Measurement of the meridional flow from eigenfunc5on perturba5ons

Transcription

1 Measurement of the meridional flow from eigenfunc5on perturba5ons Ariane Schad, Markus Roth Kiepenheuer- Ins5tut für Sonnenphysik Solar Subsurface Flows from Helioseismology: Problems and Prospects Helioseismology Workshop July 21-23, 2014, Stanford, CA

2 I. Theory: Perturba5on & coupling of p- modes > coupling of modes in a neighborhood K k of a reference mode k=(n,l,m): (Lavely & Ritzwoller 1992) perturbed eigenfunc5on unperturbed eigenfunc5on > c kk coupling coefficient between mode k,k ( coupling strength): (1. order approxima5on) advec5on of acous5c wave (Schad et al., ApJ, 2011) 2

3 I. Theory: Perturba5on & coupling of p- modes Spherical harmonic representa5on of u: radial component horizontal component s=1 s=2 s=3 s=4 conserva5on of mass: Polynomial expansion of coupling coefficients: ( Figure source: D. Hathaway, NASA) b- coefficients: (Schad et al., ApJ, 2011) > knowing b s k k one can infer the radial flow coefficient u s (r)! 3

4 II. Effect of mode coupling on global oscilla5on data SHT of full- disk Dopplergrams:, L k k - leakage matrix elements (imperfect orthogonality: line of sight projec5on, solar disk, etc.) (Fourier) amplitude ra5o: between reference mode k=(n,l,m) and coupling modes k Es5mator: complex gain > measure for cross- talk in SHT data (Schad et al., ApJ, 2011) (Schad et al., ApJL, 2013) 4

5 III. Opera5on scheme > given SH 5me series and leakage matrix es5ma5on of amplitude ra5os y kk es5ma5on of b- coefficients b s kk inversion for radial flow coeff. u s (r) reconstruc5on of horizontal flow coefficient v s (r) from u s (r) > cross- spectral analysis > LS fit > SOLA inversion method (Pijpers & Thompson, 1994) > polynomial fit 5

6 IV. Applica5on to MDI data - superimposed flow components (s even) southward outward > MDI data > s=1,...,8 > complex flow paiern in la5tude & depth How reliable are these results? northward inward > Mm depth (Schad et al., ApJL 2013) 6

7 V. Evalua5on of the method - sources of systema5c errors How can we evaluate the method and the reliability of the results? > comparison of flow measurements from different methods > forward simula5ons: e.g. computa5on of a- ra5os for a meridional flow model > sophis5cated simula5ons of the acous5c wave field (T. Hartlep et al. 2013) Sources of systema5c errors? > effects of higher order? (approxima5ons in 1. order) > other large- scale flows, e.g. rota5on? > leakage matrix? >... 7

8 V. Evalua5on: Comparison with subsurface measurements large- scale flow component (s=2) small- scale flow component (s=8) (Komm et al., 2005, ApJ) global inversion method ring- diagram analysis depth [Mm] depth [Mm] > good agreement at subsurface la5tude [deg] (Schad et al. ApJL, 2013.) 8

9 V. Evalua5on of the method: Simula5on study proof in principle flow model: s=2, horizontal flow (surface) 25 m/s (mid- la5tude) leakage matrix of MDI solar model S amplitude - ra5os + 1% Gaussian noise modes with 1 l 200 radial flow coefficient u 2 horizontal flow coefficient v 2 > inversion results in good agreement with model 9

10 V. Evalua5on of the method: Contribu5ons of higher order (with H ik ~ c ki in first order) 1. order, imaginary 2. order, real (Schad Disserta5on, 2013) for coupling modes (n=2, l=120) & (n =2, l =118) > higher order terms are negligible for the a- ra5o > real part of a- ra5o essen5ally determined by leakage 10

11 V. Evalua5on of the method: Influence of leakage Ignore influence of leakage at inversion for the flow: flow model L kk =L kk δ kk (Schad Disserta5on, 2013) 11

12 V. Evalua5on of the method: Influence of leakage Ignore influence of leakage at inversion for the flow: flow model L kk =L kk δ kk Result: > s=2 component is redistributed to other harmonic degrees (Schad Disserta5on, 2013) 12

13 V. Evalua5on of the method: Sources of systema5c errors leakage matrix Comparison between MDI leakage matrix and leakage es5mated from MDI data Assume no mode coupling due to perturba5ons (e.g. flows): > annual varia5on of B angle? > devia5on vanishes on average over large 5mes (Schad Disserta5on, 2013) > for (l =90, m =0) and (l=l +dl, m=m +dm) 13

14 V. Evalua5on of the method: Sources of systema5c errors leakage matrix nl frequency [mhz] 14

15 V. Coupling due to rota5on Toroidal velocity field of solar rota5on: (Ritzwoller & Lavely 1991) Perturba5on theory for mode eigenfunc5ons: Expansion coefficients: (Howe et al., Sci. 2000) (Schad Disserta5on, 2013) 15

16 V. Influence of rota5on and meridional flow on a- ra5os Amplitude ra5os from MDI data vs. flow model: real part imaginary part (n=1, l=180) and (n =1, l =182) > rota5on lixs azimuthal symmetry of amplitude ra5os > symmetriza5on in m can compensate this effect 16

17 VI. Restric5ons: Frequency resolu5on and mode frequency separa5on Modes should separate in frequency domain (blend into ridges): > analysis of low and medium degree (l 200) > analysis of higher frequency modes may be affected by bias > expect bias of about < 1% Accuracy: > need of long 5me series/ high frequency resolu5on dof degrees of freedom of spectral es5mator smoothing kernel width 17

18 Summary - uses full- disk data and exploit spherical geometry - allows analysis of modes with low and medium degree (deep inversion kernels) - take into account horizontal and radial component of the flow - higher order contribu5ons are negligible - the instruments leakage matrix is important - what is with the horizontal- to- ver5cal ra5o? - rota5on affects a- ra5os - can be compensated - long 5me series should be used (frequency resolu5on, accuracy) 18