Excitation and damping of kink waves
in the solar corona
David Pascoe
Centre for Fusion, Space and Astrophysics
University of Warwick
I. De Moortel, A. W. Hood, A. N. Wright
Warwick-Monash Meeting
25 February 2015
1
10 k are the frequency and the longitu
dralthecoronal
longitudinal
wave respectively;
where
ωthe
and
obtain the loop
≈1.9
×
10
cm.for internal
thelength
indices
α=
0,
ederivative
are
loops number,
shortly
of a function with respect to its ar
αn=the
0, eadjacent
are for internal
active
reand external
media, respectively;
Im (x)
andtrapped
Km (x) modes,
are
For
the
which areαevanescent
out
number,
respectively;
the indices
= 0, e are
seismology
with
standing
kink
Im (x)
and K
(x)
nal
sequence
with
theare
ca- Bessel
mCoronal
tube,
the condition
memodes
> 0 has to be fulfilled. The
modified
functions
of
order
m;
the
prime
denotes
and
external
media,
respectively;
Im (x) and
2.3. Global kink modes
m;the
the exposure
prime denotes
nd
time
of
m
determines
the
mode
structure,
and
for the kin
the derivative of a function with respect to its argument.
modified
Bessel
functions
of order
prim
m
= 1.the
Inisaddition,
therem;
existthe
a disper
es of the
espect
to oscillations.
its argument.
The natural
interpretation
ofconsidered
these
phenomena
that
For The
the trapped
modes,
which are evanescent
outside
kink oscillations
the
derivative
of
a
function
with respect
its
torsional
Alfvén
mode propagating
with theto
Alfvé
tion
in four outside
neighbouring
the
loops
experience
kink
global
mode
MHD
oscillations.
evanescent
the
tube, the condition me > 0 has to be fulfilled. The number
Inthe
thekink
observational
examples
abo
opfulfilled.
apex showed
synphase
As the
thetrapped
displacement
is observed
A loopare mentioned
be
The number
For ofthe
modes,
which
evanescent
m determines
themain
modeamplitude
structure,
and
for
modes
the
global mode
was seen, with the longitudina
sidering
loop near the loop apecis and
displacement
is synphase,
Flare the
epicentre
and for the
the averaged
kink modes
considered
m = 1. In addition,
there
exist
a dispersionless
tube,
the
condition
m
>
0
has
to
be
fulfilled.
T
e
length
of
double
the
loop
lengths.
For
the
observ
meexist
and approximating
the we conclude that the oscillations are the global standing
re
a dispersionless
torsional Alfvén
mode propagating
with
the Alfvén
speed.
Fig. 2. then
Theand
magnetic
fieldi
m determines
the
mode
structure,
for
the
widths
which
are
much
smaller
loop
lengths,
pendence with an exponen- mode of the loop, with
the
wave
length
double
the
loop
with the Alfvén speed.
In theFig.
observational
examples
mentioned
above,
only
plasma
density
inside
the
2. The
magnetic
field inside
a≪m
coronal
loop
aslengths
function
of
ka
1,
which
allows
us
to
simplify
the
dispersi
L
footpoint
on, the period of the oscilconsidered
=
1.
In
addition,
there
exist
a
dis
length.
Taking
the observed
periods
P
and
loop
Blast wave
mentioned
above,the
only
modes was
seen,
with
thefor
longitudinal
ratio
is 0.1. The solid curv
inside
the
The
external
towaveinternal
density
10loop.
tion.
∼256 s (Nakariakov
etglobal
al. plasma
L (256density
and 1.30
× 10
cm
14th
of
July,
1998;
360
s
torsional
Alfvén
mode
propagating
withCkthe
Al
the
longitudinal
wavelength
of
double
the
loop
lengths.
For
the
observed
loop
the
kink
speed
=
1030
ratio
is
0.1.
The
solid
curve
corresponds
to
the
central
value
of
10
There
are
two
kink
(m
=
1)
modes
in
the
limit
footpoint
he loop footpoints was esti- and 1.9 × 10 cm for 4th of July, 1999),
we estimate the
−1
In
the
observational
examples
4th
of
July,mentioned
1999),
and
ta
For
observed
loop
the
kink
speed
C
±
410
km
s
(for
the
event
of
the
widths
which
are
much
smaller
then
loop
lengths,
it
makes
k = 1030
slow
and
fast.
The
slow
kink
mode
has
the
phas
h,
forthe
a semi-circular
loop
phase
speed
required
as
TRACE observation 14 July 1998
Figure
1.global
Adashed
possible
mechanism
forseen,
excitation
of
kink
upper
and
the lower
possi
10 lengths, it makes
of July,
1999),
and
the
curves
correspond
towith
the
ka ≪ 1, 4th
which
allows
us
to
simplify
the
dispersion
relaloop
about
the
mode
was
the
longitud
×n10
cm.
!
oscillations
of
loops
by a coronal
blastlines
wavegive the
−1coronal
Figure
dotted
limits
and the 1020
lower
values
of
the
speed.
The
vertical
ω
2L
±possible
132
km
s
(14th
July,
1998),
ify the dispersiontion.
rela- upper
length
of
double
the
loop
lengths.
For
the
obs
ω
C
C
s0 A0 (1) k = π/L
excited by a flare.
=
≈
coronal
−1
TRACE
171
Å
and
195
Å
≈
CT0
≡ ka
·
lines
give
the
of the
loop
density
estimation
using
1030
410
km
s in
(4th
July,
1999).
Theredotted
are
kink
(m
=±limits
1)
modes
the
limit
≪
1:
k two
P
2
2
1/2
k
July, 19
(Cs0 + C
widths
which
are much
smaller
then loop length
A0 )
TRACE
171
Å
and
195
Å
images.
des in the limit kaslow
≪ 1:and fast. The slow kink mode has the phase speed
long
wavelength
limit: ofand
ka
≪
1,
which
allows
us
to
simplify
the
disp
Shrijver
et
al.
(2002).are
In particular,
it
found
Possible
errors
these
measurements
discussed
inwas
The
phase
speed
of
the
fast
kink
mode
is
The
Alfvén
speed
i
are-generated
loop
oscillaabout
e has the phase speed
that the kink oscillations do not always have a simple
5.
which i
tion.
& Brown (2000), was less Sect.
!
"
strength
and
the
density
The
Alfvén
speed
is
defined
by
the
magnetic
field
1/2
(or global) 2mode and there can
ω
Cs0 CA0 form of a principle
ω
derivati
rring at 8:20
≈ CofT0strength
≡
·
(4)
can
estimate
the
value
o
fast
There
are
two
kink
(m
=
1)
modes
in
the
lim
be
higher
spatial
harmonics
observed.
≈
C
≡
C
,
slowUT, 14th
and
the
density
of
the
medium.
Consequently,
we
k
A0
2
2
1/2
analysis
(Cs0 + CA0 )
k
1 + ρe /ρ0
√
passing through thek South
(4) can
estimate
the The
value
of
the
magnetic
field
in
the
loop:
observe
and
fast.
The
slow
kink
mode
has
the
p
3. MHD
modes
ofslow
coronal
loops
event on 14th of July (Aschwanden et al., 1999; 1/2
ely, the TRACE 171 Å ob√mode
than
where
Ck may
is a so-called
kink
speed.
B0 =
(4πρ
CA0
= th
The phase speed of the fastNakariakov
kink
is #
0)
et
1999)
be considered
as
a typi3/2al.
about
2 πofofmagnetic
L oscillations
8:17 and 8:33 UT and only The theory of
observe
MHD
modes
structures
is wellloops. The
1/2
cal
example
kink
of
coronal
B0!= (4πρ0 ) "1/2
CA0 =
ρ0 (1 + ρe /ρ0 ).
(6)
mode
is
theofloo
oscillations
was registered.
developed
(see,
e.g.
Roberts
1991).
Considering
a
coroP
The
determination
th
analysis
of
the
loop
displacement
shows
that
the
osω
2
ω
C
C
s0
A0
the dist
4.almost
Determination
of
the
magnetic
≈ Ck ≡nal loop as a straight
CA0magnetic
,≈ C
(5)
evolution of the intensity
cylinder
of width
a, the
one
cillations
areT0
harmonic
with
period
of thefield
≡
·
to
errors
in
determi
ofkthe
magnetic
field
weakly
sensitive
2 is
2 )1/2
(Nakari
k taken The1determination
+ ρe /ρ0
(C
+
C
about
256
s
(the
frequency
about
4
mHz).
About
ee neighbouring slits
can
connect
properties
of
MHD
modes
of
the
cylinder
= 13 ± 9G
s0of slow
A0
(5) to errors in the B
The
speed
waves,
Cthe
below
both
theo
magnetic
field is
propo
speed
T0 is
determination
of
the
density,
because
Nakariakov
&
Ofman
(2001)
three
periods
of
oscillation
were
observed.
