1 part 1: inferring flare loop parameters with measurements of standing sausage modes bo li...
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Part 1: Inferring flare loop parameters with measurements
of standing sausage modes
Bo LiInstitute of Space Sciences
Shandong University, Weihai
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Inferring flare loop parameters with measurements of standing
sausage modes
ContentsObservational motivationsStanding modes in magnetized loopsInferring active region loop parameters with kink oscillations Inferring flare loop parameters with sausage oscillations
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Motivation: What MHD seismology is interested in?
Magnetic field strength via Alfven speed– Kink (Nakariakov & Ofman 01, ….)– Sausage
Density structuring– Longitudinal (e.g., period ratios, Andries+05 …)– Transverse
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Motivation: density structuring across loops
Remains largely unknown, even with spectroscopic measurements– insufficient spatial resolution– LOS effect (optically thin radiation)
Important in that it is key to – understanding loops – building blocks of corona– Profile steepness important in determining efficiency of heating mechanisms
like • phase mixing (Heyvaerts and Priest, 83)• resonant absorption (Hollweg & Yang 88; Goossens+02; Ruderman & Roberts 02)
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Standing modes in magnetized loops
Kink mode
Loop axis displaced
Sausage mode
Axisymmetric, Loop axis not displaced
animation from Nakariakov & Verwichte 05 (LRSP)
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Kink oscillations in active region loops
kink mode measurements– TRACE(Aschwanden+99, Nakariakov+99, ……) – Hinode/EIS(van Doorsselaere+08, Erdelyi & Taroyan
08…)– STEREO/EUVI(Verwichte+09, …)– SDO/AIA(Aschwanden & Schrijver 11, ...)
tend to be strongly damped
1998 Jul 14, TRACE, Nakariakov+99
7Tra
nsitio
n
layer
Un
iform
co
rd
Uniform external medium
A couple of definitions Geometrical
– L: Loop length– R: mean radius– l: transition layer width
Physical– vAi: internal Alfven speed
– :density contrast
8Tra
nsitio
n
layer
Un
iform
co
rd
Uniform external medium
Radial (transverse) density structuring
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kink mode damping due to resonant absorption
other damping mechanisms also available (Nakariakov+99 Sci)
Resonant Absorption: (collective) kink mode energy resonantly converted to localized azimuthal motions in transition layer (Goossens+09 SSRv)
Tra
nsitio
n
layer
Un
iform
co
rd
Uniform external medium
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Inferring active region loop parameters with kink oscillations
Alfven speed can be inferred to some extent
density contrast + lengthscale range rather broadSoler+14
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Quasi-periodic-pulsations(QPPs) in solar flare lightcurves
QPPs seen in flare lightcurves– in all passbands– all phases– compact and 2-ribbon flares
(Nakariakov & Melnikov 09)
Second-scale QPPs– often attributed to sausage
modes in flare loops– a multitude of events
(Aschwanden et al.04)– sometimes temporal damping is
seen
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apparent damping of sausage modes
non-ideal mechanisms (ion viscosity, resistivity, electron conduction) may be inefficient (Kopylova+07)
Two regimes of sausage modes– trapped (for thick loops) – leaky (thin, Cally 86, Kopylova+07 …)
• an ideal mechanism• wave energy emitted into surrounding fluids
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(lateral) leakage of sausage modes
thick loops
thin loops
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Inferring flare loop parameters with sausage oscillations
Assumptions: – beta=0 (cold plasma)– density structuring across flare loops
Q: with period + damping rate known– possible to infer vAi, l/R, ρi/ρe?
Key is to establish F and G
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F & G from an eigenmode analysis
CORD external
Transition Layer
recurrent relation
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recurrence relation for the coefficients in the series expansion
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Dispersion relation
perturbation eq. solved in cord + Tran. Layer + external medium
continuity of Lagrangian displacement + total pressure
F and G
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Numerical Results tau more sensitive to
density profiles for sufficiently thin
loops
(also Nakariakov+12, Chen, Li,+15)
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Application 1: spatially unresolved QPPs
P = 4.3 s, tau/P=10 Attributing
attenuation to leakage, the best one can do is
Culgoora 230MHz, 1973 May 16, McLean & Sheridan 73
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Application 1: spatially unresolved QPPs
R/vAi: max/min = 1.8 den. contrast:
max/min = 2.9 l/R: [0, 2] possible
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Application 2: spatially resolved QPPs
Counting knowns and unknowns– knowns : L, R; P, τ
– unknowns: vAi , l/R, ρi/ρe
Problem remains under-determined
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spatially resolved multi-mode QPPs
Geometrical parameters known– L=4.e4 km & r_e = 4.e3 km
Two modes identified– saus: P = 15 s & tau = 90 s– kink: P = 100 s & tau = 250 s
Klotkov+15 AA
NoRH, 14 May 2013
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spatially resolved multi-mode QPPs
(F, G, H)– saus: analytical DR– Kink: linear resistive
computation Knowns
– L, R;
– Pkink, τkink; Psaus, τsaus
unknowns– vAi , l/R, ρi/ρe
problem over-determined
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spatially resolved multi-mode QPPs
Inversion Procedure
1. choose a den. profile
2. τsaus /Psaus constrains [l/R, ρi/ρe]
3. Psaus constrains vAi
4. τkink yields a unique [vAi, l/R, ρi/ρe]
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spatially resolved multi-mode QPPs
Possible to tell which profile best describes the flare loop?– Not for this one
Nonetheless, flare loop parameters constrained to rather narrow ranges
measured kink period = 100 sec
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How robust is this inversion?
