Master Thesis

Properties of Magnetohydrodynamic Waves in the SolarPhotosphere Obtained with Hinode

「ひので」によって発見された、太陽光球から発生するMHD 波動の性質

Daisuke Fujimura

Space and Planetary Science GroupDepartment of Earth and Planetary Science

Graduate School of Science, The University of Tokyo

January, 25, 2010

Abstract

There are two major problems in solar physics: solar wind acceleration and coronal heat-ing. The source which supplies sufficient energy to the corona is still unknown. Weconcentrate on MHD (Magneto-Hydro Dynamics) waves, which would play a key role insolar wind acceleration and coronal heating, and perform observational studies for detect-ing the MHD waves using the data obtained with theHinodeSatellite. section 1 is thegeneral introduction for important roles of MHD waves, previous studies in MHD waves,theHinodeSatellite.The observation is performed with the spectro-polarimeter (SP) of the Solar Optical Tele-scope (SOT) aboard theHinode. We observe pores and plages, and make the time profilesof line-of-sight (l.o.s.) magnetic flux, l.o.s. velocity, and photometric intensity. In section2, we mention about the calculation of the magnetic flux, the velocity, and the intensity,estimate the detection limit of the magnetic flux and the velocity by SP, and show that theprecision of measurement by SP is good enough to detect MHD waves. In section 3, wemention about the data analyzed in this study, calculation of the magnetic flux, the veloc-ity, and the intensity, and the method to select the region-of-interest. We performed FastFourier Transform, and identified isolated strong peaks in pores and plages of positivepolarity in the power spectra of the l.o.s. (line-of-sight) magnetic flux, the l.o.s. velocity,and the intensity. The oscillation periods are located in 3−6 min for the pores and in 4−9min for the plages. These peaks correspond to the magnetic, the velocity, and the intensityfluctuation in time domain with r.m.s. (root mean square) amplitudes of4−21 G (0.3−1.5%), 0.03 − 0.12 km s−1, and0.1 − 1%, respectively. Phase differences between the l.o.s.magnetic flux(ϕB), the l.o.s. velocity(ϕv), the intensities of the line core(ϕI,core), andthe continuum intensity(ϕI,cont) have striking concentrations at around−90◦ for ϕB −ϕv

andϕv−ϕI,core, around180◦ for ϕI,core−ϕB, and around10◦ for ϕI,core−ϕI,cont. Here, forexample,ϕB − ϕv ∼ −90◦ means that the velocity leads the magnetic field by a quarterof cycle. The properties of the common peaks are summarized in section 4.1. In section4.2, we analyze the dependence of such strong fluctuations on the position of a pore.In section 5, we interpret that the fluctuations with low intensity can be considered to bethe kink mode (transverse) MHD waves, while those with high intensity can be consid-ered to be the sausage mode (longitudinal) MHD waves. We also show that the observedphase relation between the magnetic and the photometric intensity fluctuation would notbe consistent with that caused by the opacity effect, if the magnetic field strength de-creases with height along the oblique line of sight. In sections 6 and 7, we show that theobserved phase relation between the fluctuations in the magnetic flux and the velocity isconsistent with the superposition of the ascending wave and the descending wave reflectedat chromosphere/corona boundary, which has a property of a standing wave. Even withsuch reflected waves, the residual upward Poynting flux is estimated to be2.7 × 106 ergcm−2 s−1 for a case of the kink wave. Seismology of the magnetic flux tubes is possible toobtain various physical parameters (mass density inside and outside the flux tube, plasmaβ, and Alfven velocity) from the observed period and amplitude of the oscillations.Sausage (longitudinal) wave may be more easily observed at the disk center, while kink(transverse) wave may be more easily observed at the limb. In section 8, we observemagnetic concentrations from the disk center to the limb in order to distinguish the twomodes.

In section 9, we perform statistical analysis for magnetic structures of both positive andnegative polarity and compare them. The result shows that the phase relations associatedwith the magnetic field (i.e.ϕB −ϕv andϕI,core−ϕB) differs by 180◦, while others do notshow any significant changes. The result support that the observed fluctuations are due tothe standing kink and/or slow sausage waves.Note that this thesis is the extensive work of Fujimura and Tsuneta (2009, ApJ, 702,1443).