The
pemost identically, suggesting
with
physical conditions
andand
outside
the cylinder
2inside
where Ckmagnetic
is a so-called
kink
speed.
C
the
sound
Cs0 density.
speeds.
For
the coronal
lok
Figure
2 called
shows
A0
remains almost
constant
these
oscillafield is riod
proportional
to the
squareduring
root
of the
(fast magnetoacoustic)
Magnetic field estimates using standing kink modes
Table from Wang (2011)
•
Magnetic field strengths from seismology and
magnetic extrapolation are consistent
•
Combining seismology with other methods leads to
more powerful inversions e.g. density contrast and
inhomogeneous layer thickness (Verwichte et al.
2013)
•
Also SDO observations (e.g. White et al. 2012)
•
Low amplitude decayless oscillations seen by
Nisticò et al. (2013) using SDO
Verwichte et al. (2013)
3
Ubiquitous propagating kink waves
200
0.00
1.00
0.00
5.00
10.00
2.00 −10.00 −5.00
A) CoMP 10747Å Intensity [log10 μB]
B) 3.5mHz Velocity Image [km/s]
−0.50
0.00
0.50
C) Outward/Inward Power Ratio
Solar Y [arcsec]
0
−200
−400
−600
−800
−1000
−1200 −1000 −800 −600 −400
Solar X [arcsec]
−1200 −1000 −800 −600 −400
Solar X [arcsec]
−1200 −1000 −800 −600 −400
Solar X [arcsec]
•
Ubiquitous transverse velocity perturbations propagating along field lines
•
Broadband power spectrum centred on 5 minutes
•
Strongly damped
Tomczyk et al. (2007)
Tomczyk & McIntosh (2009)
4
1 þ $Þ ! ð1 ! $Þ sin
;
i
right-hand side of equation (65) can be neglected in compar-
tive layer would be quasi-stationary and described by the
y.
. The
two inregions
arebrackets
connected
by!1
a thin layer,
is "ethe
ison with
first term
the square
for t4!
same formulae as in the case of stationary resonant absorpk . As
e that propagating compressive waves have a result,
in the following
approxition (e.g., Mok & Einaudi 1985).
a " we
‘ <rewrite
r < a equation
with ‘5(65)
a, where
the plasma
density varies
mate
form:
ed in coronal loops (Berghmans & Clette
monotonically from "i to "e , with "i > "e . The equilibrium
(field BZ is everywhere uniform and in the
l, Ireland, & Walsh 2000; Robbrecht et al.
magnetic
z7. APPLICATION
TO CORONAL LOOP OSCILLATIONS
!
t#!
!!t
ie
!i!
t (see Fig. 1).
!1The nonuniformity in plasma
al. 2001) and in polar plumes (Ofman et al.
kz
direction,
B
¼
B^
< Se
exp i#A ðr ! rA Þs sgn D
v¼!
Formula (56) gives the calculated decay rate of oscillaa"#! "ðrÞ produces
akariakov, & DeForest 1999; DeForest &
0
density
a nonuniform Alfvén speed, allowing
tions in the kink mode. Its use may be conveniently illus) to occur. It is in such nonuniform
Furthermore, prominences are also
resonant
wave
effects
"
tratedetc.
by taking the density profile in the annulus in the
• Mode coupling studied by Hollweg
& Yang, Goossens et al. 1992
3
late and
indeed
may
exhibit
decaying
oscil!
s
=3
ds
:
ð68Þ
þ
s!=#
layers
that
viscous
effects
are
likely
to
be
most
important.
!
form
i
y-Hobas et al. 1999; Terradas, Oliver, &
In what follows we adopt the cylindrical coordinates r, ’,
#
$
"
%ð2r
þ
‘
!
2aÞ
i
hat have some similarity
to
the
decaying
and(2002)
z withthe
the
z-axis alignedand
with
the
equilibrium
magnetic
• Ruderman & Roberts
considered
the
context
the
rapid"ðrÞdamping
of!coronal
ð1 þ $Þ
ð1 ! $Þ sinloop
;
¼
A
i
Let us estimate
ratio of the thirdin
second
terms inofthe
2‘
2
illations discussed here. It may be that an exponent
field.inInequation
the dissipative
layer,
there
are
large
gradients
in
the
oscillations
by TRACE(68). We have
the decay in prominence
oscillations observed
can
a!‘<r<a;
ð71Þ
radial direction only. This observation enables us to use the
3
3
!1
2
2
2
3
2
3
lines similar to those proposed here for s =3
10 !
R #@ v=@r . 4Then
t #!
v k#
approximation
2 8 (#r
2 2 the
!1 linearized MHD
2‘
Resonant
a damping mechanism
a!‘<
r < a ; absorption asð71Þ
=" . Using equation (47), we obtain
nd " ¼ " ð1 þ $Þ=2. Then it follows from
and (56) that
't
&
' 3 ( 10 t ! R
:
ð69Þ
where
$ ¼ "e ="i .
Using
equation
(47),
k
s!=#of
3! modes
3ð‘=aÞ!
! kink
k
• Modelled damping
is due
to resonant
absorption,
acting
the
a ! ‘=2 in
and
"A inhomogeneous
¼ "i ð1 þ $Þ=2. Then it
rA ¼
absorption considered in this paper is
equations (50) and (56) that
!2 !1 1=2 .
of the tube
kregions
Now we consider times t such that !!1
chanism that might explain
the damping
k 5 t5 10 !k R
! ‘ð1 ! $Þ
:ance with
! Another
¼ possible damping Note
. Then, in accordthat 10 R ' R for R ' 10 zð72Þ
e oscillations.
equation (69), we can neglect the term s =3 in the
þ $Þ
adial wave leakage4að1
(e.g., Cally
1986;
exponent in equation (68) in comparison with the term
!2
1=2
1=3
12
3
!¼
!k ‘ð1 ! $Þ
:
4að1 þ $Þ
we obtain
follows from
ð72Þ
s, & Goossens 1998; Stenuit et al. 1999). s!=# , and the integral in equation (68) is easily calculated.
In terms of the period & ¼ 2L=ck of the fundamental kink
!
a
ge occurs when the solution in the envi- As a result we arrive atl
mode with wavenumber k ¼ %=L and kink speed ck
be has the form of a propagating wave.
(=!k =k), we obtain an oscillation decay rate &decay (=! !1 ) of
(
k
!i!
t
echanism is not applicable to the particui
Se k
2 a "i þ "e
<
v
)
!
oscillations of a coronal loop. This is
B a"
&decay ¼
&:
ð73Þ
! þ iDðr ! rA Þ=ð2!ρk Þ k
ρ
% ‘ "i ! "e
i
e
age can only happen if the phase speed
)
!1 !!t
illation is larger
than the Alfvén
speed
We consider this result in relation to the observational data
Exponentially
damped
( ½exp
iðr
!
r
ð
ÞDt=ð2!
+ :
ð70Þ
A
k ÞÞ ! e
decay
However, as we will see in what follows,
reported by Nakariakov et al. (1999). These authors
oscillation
with
reported a coronal loop oscillation with frequency
nsity in the magnetic tube is larger than
estimate v ' u0 a=‘,
!k ) 0:024 s!1 (& ¼ 256 s) and decrement ! ) 0:0011 s!1
ide the tube, as in coronal loops, then Using equation (60b), we obtain the !1
!1
!1
!2
i
R1=2 for R4106 ).
(&decay ¼ 870 s). Taking "i ¼ 10"e , we obtain from equation
not satisfied. Hence, kink oscillations
ofe valid for te! (note that ! 5 10 !k !1
of
the
globalHence,
during
the
characteristic
time
!
(72) that ‘=a ) 0:23.
re alwaysdecay
nonleaky.
mode damping, the amplitude of the wave motion in the disIt follows from equation (72) (and more generally from
rganized as follows. In the next section
wee sipative layer increases from a value of order u to a value of
i
eq. [56]) that the condition !5 !k is equivalent to ‘5 a. It
0
oblem and in x 3 derive the governing equa- order u0 a=‘. This increase occurs because of the energy flux
can be shown that, in the general case where the density
urbation of the magnetic pressure, obtain- from the global motions into the dissipative layer. Then the
varies through the whole tube cross section, the condition
ϕ
o this equation in the form of a Bromwich amplitude remains of the same order of magnitude at least
!5 !k is equivalent to jDj4!2k =a. This inequality means
!1
r that the characteristic scale of the density variation near the
ution is used in x 4 to study the fundamen- up to the time satisfying t5 10!2 !k R1=2 .