Results similar for density profiles not involving a uniform cord (Guo+15, SoPh, in press)
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Summary A substantial fraction of QPPs attributed to standing
sausage modes in flare loops We derived a DR for general transverse density
distributions If only one sausage mode is involved, inversion
problem under-determined If more than one mode involved
– flare loop parameters constrained to narrow ranges, even if specific density profile remains unknown
Multi-mode measurements worth pursuing
Chen, Li*, Xiong, Yu, Guo 2015, ApJ, 812, 22
Guo, Chen, Li*, Xia, Yu 2015, SoPh, in press
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Part 2: Kink and sausage modes in coronal slabs
Bo LiInstitute of Space Sciences
Shandong University, Weihai
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Kink and sausage modes in coronal slabs with continuous
transverse density distributions
ContentsObservational motivationsCollective modes in a magnetized slabObservational implications
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A slab geometry sometimes more appropriate
Fast sausage waves in slabs employed to account for– Sunward moving tadpoles in post-flare TRACE supra-
arcades (Verwichte+05) – Fine structures in type IV radio bursts (Karlicky+13, …)
Fast kink waves in slabs employed to account for– Waves in streamer stalks
Chen, Kong, Li et al. 2010 ApJ
Chen, Song, Li et al. 2011 ApJ
Feng, Chen, Li et al. 2011 SoPh
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Questions to address Many studies on slab waves consider a
piece-wise constant transverse density profile (Edwin & Roberts 82, …)
Consequences of a more “realistic” continuous one? (Yu et al. 15)
Geometrical– R: mean half-width– l: transition layer width
Physical– vAi: internal Alfven speed
– :density contrast
Un
iform
core
Tra
nsitio
n
Laye
r
Uniform external medium
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DR for waves in slabs Assumptions
– beta = 0 (cold plasma)– out-of-plane propagation neglected
method– perturbation eq. in nonuniform portion solved as a regular series
– continuity of Lagrangian displacement + total pressure
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DRs for waves in slabs
key: after a profile is chosen,
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Dispersion diagrams
Edwin & Roberts (82, step-function profile) remain valid– Lowest-branch kink mode: always trapped– Sausage modes/other kink branches: leaky at small k
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Continuous structuring: effects on branch I kink modes?
Not too much; good news for magnetic field inference with streamer waves (Chen+10, 11)
maximal fractional deviation of period when kR < 0.2 pi
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effects on branch II kink modes?
Period & damping sensitive to density lengthscale– l/R ↗, period ↗by up to 40%, tau/P ↘by up to 50%
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effects on branch II kink
modes?
they are sensitive to profile choice as well
for “parabolic”– l/R ↗, period ↗by up to
60%
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Observability of branch II kink modes
branch II kink modes observable only for flare loops
R/L required for tau/P=3
0
always observable
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Observability of branch II kink modes
branch II kink modes observable only for flare loops
R/L required for tau/P=3
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effects on branch I sausage modes?
Period & damping rate sensitive to profile
period– up by 60%, or down by
12 % tau/P
– somehow less sensitive to l/R than kink modes
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Observability of branch I sausage modes
R/L required for tau/P=3
observable only for flare loops
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Summary Slab geometry more suitable for describing
collective waves in some situation We derived a DR for general transverse density
distributions lowest-order kink modes always trapped, their
periods not substantially influenced by a continuous density profile
Effects of continuous density structuring need to be incorporated in studies of sausage modes and higher-order kink modes
Yu, Li*, Chen, Guo, 2015 ApJ, 814, 60
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BKUP SLIDES
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Comparison with Soler+13
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Comparison with Soler+13