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要旨

太陽物理学における大きな謎として「太陽風加速問題」と「コロナ加熱問題」が挙げられる。「太陽風加速問題」とは太陽極域で観測されるプラズマの流出速度がパーカーの古典的理論では説明できないほど高速である問題で、「コロナ加熱問題」とは太陽光球における温度が約 6000度であるにも関わらず、上空のコロナでは 100万度～1000万度に達するという問題である。我々はコロナにエネルギーを供給する候補としてMHD（電磁流体力学的）波動に着目し、太陽観測衛星「ひので」を用いて観測的研究を行った。1章ではMHD波動のコロナへのエネルギー供給に対する役割や先行研究について説明した後、「ひので」の概要について説明する。我々は「ひので」の SOT(可視光望遠鏡)に搭載された SP(偏光分光計)を使用して、活動領域のポア・プラージュの磁束管を観測し、視線方向の磁気フラックス・視線方向の速度場・輻射強度の時系列データを作成した。まず 2章では磁場・速度場・輻射強度の計算方法について説明したあと、磁場・速度場のそれぞれについて測定限界を見積もり、SPがMHD波動を同定するのに十分な観測精度を持っていることを示す。次に 3章では解析に使用したデータや解析したターゲット領域の取り方について説明する。そして得られた時系列データにフーリエ変換を適用した結果、これらが共通して強いピークを持っていることが分かった。4章ではこのような共通ピークの振幅・位相関係・振動周期などの性質をまとめて示す。共通ピークについて統計解析を行った結果、磁気フラックス・速度場・輻射強度の位相差、共通周期が一定値に集中していることが分かった。磁束管を伝播するMHD波動には kink mode（非圧縮波）と sausage mode（圧縮波）の 2種類があるが、とくに位相関係はMHD波動のモードを同定するのに重要な役割を果たす。またこの章では、ポアにおける振動の場所依存性についても触れる。5章で輻射強度の振幅比率を求めた結果、これが幅広く分布していることが分かった。この振幅が大きいものは sausage mode、小さいものは kink modeと見なせる。また観測された磁場と輻射強度の位相関係が opacity effectと呼ばれる見かけ上の振動とは整合しないことも示す。6・7章では観測された位相関係は進行波とは整合しないが、定常 kink waveと定常 slow sausage waveのいずれとも整合することを示す。これは光球で発生した波が遷移層で反射し、上昇波と下降波が共存するためと解釈できる。また kink waveの一例について（上昇－下降）の差分ポインティングフラックスを見積もることができた。さらに観測された磁場・速度場の振幅や磁場強度から、光球における磁束管の物理量（プラズマ密度・プラズマベータ・アルベン速度）を見積もった。視線方向磁場を見ているので、太陽中心では縦波の sausage modeが、リムに近づくほど横波の kink modeがより観測されると考えられる。8章では太陽中心からの角度に対してMHD波動が観測される磁束管の確率がどのように変化するかを調べた。今までの解析は正極磁場についてのみ行ってきたが、9章では負極磁場についても統計解析を行い、両者で比較した。その結果、磁場に関する位相関係が反転していることが分かった。これは、定常 kink wave・定常 sausage waveのいずれとも整合する結果である。なおこの修士論文は Fujimura and Tsuneta (2009, ApJ, 702, 1443)の内容を拡張したものである。

Contents

1 Introduction 11.1 Important Role of MHD Waves . . . . . . . . . . . . . . . . . . . . . . . 11.2 Previous Studies about MHD Waves . . . . . . . . . . . . . . . . . . . . 21.3 The Hinode Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Measurement by Spectro-Polarimeter 62.1 Method to Calculate Parameters . . . . . . . . . . . . . . . . . . . . . . 62.2 Detection Limit ofΦlos . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Detection Limit ofΦpar . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Detection Limit ofδvlos . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Summery and Choice of Stokes V . . . . . . . . . . . . . . . . . . . . . 11

3 Observations and Data Analysis 173.1 Hinode Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Time-Profile Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Power Spectrum and Phase Relation 214.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Dependence of Fluctuations on the Position of a Pore . . . . . . . . . . . 22

5 Intensity Fluctuation 28

6 Kink Mode MHD Waves 326.1 Phase Relation of Propagating Kink Waves . . . . . . . . . . . . . . . . 326.2 Phase Relation of Standing Kink Waves . . . . . . . . . . . . . . . . . . 336.3 Interpretation for Standing Waves . . . . . . . . . . . . . . . . . . . . . 346.4 Leakage of Poynting Flux to the Corona . . . . . . . . . . . . . . . . . . 356.5 Seismology of Photospheric Flux Tubes . . . . . . . . . . . . . . . . . . 35

7 Sausage Mode MHD Waves 427.1 Phase Relation of Propagating Slow Sausage Wave . . . . . . . . . . . . 427.2 Phase Relation of Standing Slow Sausage Wave . . . . . . . . . . . . . . 447.3 Seismology of Photospheric Flux Tubes . . . . . . . . . . . . . . . . . . 44