Note
that,
in
accordance
with
equation
(70),
the
characresonant position is much smaller than the tube radius.
of oscillation of a magnetic tube. In x 5 the
When this condition is not satisfied, all solutions to the disof the oscillation in the magnetic tube teristic scale of variation of v in the dissipative layer
1=t. This decrease
corresponds
phase mixing
persion
Fig.as1.—Sketch
of the equilibrium
state,to
showing
a magnetic flux
tube equation corresponding to the kink oscillations
ed for times much larger than the period of decreases
depends
on
r
of Alfvén
oscillations
that
occurs
because
!
have
with plasma density "i embedded in a plasma
A with density "e . The equili-imaginary parts of the same order as real parts. As a
In x 6 the wave motion in the dissipative and the
brium
magnetic field
everywhere
has strength
B.with
The differequilibrium density
neighboring
magnetic
field lines
oscillate
result all tube perturbations damp aperiodically or almost
!1
he ideal resonant position is studied. In x 7 ent frequencies
to
"
.
The
dashed
lines
varies in the annulus
region
a
"
‘
<
r
<
a
from
"
(e.g., Heyvaerts & Priest 1983;
Wright
aperiodically,
an external perturbation does not cause
i
e
•
Transfer
of
energy
from
kink
mode
to
Alfvén
(azimuthal)
oscillations
withinand
inhomogeneous
layer
theoretical results with the observations of 1992a,
show
the perturbed
magnetic&tube
in its
kink mode of oscillation.
1992b;
Mann, Wright,
Cally
1995).
pronounced tube oscillations.
period & ¼ 2L=c of the fundamental kink
avenumber k ¼ %=L and kink speed c
tain an oscillation decay rate &
(=! ) of
&
2a" þ "
¼
&:
%‘" !"
ð73Þ
is result in relation to the observational data
Nakariakov et al. (1999). These authors
oronal loop oscillation with frequency
(& ¼ 256 s) and decrement ! ) 0:0011 s
The behavior of
v given byequation
equation (68) was studied by
Hollweg & Yang (1988) also discussed damping of the
Taking "i ¼ 10"e , we Ruderman
obtain
from
& Wright (2000). It was shown that, in the case
kink mode by resonant absorption, in the ideal case. Their
• Only loops with small
inhomogeneous
layers
arevalue
able
to¼ support
coherent
R
,
jvj
takes
its
maximum
at
r
r
where
!4!
analysis
was for aoscillations
plane, but they applied it to a cylinder by
0:23.
when t ¼ t ) 3! lnð!=# Þ. Since # ' 20!ða=‘ÞR
, we
replacing the perpendicular wavenumber by 1/a. Surpris5
obtain t ) 15! for a=‘ ' 0:1 and R ¼ 10 10 . This
ingly, their procedure gives results identical to equations
om equation (72) (andmaximum
more
=‘. After reaching its
valuegenerally
is of the order of au from
(56) and (73). They concluded that under coronal conditions
k
m
m
!1=3
!1
!1
!
!
12
!1=3
14
A
Propagating kink waves
•
Transfer of energy from kink mode to Alfvén (azimuthal) oscillations within inhomogeneous layer
•
Alfvén wave component is harder to observe so only damped kink component is seen
6
used in the numerical solutions (ζ ≥ 1.5).
L (108 m)
1.68
0.72
1.74
2.04
1.62
3.90
2.58
1.66
4.06
1.92
1.46
ε = πR/L P (s) τd (s) P/τd
l/R
τAi (s)
VAi = L/τAi (km s−1 )
0.067
261
870
0.30 0.20–0.83 ⋆ 155–190
880–1080 ⋆
0.146
265
300
0.88
0.58–2.0
378–423
170–190
0.075
316
500
0.63
0.41–2.0
225–259
670–770
0.061
277
400
0.69
0.45–2.0
163–183
1110–1250
⋆
0.071
272
849
0.32 0.22–0.90
170–210
770–950⋆
• 0.068
Analytically
derived
asymptotic
state e.g.
Ruderman & Roberts (2002), Goossens et al. (2002)
522
1200
0.44 for0.30–2.0
327–386
1010–1190
0.043
435
600
0.73
0.47–2.0
181–203
1270–1420
0.059
143
200
0.72
0.46–2.0
85–93
1780–1950
0.036
423
800
0.53
0.35–2.0
140–165
2460–2900
• Inversions produce curves in parameter space e.g. Arregui et al.856
Goossens et al.: Analytic approx
(2007), Goossens et al.M.(2008)
0.056
185
200
0.93
0.57-2.0
105–113
1690–1830
0.169
396
400
0.98
0.62–2.0
663–730
200–220
Table 2
Seismology with exponential damping profile
the damping rate of loop oscillaestablish a one-to-one relation bers of interest. As Fig. 1 shows, there
erent equilibrium models, with dift yield the same observational value
impossible to discriminate between
s is fixed the other can be obtained.
ished by Aschwanden et al. (2003).
period and damping rate of loop
ping rate provide us with important
ces of information about the physcoronal loops. The analysis in this Fig. 2. Curves of constant period (P = 185 s) and constant damping rate
Fig. 1. Analytic inversion (solid lines) and corrected numerical inver(P/τd = 0.93)
on thehave
(ζ, l/R)-plane
corresponding
to loop oscillation
ed period and•damping
for the
One rate
fitted
parameter
Ld can
various
combinations
of inhomogeneous
sion (filled circles) inlayer
the (ζ, l/R,width
τA,i )-spaceand
for loop oscillation event #5
event
#10,
for
two
different
values
of
the
internal
Alfvén
travel
time;
rium models that reproduce obserin Table 1.
density
contrast
a new observable,
the period,
but τAi1 = 113 s (solid lines) and τAi2 = 106 s (dashed lines). The filled
vén travel time. As we do not make circle indicates the only valid equilibrium (ζ ≃ 18.8, l/R ≃ 0.57) that
at the numerical ζ ≃ 20. We have also calculated zmin = 1/C for
n any equilibrium parameter, again reproduces the observed wave characteristics for τAi1 . When τAi = τAi2 ,
the events studied by Arregui et al. (2007) and the results agree
two different
equilibriumsolutions,
models (filledthough
squares) arebounding
possible; (ζ ≃ values
4.2,
•
Inversion
problem
has
infinite
can
estimated
well
withbe
the corresponding
values of (l/R)min in their Table 1.
btained in the new (ζ, l/R, τAi ) pa- l/R ≃ 0.8) and (ζ ≃ 2.48, l/R ≃ 1.44).
observation of damped transverse
• Also
d of the last century
(Aschwanden
P/τd depends
each
of the form,
= P (ζ, l/R, τAi ) andthat
= P/τd (ζ, l/R)
have a constant
of Pproportionality
onfordensity
. 1999) several similar events have loop oscillation event. These data-cubes can now be combined
layer (sinusoidal, linear etc.)
s and derived physical properties with the observed periods and damping rates by requiring that
d in Aschwanden et al. (2002) and
(5)
ction of 11 of those events was used P (ζ, l/R, τAi )! = τ"Ai f1 (ζ, l/R) = Pobs ,
002) and Goossens et al. (2002) in P
P
7
(ζ,
l/R)
=
·
(6)
g and resonant absorption as damp- τ
τ
d
d obs
As an illustration that any of the three seismic quantities can
be used as parameter we take z as the parameter. We let z vary
and compute for each value of z the corresponding value of ζ by
the use of the
functioninhomogeneous
G2 defined in (29),
profile
inside
ζ=
9.81z + 1
·
9.81z − 1
(33)
Subsequently we compute the corresponding value of y by the
use of the function F1 defined in (26),
!
"1/2
1
ζ
y= √
·
(34)
variab
of τA,i
part of
functio
inversi
lytical
seismi
with tw
The fa
oscilla
by ext
ipated
schem
5. Co
In thi
Full solution for mode coupling damping envelope
A&A 551, A39 (2013)
•
Exponential damping profile describes
changes the coefficient of the Gaussian term at small Z and reasymptotic
stateHowever,
of theitsystem
sults
in too much damping.
still remains Gaussian in
nature. For larger Z, Approximation 1 is more or less parallel to
the numerical solution to Eq. (47). Approximation 2 damps significantly faster. Approximation 3 includes the inhomogeneous
• term
Does
not
(and assumption
is not intended
to) the intebut the
simplifying
of taking η̃ outside
gral predicts a slower damping rate at large Z. However, it does
describe the behaviour at early times
have the correct behaviour for small Z.
The best approach in understanding the spatial dampetmode
al.: The
Damping
of Propagating Kink
ing of the kink modeHood
through
coupling
to the Alfvén
in
the transition solution
layer, is to solve
Eq. (47) by
numerically.
• mode
Full
analytical
derived
Hood
Keeping in mind the results shown in Fig. 3, the error in using
Hence,
the(2013)
second
order
derivative
beatneglected
to leading
et al.
describes
behaviour
all
Approximation
3 is not
too significant.
