
intensity
Doppler
velocity l
¯
s
bg
0
2
2
&
'
firstly,
a
Doppler
shift
in
the
line
by
v
=−
f
v
to
the
blue
¯
lustrates
that
for
two
oscillation
phases,
φ=0,
π,
the
effect
of


¯
D
1
th,0
a
c
cos
α
a
f
recent
observations
by
De
Pontieu
&
McIntosh
(2010),
except
s
2
2
(γ
+
3)
,
f
=
with coefficients
stronglythecorrelated.
instance,
aisDoppler
5Ibgkm
8−
3(γm− 1)2 −
=41 +
+
 secondly1f¯0a−non-thermal
f¯3 ≈10a
. byTherefore,
aof+slow
with a
variations
are
reduced
a factor
1/(1
/I0wave
). Therefore,
slow
wave
on For
the combined
linein
variations
invelocity
intensity,
¯1/2,vth,0
4
v
f
(x,
t)
d
F(s)
wing,
line
broadening
∆v
=
f
for
the
half-period
oscillation
the
line-width.
However,
the
th,0
2
16
4
v
−1

m
NT
th,0
2
2


s
has
an
associated
non-thermal
line
width
of
approximately
Doppler
shift,
line
width
and
line
asymmetry,
the
strength
of


!
"
!
"
!
"
!
"
≈and,
n0  fthirdly,
+2 2 " through the
(6)
even
though
the
slow
wave
may
have
large
amplitude,
0 (x, t)F(s)
2 !, involving
of
5%
is likely
tovelocity
produce
an
averagethe
line
an" asymmetry
addition
of
a modest
steady
upflow
as Doppler
reported
could adiminish
" 2 "2 the "
−1" depends
α "m 2aterm
an m! c3s cosT
which
on Iwith
The
intensity
and
ds
bg /I0 . observations
T
n
n
T
1
c
cos
α
a
20
km
s
,
consistent
reported
by
Hara
et
al.
¯
s
(γ + 3)Eq. (9) ,it can be seen that for
fF.
= Using
resulting
intensity
and
Doppler
velocity
may
be
small.
Again,
1 m=1
third
derivative
of
the
excursions
into
the
red
wing
of
the
static
line
and
cause
the
¯
reduced by
a factorof
1/(1a+few
Ibg /I0 ).percent.
Therefore, Figure 2 shows the
order
2 4− 1)(3γ
+ v+th,01) + 2!+ v " + ,(2008).variations
, are
f0 = 1 + f2 = +4 (γ
2 using Eq.
Furthemore,
for
an
iron
emission
line,
(9),
the
contributions
of
order
O(a
) in Eq.
stronger
even
though
the
slow
wave
may
have
a
large
amplitude,
the (6) introduce
th,0
2 T 0 ¯ an2 0 8 T 0 a2 cs cosnα0 2
n
20
ith coefficients
0
¯fT
broadening
and
line
asymmetry
as
aoffuncti
≈10a
.
Therefore,
a
slow
wave
with
a
relative
amplitude
resulting
intensity
and
Doppler
velocity
may
be
small.
Again,
3
emission
in
the
blue
wing.
Figure
3
shows
an
example
the
2 (γ − 1)(3γ + !1) +
2,
f2 =
"
2
3a
c
cos
α
3a
!
"
!
"
!
"
!
"
the
contributions
of
order
O(a
)
in
Eq.
(6)
introduce
stronger
s
4
2
v
"
"
"
2
th,0
2
2
5%
is likely
to produce
an average
line asymmetry
the
¯4" T
"
spectral
signatures
aofsingle
fit fo
(2)-(5).
Figure
2 shows
vD Gaussian
and ∆vNT
= 2 (γn− of
− 1)
1) v
. cos
(9)
α insimulated
n" 3f¯3 =T " 2 (γ
Tα" , nf1
1 T"
emission
the blue wing.
Figure 3 line
shows
an example
offrom
the that
2 n
3a
c
cos
3a
v
2
s
order
of
a
few
percent.
Figure
2
shows
the
Doppler
shift,
line
+2 +
+th,0 1 +¯
, + 2 2 (9)
f0 = 1 +
f1 =
, spectral
to the line
linesignatures
as a function
of time
slow wave
with a pe- am
¯ +
simulated
from a single
Gaussian
approximations
in for
thefitarange
of observed
2 T0
n0 8 T 0f3 = 2 (γn0− 1) vth,0T 0 , f4n20= T20 (γ − 1)n.0 broadening
v
to the and
line riod
asline
a function
of
time
for
a
slow
wave
with
a
peasymmetry
as
a
function
of
a
using
Eqs.
th,0
of 5 minutes and a=0.15, and a static plasma component
" " is defined as
The argument! of the!Gaussian
asymmetry
is
characterised
riod
of 5 minutes
and
a=0.15,
and
aline
static
plasma
component
" Gaussian
" profile
"
"
"
!
"
!
!
The argument of
the
profile
is
defined
as
(2)-(5).
Figure
2
shows
that0 .The
vDOscillations
and
∆vNT
follow
the analytical
with
Ibg =2I
of are
reduced
amplitude
are seen usin
in
1 T " 2 n v "cos α 2
"
"
"
with
I
=2I
.
Oscillations
of
reduced
amplitude
seen
in
bg
0
f1 T
= 1 +3s +Tf¯+ ¯2
,
v
cos
α
T
n
R,
which
are
the
integrated
intensity
betw
approximations
in
the
range
of
observed
amplitudes.
the
intensity
and
Doppler
velocity.
Also,
the
line
width
has
sv
the intensity
Doppler velocity. Also, the line width has
sv
2= T-0s +1 f1 =
n+
vth,0th,0− v−D vD . + 2
f2 = s∗ +
, theand
=0 .
. (10)The(10)
= s∗ . th,0
the
tendency
to
show
