1 Part 1: Inferring flare loop parameters with measurements of standing sausage modes Bo Li Institute of Space Sciences Shandong University, Weihai

2 Inferring flare loop parameters with measurements of standing sausage modes Contents Observational motivations Standing modes in magnetized loops Inferring active region loop parameters with kink oscillations Inferring flare loop parameters with sausage oscillations

3 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

4 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)

5 Standing modes in magnetized loops Kink mode Loop axis displaced Sausage mode Axisymmetric, Loop axis not displaced animation from Nakariakov & Verwichte 05 (LRSP)

6 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

7 Transitionlayer Uniform cord Uniform external medium A couple of definitions Geometrical – L: Loop length – R: mean radius – l: transition layer width Physical – v Ai : internal Alfven speed – :density contrast

8 Transitionlayer Uniform cord Uniform external medium Radial (transverse) density structuring

9 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) Transitionlayer Uniform cord Uniform external medium

10 Inferring active region loop parameters with kink oscillations Alfven speed can be inferred to some extent density contrast + lengthscale range rather broad Soler+14

11 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

12 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

13 (lateral) leakage of sausage modes thick loops thin loops

14 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 v Ai ， l / R ， ρ i / ρ e ? Key is to establish F and G

15 F & G from an eigenmode analysis Fig. here CORDCORD external Trans ition Layer recurrent relation

16 recurrence relation for the coefficients in the series expansion

17 Dispersion relation perturbation eq. solved in cord + Tran. Layer + external medium continuity of Lagrangian displacement + total pressure F and G

18 Numerical Results tau more sensitive to density profiles for sufficiently thin loops (also Nakariakov+12, Chen, Li,+15)

19 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

20 Application 1: spatially unresolved QPPs R / v Ai : max/min = 1.8 den. contrast: max/min = 2.9 l / R : [0, 2] possible

21 Application 2: spatially resolved QPPs Counting knowns and unknowns –knowns ： L, R ; P, τ –unknowns: v Ai ， l / R ， ρ i / ρ e Problem remains under-determined

22 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

23 spatially resolved multi-mode QPPs ( F, G, H ) –saus: analytical DR –Kink: linear resistive computation Knowns – L, R ; – P kink, τ kink ; P saus, τ saus unknowns – v Ai ， l / R ， ρ i / ρ e problem over- determined

24 spatially resolved multi-mode QPPs Inversion Procedure 1.choose a den. profile 2.τ saus / P saus constrains [ l / R, ρ i / ρ e ] 3.P saus constrains v Ai 4.τ kink yields a unique [ v Ai, l / R, ρ i / ρ e ]

25 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

26 How robust is this inversion? Results similar for density profiles not involving a uniform cord (Guo+15, SoPh, in press)

27 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

28 Part 2: Kink and sausage modes in coronal slabs Bo Li Institute of Space Sciences Shandong University, Weihai

29 Kink and sausage modes in coronal slabs with continuous transverse density distributions Contents Observational motivations Collective modes in a magnetized slab Observational implications

30 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 ApJ Chen, Song, Li et al ApJ Feng, Chen, Li et al SoPh

31 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 – v Ai : internal Alfven speed – :density contrast Uniform core Transition Layer Uniform external medium

32 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

33 DRs for waves in slabs key: after a profile is chosen,

34 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

35 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

36 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%

37 effects on branch II kink modes? they are sensitive to profile choice as well for “parabolic” –l/R ↗, period ↗ by up to 60%

38 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

39 Observability of branch II kink modes branch II kink modes observable only for flare loops R/L required for tau/P=3

40 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

41 Observability of branch I sausage modes R/L required for tau/P=3 observable only for flare loops

42 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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44 BKUP SLIDES

45 Comparison with Soler+13

46 Comparison with Soler+13