It has can
the advantage
of
having a solution in a closed analytical form. The functions Si
times
density
assumption
κ ≪ 1algebra
leadspackages,
to
and
Civariation
are rapidly obtained
from computer
such as!
Maple, and it is easy to"use a package to numerically #
in Eq.
expansion,
using
ϵ the
a(1terms
− cos
Z)(54). TheZ small Zsin(Z
− u)
dη̃ integrate
= (54) gives the correct Gaussian
− behaviour.
η̃(u)
du .
Eq.
dZ
2
Z
Z −u
0
Waves
13
order in κ and the weak
(48)
Comparison with numerical results
Eq.5.(48)
is the important equation that can be used to investigate the spatial damping
The
numerical solution to the equation
linear MHD equations
Integro-differential
allowsremains
the most
accurate
description kink
of the damped
kink
mode. In this
of the
the
propagating
mode,
whenever
the density contrast is not too large. For
precise
testing
of
numerical
results
section, the approximate solutions derived (and the methods illustrated)good
in the previous
sectionisarefound
compared
withthe
the actual
example,
agreement
with
numerical solutions (see Section 5) whenever
numerical results. Two different numerical codes are used, a
second order Lax-Wendroff scheme and a fourth order, finite
•ρi /ρ
equivalently
κ results
≤ 1/2.
Thewith
advantage
of Eq. (48) over the full expression in
Inconvenient
seismological
e ≤ 3 or
difference
method.
Thefor
numerical
obtained
the two
different
methods for
are consistent
and indicate that the results obinversions
observations
Eq.tained
(27)are
is not
itsdependent
relativeonsimplicity.
This
equation
be solved numerically. We can use it
the method used
to solve
the linear can
Fig. 4. Amplitude of η = ξr and logarithm of the modulus of η at the
MHD equations. We consider two examples for a small tran- centre of the tube shown as functions of distance z. The solid curve
layer, namely small
(ϵ < κ standard
< 1) and a small
density represents
Hood
et numerical
al.
alsosition
to investigate
howϵ some
approximations
compare
the
full solution.
In(2013)
the numericalwith
solution,
the dashed
curve
is the
so8
contrast (κ < ϵ < 1).
lution of (47). The period is 36 s, the ratio of the width of the transition
•
General spatial damping profile
•
Damping profile for non-asymptotic “early” regime is Gaussian rather than exponential
•
“Early” times depends only on density contrast, for low density contrasts may be entire signal
•
For the general spatial damping profile we have 3 fitted parameters: Lg, h and Ld
•
Can obtain seismological results for both layer width and density contrast simultaneously (for
data of sufficient quality to identify these 3 parameters)
9
Hood et al. (2013)
Pascoe et al. (2013)
(2)
exponential damping profile occurs (1/κ) as a function of the
density contrast ratio ρ0 /ρe . For ρ0 /ρe < 2 the Gaussian damp(3)
Forlow
thedominate
modifiedquality
Gaussian
profile
inFor
Eq. 2, the equ
Seismology
for
signal
oscillations
ing
profile
is
expected
to
the
observed
signal.
and Lg
relationship
is given exhibit
by Eq. 7.
ρ0 /ρe > 3 only the first
1–2 wavelengths
theHowever,
Gaussiandue to the pr
ds
upon
of the constant term A0 /2, this profile does not e
ised) inhomogeneous
layer
damping
behaviour.
• Many
observations have a signal quality of τ/P ~ 3
is a ratio of densities, and match the gradient of the exponential profile at z
r a wavelength λ. The con- Despite this, we find the modified Gaussian profile pr
a slightly
better
to our
data
(see also Hood
(3) the
For
the contrasts
modified
profile
inoverall
Eq. 2,fitthe
equivalent
ds upon
chosen
density Gaussian
Low
density
Medium density
contrasts
High
density
contrasts
<2
~2–10
and
so use it in
spatial damping
is given2012)
by Eq.
7.
However,
dueour
to general
the presence
yer and hererelationship
we choose
pro(Eq.A2).
r ≤ (Rof− the
l/2)constant
to ρe at term
0 /2, this profile does not exactly
usat layer
we substitute
3 and
Eq. 7, we obta
exponentialEqs.
profile
at 5zinto
= h.
ies, and match the gradient ofIfthe
Mainly exponential damping
Despite
this,
we
find
the
modified
Gaussian
profile provides
Gaussian
damping
profile
Both
damping
profiles
present
g profile
The
con- is the exponential
profile
1 our data (see also Hood et al.
a slightly better overall 2π
fit to
0)density
of the form
h=
k κgeneral spatial damping profile
ose pro- 2012) and so use it in our
(4)
to ρe at (Eq. 2).
If we substitutewhich
Eqs. 3can
andbe5 rearranged
into Eq. 7, as
we obtain
mping
length scale given by
onential
Gaussian
h
1
ρ0 + ρe
2π 1
= =
.
h=
(8)
κ
ρ0 − ρe
k κ
(5) λ
(4)
which can be rearranged
as that the height of the switchExponential
This
shows
in profiles (in
ality
has
again
been
chosen
given by
of the number of wavelengths) depends only upon t
e inhomogeneous layer.
h
1
ρ0 + ρe tio of densities. This information can be used to info
file, the gradient
= of=the pro- .
(9)
10 to which analytical formula is most app
κ
ρ(neg0 − ρe observer as
es to (5)
some λmaximum
Damping profile for seismology
Error in seismologically inferred inhomogeneous layer
width from fitting to first 3 wavelengths (ρ0/ρe = 2)
Gaussian profile
exponential profile
Gaussian fit better for low signal quality observations
(for low density contrasts)
Hood et al. (2013)
Pascoe et al. (2013)
11
Gaussian damping regime for standing kink modes
•
Same physical mechanism applies to standing modes
Ruderman & Terradas (2013)
Pascoe et al. (2013)
•
For standing kink modes, a large density contrast is usually assumed (typically 10, e.g.,
Nakariakov et al. 1999; Ruderman & Roberts 2002; Goossens et al. 2002)
•
This density contrast is consistent with an exponential damping profile (Gaussian profile for ~
1.2 oscillations only)
12
Seismology of standing kink modes: observation
•
Van Doorsselaere et al. (2008) observed a weakly damped standing kink mode with Hinode/EIS
T. Van Doorsselaere
et al.:
field
measurement
T. Van Doorsselaere
et al.: Magnetic
fieldMagnetic
measurement
using
Hinode/EIS using Hinode/EIS
L19
Alfvén modes assume a torsional character in cylindrical
structures, such as loops, which are not necessarily of circular
cross-section (Van Doorsselaere et al. 2008). With EIS’s spatial
resolution, they cannot be observed by Doppler shifts but only by
a periodic line broadening. Since the waves are clearly observed
Gaussian
by Doppler shifts, this implies that they cannot be interpreted in
terms of Alfvén modes.
The sausage mode is a periodic transverse contraction and
expansion of the flux tube, which does not produce a net Doppler
shift. Furthermore, a sausage mode has strong intensity variations, which is not in the current observation. We are there-Exponential
fore unable to interpret our observation in terms of fast sausage
modes.
Slow magneto-acoustic waves have a strong longitudinal velocity component and show intensity variations. For the velocity
amplitude that we measure here, the intensity variations would
Overview of the observed active region on 2007 July 9 at
however be below the noise level, which is a few percent of the
T (just after the oscillation) and 18:13 UT, respectively, acbackground intensity. Low harmonic, standing slow magnetoy Hinode/EIS. The colour code is the intensity in the 195 Å
Fig.
2.
Top
panel:
The un-smoothed
(dashed
line)
smoothed (full
Top panel:
The
intensity
in theof the
195slit.ÅThe
line
along the slit acoustic
vertical
black line
indicates
the position
white
oscillations
can
be=and
dismissed
easily, because the phase
′′
line)
velocity
measurements
in
the
195
Å
line
for
y
−72.8
(indicated
−1
indicatesdata,
the pixel
oscillations.
The white
line traces
riginal
—:showing
smoothed
data). The
position
with the oscilla- speed (2600 km s ) of the observed wave would be much higher
with the diamond in Fig. 1). Bottom panel: The Scargle periodogram of
scillating
loop.
shown with the
line)profile
is associated
the density
en- than
• vertical
Density
fromwith
CHIANTI
(FeXII
186/195
line
ratio)
gives
density
of 1.2exthe un-smoothed
signal.
the local
sound
speed
(a few
100 kmcontrast
s−1 ). We cannot
ment. Bottom panel: The density derived with CHIANTI using
clude completely a running slow magneto-acoustic wave, but
XII 186/195 line ratio. The densities used for the external denthere are
two important
indications
it would be the incorrect
Thethe
un-smoothed
line-of-sight
velocities
measured that
by the
eervational
underlinedresults
with a full line, whereas the densities used for
• Gaussian damping profile predicts interpretation.