oscillation
withquantities
an
ampliline
asymmetry
isa half-period
characterised
using
B andin
tendency
to show
a half-period
oscillation
with
ampliwidths
from
the
line
centre
thean red
and
! " "2T 0 ! " 2 1 T
"2v! +th,0
" NT )2 2
T
n
1"02+
¯2 f¯2 ! "vv2th,0
0
0 of approximately 1 km s−1 and is caused by an
(∆v
"
" (∆v
+
f
−1
tude
excursion
+
)
T
3 T
v cos α
T th,0n
NT
R, which
are
the
intensity
between
1wing
and
3 line
tudeintegrated
of approximately
1 km
s and
is
caused
by −
an B)/(R
excursion
respectively.
(R
+B
!
"
of the
wave contributed line,
line towards
both blue and red Hence,
f2 =
+
+
+
2
,
2
"
"
"
of
the
wave
contributed
line
towards
both
blue
and
red
wing
widths
from
the
line
centre
in
the
red
and
blue
wings
of
the
The
line
deformations
imposed
by
the
slow
wave
cause,
signatures
are consistentin
with
T0 2 T0
vth,0
T 0 v cos
n α
Tline
Tof the static line. The spectral
ofPontieu
the
asymmetry
the
wings
of thewith
line
The line
deformations
imposed
byvD0the
wave
cause,
firstly,
a Doppler fshift=
in the
by
=−
f¯1 vslow
tofthe
blue
th,0
recent
observations
by
De
&
McIntosh
(2010),
except
of
the
static
line.
The
spectral
signatures
are consistent
line,
respectively.
Hence,
(R
−
B)/(R
+
B)
gives
a
measure
3
,
=
3
,(7)
!
"
3
4
2
1/2
"v =−∆v
secondly
a non-thermal
line broadening
= f¯2 to
vth,0
for the half-period
invalues
the
However,
firstly, awing,
Doppler
line
by
f¯1NT
vth,0
the
tive
representing
an
excess
the b
T " v" shift
cos
αin the T
Tvth,0
D
Tasymmetry
recent
observations
by
Pontieu
McIntosh
(2010),in
except
of
theblue
in oscillation
the
wings
of line-width.
theDe
line
profile&the
with
nega0
term
involving the¯1/2 addition of a modest steady upflow as reported could diminish
f3 and,
= 3thirdly, an asymmetry
, f4 through
= 03 the
,(7)
wing, secondly
broadening
∆vthat
=forf2 values
vth,0
forinto
thethe
half-period
oscillation
incause
the
line-width.
the
tieu
et
al.
2009).
Anthe
alternative
measure
representing
an
excess
in
the
blue
wing
(De
Pon- However,
NTtive
T 0 a non-thermal
vth,0
Tit0 can be seen
third derivative
of F. Using line
Eq. (9)
the excursions
red
wing
of the
static
line
and
,
2
and, thirdly, an /2).
asymmetry
through the term
involving
addition
of
steady upflow
reported could diminish
3
tieuspeed
et the
al. 2009).
An alternative
measure
of line
asymmetry
and F(s)=exp(−s
The equilibrium
thermal
is re,isa modest
, as
skewness,
defined
as
((s
−
s
)/σ)
I(s)
2
0
3 wing of the static line and cause the
nd F(s)=exp(−s
/2).
The
equilibrium
thermal
re-be
third
of F. Using
Eq. speed
(9)as
it iscan
that
for
the excursions
into
the red
is
skewness,
defined
as
((s
−
s
)/σ)
I(s)ds/
I(s)ds
where
lated to
thederivative
equilibrium
line
width
vth,0
/cseen
=
∆λ
/λ
.
Ex0
0
0 0
s and σ are the mean and standard devia
Multiple component spectral line
We consider the line-of-sight integration of plasma with a slow wave and
a static background plasma component.
Multiple component spectral line
Intensity amplitude and Doppler shift reduce by factor 1/(1 + Ibg/I0).
The real wave amplitude may be quite large.
Appearance of line width and line asymmetry
variations
Line width may have half period.
Multiple components may be integation along
loop where amplitude changes with height.
Conclusions
Slow wave can naturally produce (periodic) signatures of spectral line asymmetry
and line-width.
The flow model requires always the presence of a static plasma component in the
line-of-sight everywhere whilst a slow wave model does not need it to produce
intensity, velocity and line-width perturbations!
Slow waves can naturally explain:
- Sonic propagation speed
- Damping
- Persistence of quasi-periodic signals
The claim more careful spectral observations are needed to distinguish the two
models may be a red herring as these spectral observations are done at
(beyond?) the limit of spectrometer capabilities!
In order to test models, a physical model of periodic upflows needs to be
formulated!