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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