4x
weaker
damping
exponential
′′ loop is rooted in a sunspot.
Firstly,
observed
of the
195 Å spectral
linethe
at ythan
= −72.8
(indialysed
density
are underlined
with a dashed
line.
data from
the EIS instrument
(Culhane
et al. 2007) Doppler shift According
to 1)
DeareMoortel
etFig.
al.2 (top
(2002),
slow magneto-acoustic
cated
with
the
diamond
in
Fig.
plotted
in
panel,
d Hinode (Kosugi et al. 2007). EIS has both imaging
These un-smoothed
velocities
using period in general than
in these
loopswere
havederived
a shorter
ot) and spectroscopic (1′′ and 2′′ slit) capabilities, in the dashed lines). oscillations
a macro-pixel
approach,
i.e.
average
value300
in as).
pixel
with its running slow waves
• Gaussian
damping
estimate
consistent
with
weakly
oscillation
we
study
here
(180thesdamped
vs.
Secondly,
range
of h170−210
Å andthe
250−290
Its spectroength
taken
4.5
later. From
tracedÅ.loop,
we can two
estimate
neighbouring pixels. This reduces the noise level on the data
are excited near the loop foot-points. The detection lengths, over
mode
canloop
operate
in a and
rastering
mode it
(repeated
expoojected
length
de-rotate
to correct
for the
effect
and corresponds to an error in the velocity of ∼0.9 km s−1 .
hile scanning over the observation target) or a sit-andwhich
the amplitude
decreases
It isthe
apparent
fromdistance
the time series
of the velocity
that onlyby a factor of four, for
solar
longitude,
i.e.
divide
the
projected
loop
length
by
ode (repeated exposures at the same spatial location).
these are
oscillations
are
below
(De Moortel et al. 2002).
three loop
or four periods
present13
in the
current
data25
set.Mm
However,
e used
of thesit-and-stare
solar longitude
to derive
length.
observations
of its
the true
active
re- The
this implies that
is ofcurrent
relatively
high quality,
At the
the observation
height of the
velocity
measurement (∼100 Mm),
2Lg
2 a
which,
unlikethis
Sect. 2,
times
can
be described
profile. Gaussian
Subsequently
the frequencystratification
dependence
of the
dampingin rate
which
act
exp ´zby
{Lg2 Gaussian
. An alternative
function
without
causes
an increase
amplitude
ofmay
the velocity
Hood
et al.& (2013)
derived
expressions
for be
theused
damping
be-Eq. 1particular
as a low-passcomponent
filter for(e.g.
observations
broadband
oscillations
term
isfunction may
This Gaussian
in place of
in the gen-damping profi
De Moortelof
et al.
1999; De
Moortel
Hoodconstant
of this method u
haviour
all times.
integro-differential
equation
for the cation
eralThe
spatial
damping profile (Eq.
5) when
is strong.
2004) ofSoler
outwardly
andextended
so
can actfor
to oppose
(Verth et al. 2010).
et al.propagating
(2011a,b,waves
2012)
these
˜ the damping
¸
Sect.
5 we
continuous
variation in In
amplitude
with
height
z can bespatial
approxiSect. 2 we
apply
the Gaussian
profile
to consider the
the attenuation
due to
coupling. background
2 damping
studies to consider
the effects
ofmode
longitudinal
flows,
z
the broadband
oscillations
CoMP.
In. Sect.
we de- oscillation
Pascoe et al. (2012) considered stronglymated
damped
the3observed
bypropagatGaussian and
exponential
functions
large
A pzq “observed
Aat0 small
exp byand
´
(7)
stratification ing
andkink
partial
ionisation,
respectively.
Gravitational
2
2L
velop
a
new
method
for
analysing
broadband
kink
oscillations
waves and found that the damping z,
behaviour
for
early
respectively. This was employed by Pascoe et al. (2013) who
g
stratification times
causes
an
increase
in
amplitude
of
the
velocity
which,
unlike
Sect.
2,
does
not
rely
upon
the
can be described by a Gaussian profile.
Subsequently
produced
a simpler, piecewise spatial damping profile suitableassumption of a
No. 2,1999;
2010
RESONANTLY
DAMPED
PROPAGATING
KINK
WAVESdamping
IN THE
SOLAR
CORONA
L103
Gaussian
function
may
used
in
place
Eq.
1appliin the
gen- fit
Least
squares
component (e.g.
Deal.Moortel
et al.
De
Moortel
&
Hood
particular
profile.
In be
Sect.
4 we
demonstrate
the
Hood et
(2013) derived
expressions
forfortheuse
damping
be-This
for
seismological
inversions.
The
initial
stage
of
the of 2.
•
Mode
coupling
causes
kink
waves
damping
mechanism
more complex
e.g., the
multiof this
methodprofile
using data
from
numerical
haviour propagating
for all times. The
integro-differential
equation
for
eralcation
spatial
damping
(Eq.
5) when
thesimulations.
damping isInstrong.
2004) of outwardly
waves
and soin can
act
togeometries,
oppose
damping
envelope
thread loop structures (Terradas
et al. 2008), as
an illustration is described by a Gaussian function,
In driver
this section
we reco
Sect.
5 we consider
the effect
of the spatial
form ofdamping
the
on
continuous
variation
in amplitude
with
heighta coronal
z canloop
betoapproxiof the basic process
we assume
be a cylindriIn
Sect.
2
we
apply
the
Gaussian
spatial
profile
to
the attenuation
due
to
mode
coupling.
to mated
damp
with and
a signal
quality
τ/P
that
« oscillations.
˜
¸ff
cal
magnetic flux tube.
Then the kink
phase speed
inlarge
the long the observed
Conclusions
are presentedformed
in Sect. by
6. Verth et al. (
by Gaussian
exponential
functions
at small
and
2
the
broadband
oscillations
observed
by
CoMP.
In
Sect.
3 we
Pascoe etz,al.
(2012)
considered
strongly
propagatwavelength
limit for andamped
axially homogeneous
magnetic cylinder
A0
z
respectively.
Thisthe
waswith
employed
by and
Pascoe
al. density
(2013)
instead of
the deexponent
differing internal
external et
plasma
and who
constant
depends
on
loop
parameters
pzq
A
“
1
`
exp
´
,
(1)
ing kink waves
and found
that piecewise
themagnetic
damping
behaviour
for early
field
strength
is
2 analysing broadband kink
produced
a simpler,
spatial
damping
profile
suitablevelop a2 new method for
tio ofoscillations
outward and inw
Lto
g
2.
Least
squares
fit
CoMP
data
!
unlike Sect. 2, does not rely upon the of
assumption
times can befordescribed
by a Gaussian
profile.
use for seismological
inversions.
The Subsequently
initial stage of thewhich,
frequency of
andaintegr
Figure 1. Simple illustration of the observed semi-circular annulus geometry
2
of
the
coronal
loop
system.
The
integrated
wave
paths
analyzed
by
Tomczyk
&
v the
,
= B where
damping derived
envelope is
described by for
a Gaussian
function,
particular
damping
In
Sect.
we
theexpression
appli- (th
Hood et al. (2013)
expressions
damping
be-the (1)
InMcIntosh
this
section
reconsider
the
analysis
data perinitial
amplitude
of profile.
the
oscillation.
L4galong
is ofdemonstrate
theCoMP
µ(ρ
+ ρ ) A0 is
resulting
(2009,
shaded inwe
gray)
are
approximately
only
half the
length
the total loop system (L ≈ 250 Mm) since the data in this segment had the best
«
˜
¸ff
formed
by Verth
etandal.inward
(2010)
using
ashown
Gaussian
damping
profile
Gaussian
length
which
depends
upon
the
this
method
using
data
from
simulations.
In
haviour
all times. The
integro-differential
equation
for damping
the cation
S/N.of
The scale
direction
of
outward
wave propagation
is
by numerical
the loop
• for
2 and external plasma density, B is
Damping
rate
depends
on
where
ρ and ρ are the internal
solid
and
dashed
lines,
respectively.
A
z
instead
of
the
exponential
profile
used
by
those
authors.
The
ra0
magnetic
field
andparameters
µ is
magnetic permeability.
as (1)Sect. 5 we consider the effect of the spatial form of the driver on
continuous variation in amplitude
with
zthecan
bethe
A pzq “ the
1 (1)
`height
expstrength,
´
, isapproxi2
Equation
shows
that
kink
speed
intermediate
between tio of outward and inward power was calculated as a function xP p f qyratio
2
Lg
(for both
Gaussian
and
(2010) that the oscillations.
TGV relation is simplyConclusions
given by
observed
are presented in Sect. 6.
the external
and
internalat
Alfvén
speeds. and
Regarding
kink wavesthe
in
mated byfrequency
Gaussian and exponential
functions
small
large
2λ integrated over half a semi-circular loop. Their
coronal loops, it is observed that ρ > ρ , so naturally occurring of frequency and
1
z, respectively.
ThisA0was
employed
bygradients
Pascoe
ettheal.
(2013)
where
is the
initial
amplitude
of
the
oscillation.
Lwho
is the
Lg “
density
between
internal
and external
will resulting
g plasma
expression
7) is
exponential
damping
regimes)
,
(3) (2)
L ,=(their
v ξ Eq.
where L “ 250 Mm is
1{2
cause
the
local
Alfvén
frequencies
in
this
intermediate
layer
to
f
πκϵ
Gaussian piecewise
damping length
scale
which
depends
upon
the
loop
produced a simpler,
spatial
damping
profile
suitable
ˆ
˙ and ξE “ 2{πκϵ. Assum
match the global
kink frequency
at some magnetic
surface, thus
fit bytofPout
CoMP
parameters as inversions.
where f issquares
the frequency
defined
=
vp f/λq and
ξdata
is width,
an2L
causing a resonance. This generates
azimuthal
ϵofAlfvénic
“ the
l{Rmotions
is2.
theLeast
(normalised)
layer
for use for seismological
The initial where
stage
qyratio “
xP p finhomogeneous
f , profile of
(8)the form give
equilibrium parameter
dependent
on length scale ofexp
the density
in the resonant surface which grow in amplitude, damping the
v
ξ
p
q
P
f
(3) demonstrates
L is
is inversely
kink mode as energy
is transferred
mode.
ph E
in that λ
κ “ pρto0this
´localized
ρe q{pρ
` ρthis
q issection
a ratioEquation
of densities,
and
the wavedamping envelope is described byglobal
a Gaussian
0In
einhomogeneity.
2λexcitedfunction,
reconsider
the
analysis
of CoMP data perproportional
to f, i.e.,we
the rate
of damping per unit
length
will be
For kink waves
by a broadband disturbance there will
Lmany
, a loop
(2)
g “resonances
• Frequency dependence
length.
Under
the
thin
tube
approximation,
λ
“
C
P,
where
C
greater
for
higher
frequency
waves
than
low
frequency
waves.
be
within
and
smaller
length
scales
will
where
L
“
250
Mm
is
half
the
loop
length,
f
is
thedamping
frequency,profile
k
k
1{2
«
˜ πκϵto phase
¸ffmixing, causing a cascade of energyformed
by Verth
(2010)
using
a Gaussian
The parameter
ξ canetbeal.
calculated
precisely
for a chosen
be created due
to
2 dissipationisbecomes
“ 2{πκϵ.
instead
a Gaussian
the kink
speed andand
Pequilibrium
isξEthe
period
of
oscillation.
The
constantspatial dampingxP p f qyout “
model;
e.g.,Assuming
if we
chooseprofile
a thin inhomogeneous
smaller scales where
more efficient.
A
z
instead
of
the
exponential
used
by those authors. The ra0
where
ϵ
“
l{R
is
the
(normalised)
inhomogeneous
layer
width,
demonstrated
by
power
boundary
a continuous
sinusoidal
In the
follow-up
paperratio
by Tomczyk
& McIntosh
(2009), ais profile
oflayer
thewith
form
given
by Eq.profile
7density
wedecreasing
obtain
pzq
A
“
1
`
exp
´
,
(1)
of
proportionality
determined
by
the
particular
profile
ρ and ρ then
of the sameand
CoMP
set showed
κ “ pρ0 ´ ρe q{pρ
is a detailed
ratio analysis
of
λ data
is the
wave-that
tio ofbetween
outward
and
inward
power
was paper
calculated
as
aoutward
functionpower,
2 0 ` ρe q more
Lg2densities,
for
the
propagating waves
traveling a in
larger
distance
suffered greater
ż
the
inhomogeneous
layer
and
the
values
used
in
this
L
calculation
observations
length. Under thefor
thin CoMP
tube
approximation,
Ck P,andwhere
˘
ρ + ρ over half a` semi-circular
2 R1
damping.
This inspired Pascoeλet “
al. (2010)
TerradasC
etkof
al. frequency and integrated
2
2 and loop. Their
ξ
=
,P
(4)
correspond
to
a
linear
profile.
p
qy
p
q
xP
f
“
f
exp
´s
{L
(9)
(2010)
toperiod
model propagating
kink wavesThe
damped
by the prois
the
kink
speed
and
P
is
the
of
oscillation.
constant
out
out
π
l
ρ
−
ρ
g ds,
where A0(Verth
is the initial
amplitude
of
the
oscillation.
L
is
the
g
L
resulting
expression
(their
Eq.
7)
is
cess
of
resonant
absorption.
Most
relevant
to
the
present
study,
0
et
al. scale
2010)
The
later
stage of
the damping envelope is described
of proportionality
is determined
by(2010)
the particular
profile
Terradas et depends
al.
derived
simpledensity
analytical
expressions
where l is the thickness of the boundary layer (see, e.g.,
Gaussian damping
length
which
upon
the
loop
Exponential
1
ˆ along˙the loop,
forGoossens
the outward
power,
where
is2002).
the Hence,
distance
for the damping
length
of kinkby
waves,
damping function
et al.
1992, 2002;
Ruderman
& Roberts
in the inhomogeneous layer
and the
values
used
inexponential
thisthatpaper
anshowing
(e.g.
Ruderman
&sRoberts
2002;
p
qy
xP
f
“
length
is
a
monotonically
decreasing
function
of
frequency,
the
Equations
(3)
and
(4)
demonstrate
that
the
efficiency
of
damping
in
p
q
P
f
2L
parameters ascorrespond to a linear profile.
out
L
Goossens
2002;and
Terradas
et
al.
2010)
of the
Terradas–Goossens–Verth (TGV)
relation. Thiset
hasal.
a consedue to resonant
absorption
also depends
on theform,
thickness
of the
p
qy
xP
f
“
exp
f
,
(8)
ratio
quence
that
solar
waveguides
with
transverse
inhomogeneity,
non-uniform layer and the steepness
The later stage of the damping envelope is described
ż of2Lthe
v ξ
p f q gradient
Pintransverse
e.g.,
coronal loops, will act as low-pass filters for propagating
in density.
` 2 ph2 ˘E for the inward power. F
2λ
1
• Power
by an exponential
function
(e.g.
Ruderman
&
Roberts
2002;
ratioLgis“ integrated
loop
kink waves. It is the mainover
purpose of this
Letter to investigate A pzqFrom
Figure
1p(panel
(2009),
p f q exp
f qy
“in dTomczyk
´s (3)
{Lg ds,
(10)
p´z{L
qhalf
Aseen
exp
, P&path
, et al. Alfvénic
(2) al. it canL“be“
in(A))
inMcIntosh
0xP
if the
propagating
waves
observed
by Tomczyk et where
that the
integrated
wave the
inloop
the coronal
250
Mm
is
length,
f
is
the
frequency,
1{2
L
Goossens et al. 2002;
Terradas
2010)
of
the
form,
L
πκϵ
(2007), realistically interpreted as the kink mode from MHD
loop system is off limb and assumed to be in the plane of
and results cannot
distinguish
andforξsky.
“inward
2{πκϵ.
Assuming
instead
aby
Gaussian spatial damping
where Ld damping.
is the exponential
damping
length
scale
given
wave theory,
exhibit this frequency-dependent
The
approximate
semi-circular
annular
geometry
of
this
Ethe
power.
For the
ratio
we obtain
xP p f qyratio “
wave
path
is
shown
in
Figure
1,
with
the
direction
of
outward
where ϵ “ l{R is the (normalised)
inhomogeneous
layer
width,
A pzq “ A0 exp p´z{Ld q ,
(3)profile of theGaussian
formwaves
given
byby Eq.
and inward propagating
denoted
arrows.7Wewe
showobtain
the
Gaussian
or
3.exponential
THEORETICAL MODEL
4λ
κ “ pρ0 ´between
ρe q{pρ
`
ρ
q
is
a
ratio
of
densities,
and
λ
is
the
wavep
q
pΛq
P
f
erf
analyzed
wave
paths
by
the
shaded
gray
region,
which
consists
0
e
out
where Ld is the exponential
damping length scale given by
Lwaves
.ratio different
where
Λ
“ L{Lg and er
p2f qyalong
xP
“ ż coronal
,
(11)
of many
traveling
loop structures, (4)
Traditionally, since standing kink waves in coronal loops were
d “
Lp f q toerf
length. Under
the thin tube approximation,
λ“
Ck P, where
Ctimes
˘
p2Λqthe´`erf pΛq
k was
π κϵ & McIntosh
integrated by Tomczyk
improve
observed using TRACE,
the measurement
of damping
damping
1 Pin(2009)
2 ing
2 the exponential spat
signal-to-noise
ratio
(S/N); “
cf. Figure 4 (panels
(B)pand
(C))
in
of primary
interest;
et al. (2003) and
4λ see, e.g., Aschwanden
p
qy
q
xP
f
P
f
exp
´s
{L
ds,
(9)
is the kink speed and P is the period
of
oscillation.
The
constant
out
out
g
Tomczyk
&“McIntosh
(2009).
Although
Tomczyk
&function.
McIntosh
Arregui
et al. (2007).
However,Hence,
for propagating
kink
waves
it spatial
the
general
damping
profile
is
Ld “
.
(4)
where
Λ
L{L
and
erf
is
the
Error
Whereas
integratg
produces
another
expon
L
(2009) did not integrate the entire loop0 lengths, i.e., from the
π2 κϵ to study density
is more
appropriate
the damping length
L , with the
of proportionality is determined by
the
particular
profile
the
damping
$ingoutward
footpoint (s´= 0)spatial
to¯ı
the inward
footpointfunction
(s = 2L), over a loop
” exponential
expression for wave amplitude given by
tionelement
of the Gaussian d
2
for
the
outward
power,
where
s
is
the
distance
along
the loop,
Hence,
the
general
spatial
damping
profile
is
the
important
point
is
that
in
the
gray
region,
inward
waves
A
z
in the inhomogeneous layer and the values used in" this
paper
produces
another
exponential
function
(Eq.
8),
here
the
integra0
&
#
1
`
exp
´
,
z
ď
h
form
of
Error function
have
traveled
further
and
therefore
the
damping
will
be
greater
2
s
2 of the Gaussian
Lg damping
”
´ A(s)2 =¯ı
tion
profile
produces
a
result
in
the
and
A exp −
,
(2)
´
¯
relative
to
the
outgoing
waves.
correspond to a linear profile.$
pzq
A
“
(5)
L
Eq. 11fittoofobservational
& A0 1 ` exp ´ z 2 , z ď
Letof
the Error
initial outward
atz´h
s = 0 and
inward
power
at s = 2La least squares
%
h
form
functions.
We
can
perform
14
A
exp
´
,
z
ą
h
2
Lg ¯ is described
h f be denoted by
The later stage Aofpzqthe
envelope
for frequency
P (f ), respectively.
Ld P (fż) and
Figure 1 shows the
where s is the´distance along
the waveguide and A (5)
is the Eq.
“ damping
11since
to Pobservational
data2L
in
the(3)same
way`as for Eq.˘8.
Then
(f ) ∝ A (f ), by Equations
(2) and
the spatially
z´h
Damping profile for propagating kink waves
ph
i
i
e
e
i
e
D
ph E
ph
E
D
E
i
e
E
i
e
i
e
D
0
D
0
2
out
in
Damping of broadband kink waves by mode coupling
•
Transverse (kink) velocity at loop
axis, stacked in time:
•
Higher frequency components are
damped more strongly than lower
frequency components (low-pass
filter)
15
ing, whereas the most general
mping regimes as described by
Eq. 1 since the absence of the constant
term
ˆ
˙ A0 {2 makes taking
∆z
“ exp The
´2 spectral
f .
ratios of functions much simpler.
power(15)
is given by
v ph ξE
` 2 the 2power
˘ ratio deFor an exponential spatial
damping profile,
2
P pzq
9 exp
´z {Lofgz0 . ,Calculating
(16)
pends upon
the9A
chosenpzq
∆z but
is independent
Spatially resolved power ratio calculation
ooFig.noisy
toratio
distinguish
1. Power
as a functionwhether
of frequency for CoMP data. the ratio for all available z0 should give the same value. In pracThe
dashed
line
represents
the
fit
for an exponential
spatial
tice the ratio will
vary
due introduce
to errors and soL
an average
ratiof(for
al or Gaussian, it is still instrucand
for
convenience
we
can
“
1{α
with α “
g
damping profile (Verth et al. 2010). The solid (red) line repre- a given ∆z) can be calculated, thus improving the statistical acatsents
willa fitallow
us tospatial
discriminate
for a Gaussian
damping profile. πκϵ 1{2 {2C k . curacy
the result compared
with a single measurement.
Theoffrequency
dependence
of the power at some
•
Choose
a
separation
distance
Δz
(and
a
We
now
consider
a
Gaussian
spatial
damping
profile, again
ture. To test our proposed anal- height is then given by
having the form given by Eq. 7, which is more convenient than
thesesince
two curves
the noise.
limiting cases
of entirely exponendata
thisconsider
has
no
The
maximum
height
z
m) Eq. 1 since the absence of the constant term A0 {2 makes taking
` The2 spectral
˘ power is given by
tial and entirely Gaussian damping, whereas the most general ratios of functions much simpler.
2
2
might
also
be
a
result
of
averbehaviour would include both damping regimes as described by
P pzq 9 exp ´z `α f ˘ .
(17)
Eq. 5line
and discussed
in Sect.
2
2
2
the
of sight,
as 4.2.
considered
P pzq 9A pzq 9 exp ´z {Lg ,
(16)
Although the CoMP data is too noisy to distinguish whether
Considering
a to
spectral
power ratio
as in -Eq.
15, with z1 “
used
a statistical
approach
to
the
damping
profile
isSpread
exponential
or in
Gaussian,
it isdamping
still instruc• Pascoe
and
for convenience
Lg “ 1{α
f with α “
et al.: Excitation
and
of broadband
kinkdamping
waveswe can introduce
power
ratio
due
profile
tiveof
to loop
developparameters.
analysis tools that
willshall
allow us to
1{2
z0discriminate
` ∆z, now
es
We
πκϵgives
{2Ck . The frequency dependence of the power at some
between the two profiles
in the future. To test our proposed
anal- height israther
seismological
information
then given
bythan
tysis
it technique
is indeed
possible
to since
idenandThe
hence the ratio
depends
upon“errors”
z. Instead we now compare
we use
numerical data
this has no noise.
`
˘ is a fixed
2
spread in thesignals
observational
data the
mightdata
also be a result of
¯ separation
q, where
theaverratio of P pz0 ` ∆zq to
P pz
oadband
when
pzq
P´
9 0exp
´z2 α2 f∆z
.
(17)
2
aging along multiple loops along the line of sight, as considered
2
2
distance
exp
´ pz
∆zqas inαEq.f 15, with z1 “
oop
can beet considered.
0 `ratio
a spectral
power
by Verwichte
al. (2013b) who used a statistical approach to
pz1 q
P Considering
∆z, now
produce distributions for the values of loop parameters. We shall z0 ` Exponential
` 2
˘
“gives
2
2
see in the following sections that it is indeed possible to idenq ` ∆zq exp
P pzP0pz
´zpz
f
expp´2
0α
0 ` ∆zq¯{Ld q
0
´
tify the damping profile from broadband signals when the data
`“ `´ pz ` ∆zq2 α2 f 22 ˘ 2 2 ˘
adband
oscillations
is sufficientlykink
clean and
a single loop can be considered.
0 p´2z {L q
pzP0pzexp
q1 q exp
P“
0
d α f
´ 2zexp
∆z ` ∆z
.
(18)
Δz = 20 Mm
Δz = 40 Mm
0`
˘
exp
´z20 α2 f 2
p´2∆z{Ldq
“ exp
` `
˘ 2 2˘
“ exp ´ ˆ2z0 ∆zspatial
` ∆z2 α˙
f .
(18)
exponential
damping
profile,
P pz0 q
“
ow
integrating
inward
and out3. Analysis
method
for broadband
kink oscillations
Unlike the case of the
the
∆z
lot
of
information
and
made
it
In the previous section we saw how integrating inward and out“exponential
expthe´2
fof z. and
(15)
power
ratio
now
depends
upon
choice
so
an
average
Unlike
the
case
of
the
spatial
damping
profile,
the
0
v
ξ
over the and
loop lost
a lot of information and made it
nward
thepower
Gaussian
exponential
ph E
ratio now depends
upon different
the choice of zvalues
0 and so an
impossible to distinguish between the Gaussian and
exponential
value
cannotpower
be cannot
calculated
using
ofaverage
z0 . ifInstead, if
present
a
new
analysis
method
value
be
calculated
using
different
values
of
z
.
Instead,
0
wave profiles. In this section we present a new analysis For
method
anfrom
exponential
spatial
damping
profile,
the
power
ratio
dez
ranges
0
to
some
maximum
value
z
,
there
will
be two
z
ranges
from
0
to
some
maximum
value
z
,
there
will
be
two
0
m
0
m
that
does
not
integrate
over
the
loop
in
such
a
straightforward
loop in such a straightforward
pends
upon
the
chosen
∆z
but
is
independent
of
z
.
Calculating
0
boundingfor
curves
forratio.
the ratio.The
The upper
bounding
curve will
fashion. The method we present facilitates the identification
of curves
bounding
the
upper
bounding
curve will
the
identification
of
sfacilitates
a
function
of
frequency
for
CoMP
data.
correspond
to
z
“
0
and
hence
give
a
spectral
power
ratio In practhe
ratio
for
all
available
z
should
give
the
same
value.
0
0
the damping profile and is tested against numerical data for difcorrespond
to z0will
“ 0vary
anddue
hence
giveanda spectral
power
presents
the
fit for
andata
exponential
the ratio
to errors
so an˘average
ratioratio
(for
ferent coronal
loop
parameters
andfor
photospheric
drivingtice
condiagainst
numerical
dif-spatial
`
P pz1 “ ∆zq
Gaussian
2
2
2
tions.
“ expimproving
´∆z α f the
,
(19) acrth
et
al. 2010). Thedriving
solid (red)
line repre- a given ∆z) can be calculated,
thus
statistical
nd
photospheric
condiP pz0 “ 0q
First
we
examine
the
method
used
in
the
previous
section
in
˘
pz1 “
∆zq with a`single2measurement.
sian spatial damping
profile.
curacy of thePresult
compared
2
2
more detail. Taking P pzq as the spectral power for, say, a trans- while the lower
bounding
will´∆z
correspond
“curve
exp
α fto zprofile,
0 ,“ zm and
We now
consider
a 0q
Gaussian
spatial
damping
again (19)
verse velocity signal, at some height z, then for an exponential
is given
by
pz
P
“
0
used
in the profile,
previous
section
spatial damping
Verth et
al. (2010)in
obtain
having the form given
by Eq. 7, which is more convenient than
sider
the
limiting
cases
of
entirely
exponenspectral power for,2 say, a trans- whileEq.
1 since thebounding
absence ofcurve
the constant
term A˘0 {2 makes
the
correspond
“ zm and
` will
`
˘ to z0taking
1 “ zm ` ∆zq
p´2z{Lgeneral
P pzq
9A pzq 9the
exp most
(12) lowerP pz
ussian damping,
whereas
2
2 2
dq ,
“
exp
´
2z
∆z
`
∆z
α
f
.
(20)
m
ratios
of
functions
much
simpler.
The
spectral
power
is
given
by
eight
z, then
for regimes
an exponential
pz
q
P
“
z
is
given
by
0
m
lude
both
damping
as
described
by
where Ld “ v ph ξE { f and so the frequency dependence of the
` 2 2˘
al.
(2010)
power
at4.2.
someobtain
height is given by
in
Sect.
These two curves will2 converge in the limits
f Ñ 0 and f Ñ 8
P pzq 9A pzq 9 exp ´z {L ,
(16)
ˆ
˙
but will otherwise be distinct. Increasing the gseparation distance
MP data is too noisy to distinguish
whether
z
∆zzm
brings
the curves closer`together.
`
˘ 2 2 ˘α “
pzq 9 exp ´2
f instruc.
(13)
pz
P
“
`
∆zq
s exponential
it is
1
and
for
convenience
we can introduce
Lgcalculating
“ 1{α
p´2z{LdorqP,Gaussian,
exp
2 fthewith
v phstill
ξ(12)
E
Figure 2 demonstrates
this
method
of
“ exp ´ 2zm ∆z ` ∆z α specf .
ysis tools that will allow us to discriminate πκϵ 1{2 {2C
The
frequency
dependence
of
the
power
at
ratio
for
a
chosen
∆z
and
z
.
For
each
value
of
z0 , some
a
m
q
k .power
“
z
If we consider the ratio of spectral power at some height P
z topz0tral
m
16 signal is selected and the correefiles
frequency
dependence
of
the
in
the
future.
To
test
our
proposed
analsection
of
the
transverse
velocity
the spectral power at the source of the driver we obtain height is then given by
(20)
Δz = 60 Mm
Both damping regimes
Δz = 40 Mm
•
If damping is weak enough then both damping
regimes may be observed
Δz = 120 Mm
•
Exponential damping rate now provides the
lower limit but range of power ratios still exist
Δz = 240 Mm
17
Recover exponential damping
Δz = 40 Mm
•
Only consider oscillations at heights large
enough that the Gaussian damping regime is
excluded
Δz = 80 Mm
Δz = 160 Mm
18
tube in the photosphere. Figure 11 shows the radial dependence
of the radial and azimuthal velocity components as applied at
the loop footpoint z “ 0. The radial velocity is constant within
the loop up to r ď R and then decreases as r´2 . Note that the
full velocity components are defined by multiplying the timedependence f ptq and the azimuthal dependence of the spatial
profile assuming m “ 1 symmetry, i.e.
Fig. 11. Radial dependence of the
Dependence on spatial profile of driver
•
Simple kink driver (displacement of loop footpoint)
vr “ ξr prq sin pωtq cos pθq ,
vθ “ ξθ prq sin pωtq sin pθq .
(bottom) velocity components for
paces the loop footpoint. The ver
edges of the inhomogeneous layer.
(27)
Figure 12 shows the velocity components of a new driver
that is intended to set up a kink normal mode with an exponential
spatial damping profile. In such a mode we would expect there to
be a significant amount of energy in azimuthal fields, and a radial
displacement such that there is no compression of the magnetic
field to leading order (Hood et al. 2013). The profiles in Fig. 12
match these requirements, although we do not suggest that realistic photospheric driving is likely to produce such a highly
structured and tuned driver. Nevertheless, it is instructive to see
how sensitive our simulation results are to the driver. Note that
Fig. 12 is only a snapshot in time. Previously (Eq. 27) the spatial
and temporal components of the driver could be separated. In order to describe the phase-mixed Alfvén wave component of the
eigenmode driver we now consider (Fig. 13) the spatial dependence of the amplitude (solid lines) and the phase (dashed lines,
in radians) as
vr “ ξr prq sin pωt ` φr prqq cos pθq ,
vθ “ ξθ prq sin pωt ` φθ prqq sin pθq .
(28)
The oscillations are in phase near the centre of the loop for the
(collective) kink mode but vary significantly within the inhomogeneous layer which describes phase-mixing. Hence this driver
corresponds to an eigenmode comprised of coupled kink and
Alfvén waves and tuned to the parameters of the coronal loop
19
used in this particular simulation.
Fig. 12. Radial dependence of the
(bottom) velocity components for a
kink and Alfvén eigenmode. The ve
edges of the inhomogeneous layer.
structured and tuned driver. Nevertheless, it is instructive to see
how sensitive our simulation results are to the driver. Note that
Fig. 12 is only a snapshot in time. Previously (Eq. 27) the spatial
and temporal components of the driver could be separated. In order to describe the phase-mixed Alfvén wave component of the
eigenmode driver we now consider (Fig. 13) the spatial dependence of the amplitude
(solid +
lines)
and the phase Alfvén
(dashed lines,
Finely tuned eigenmode
driver (kink
phase-mixed
wave)
in radians) as
Dependence on spatial profile of driver
•
vr “ ξr prq sin pωt ` φr prqq cos pθq ,
vθ “ ξθ prq sin pωt ` φθ prqq sin pθq .
(28)
The oscillations are in phase near the centre of the loop for the
(collective) kink mode but vary significantly within the inhomogeneous layer which describes phase-mixing. Hence this driver
corresponds to an eigenmode comprised of coupled kink and
Alfvén waves and tuned to the parameters of the coronal loop
used in this particular simulation.
Figure 14 shows the spatial damping profiles produced by
the dipolar (top) and eigenmode (bottom) drivers. The top
•
Exponential damping behaviour recovered
•
BUT unlikely to represent driver found in nature
20
Fig. 12. Radial dependenc
(bottom) velocity compone
kink and Alfvén eigenmode
edges of the inhomogeneou
Summary
• Damping behaviour described by a general spatial damping profile that is
composed of
• Gaussian damping profile at low heights
• Exponential damping profile at large heights
• Height of switch between profiles only depends upon the density contrast
• Fitting general spatial damping profile to 3 parameters (Lg, h, Ld) allows unique
solution of seismological inversion
• When only a few periods are observed the Gaussian profile is the most appropriate
for seismological estimates (unless density contrast is large)
• Same process applies to standing modes (consistent with observation of low
density contrast loop)
• CoMP data suggests (frequency-dependent) mode coupling but data is too noisy
to distinguish between Gaussian or exponential profile
• Gaussian damping profile allows seismological interpretation of power ratio
21