8 Center-to-Limb Variation 50

9 Comparison between Positive and Negative Polarity 53

10 Conclusion and Future Work 56

Acknowledgements 58

References 59

1 Introduction

1.1 Important Role of MHD Waves

Solar wind is outward motion of plasma from the corona to interplanetary space. Perker(1958) theoretically predicted the existence of the solar wind considering pressure gra-dient between the corona and the interplanetary space. But observation with ULYSEESreveals that the solar wind generated from the coronal hole in polar region is acceleratedto ∼ 800 km/s (Fig. 1.1), which can not be explained by the Perker’s model. The sourcesupplying the sufficient energy to the corona is required for such acceleration, but no ob-servational evidence of such source has been detected yet.The temperature in the corona is106 − 107 K, while that in the photosphere is only 6000K. Higher temperature in higher atmosphere can not be explained by the radiative energytransport. There are two candidates that would transport the energy from the photosphereto the corona (Fig. 1.2) (1) convective motion at the footpoint of flux tubes change thetopology of the magnetic field lines, causing many small-scale magnetic reconnectionsand releasing free magnetic energy in the corona and (2) waves generated by the convec-tive motion at the photosphere propagate upward along the flux tubes, and dissipate at thecorona. But no observational evidence has been detected for either candidate.In any case, the source of the energy for coronal heating and accelerating the plasma is be-lieved to be in the surface convection. The energy at the photosphere is lifted up to alongthe magnetic fields, and its dissipation results in coronal heating and solar wind accel-eration. The Alfven waves or more generally transverse magnetohydrodynamic (MHD)wave can play an important role because they can travel a long distance to both heat thecorona and accelerate the solar wind. Low-frequency (≤0.1 Hz) Alfven waves excited bysteady transverse motions of the magnetic field at the photosphere is responsible for theplasma heating (Suzuki& Inutsuka, 2005). Suzuki& Inutsuka (2005) performed a one-dimensional MHD simulation with radiative cooling and thermal conduction, and showedthat the plasma in the corona is heated to 106 K and accelerated to 800 km/s by the foot-point fluctuations of the magnetic fields at the photosphere. They interpreted the outgoingAlfv en waves attenuated in the chromosphere by∼ 85% dissipate in the corona due tothe nonlinear generation of the compressive waves and shocks.Numerous studies about generation, propagation and dissipation of the Alfven waves havebeen carried out observationally and theoretically (e.g. Ryutova & Priest, 1993). Alfvenwaves would be generated in the high-β region of the solar atmosphere. Its precise powerspectra is, however, not observationally known. Ascending Alfven waves with wave-length longer than the Alfvenic scale height may be reflected back at the chromospheric-coronal boundary (Moore et al.1991; An et al. 1989; Hollweg 1978; Suzuki & Inutsuka,2005). It is poorly known how much Alfven-wave flux generated in the photosphere ispropagated all the way to the corona through the fanning-out flux tubes. High-quality ob-servations to obtain spectra of magnetic fluctuation is of crucial importance to understandcoronal heating and acceleration of fast solar wind.

The literatures so far introduced are mainly concerned with the pure Alfven waves.The magnetic fields in the solar atmosphere have a form of isolated magnetic flux tubesembedded in a nearly field-free fluid. Such flux tubes carry the incompressible torsionalAlfv en waves, the kink waves (transverse waves) and the sausage waves (longitudinal

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waves) (e.g. Stix, 2002), instead of the linearly-polarized Alfven waves which can existonly in the uniform media. Magnetic tension force of the flux tube is the restoring force inthe kink mode (e.g. Spruit, 1981), and is essentially incompressible. The sausage modewith the azimuthal wave number m = 0, as first defined by Defouw (1976) and discussede.g. in Roberts and Webb (1978) and Ryutova (1981), is related to aslowmagnetoacousticmode. In the sausage-shaped perturbed boundary of the flux tube, where the flux-tube areaincreases, the magnetic field decreases, whereas the plasma pressure increases; vice versa.A fast magneto-acoustic mode can also propagates along the flux tube, but we can ruleout the mode from the observed phase relation between the fluctuations in the magneticfield and the intensity as we will explain later. In this paper, we report a clear detectionof magnetic, velocity and photometric oscillations of the magnetic flux tubes with theSpectro-Polarimeter (SP) of SOT. The data is extensively analyzed in terms of both thelinearly-polarized kink waves and the slow sausage waves, while we will not discuss thetorsional Alfven waves due to our constraint in the analysis as we explain later.

1.2 Previous Studies about MHD Waves

Ulrich (1996) made the first critical observations, and reported the detection of the magneto-hydrodynamic oscillations with properties of the Alfven waves. He suggested that theobserved phase relation between the magnetic field and the velocity perturbation is con-sistent with the outgoing Alfven waves. The observing aperture of20”× 20” is, however,very large compared with the spatial scale of the flux tubes along which the Alfven wavespropagate. Such a large aperture may make it difficult to identify the weak transversewaves with different frequency and phase, which might become evident in higher reso-lution observations. Velocity and magnetic field oscillations in the sunspot umbra weredetected by Bellot Rubio et al. (2000), Lites et al. (1998), Norton et al. (1999), Ruedi etal. (1998), Ruedi & Solanki (1999), Balthasar (1999), and Settele et al. (2002). Ruediet al. (1998) and Bellot Rubio et al. (2000) obtained the phase difference of -90◦ and90◦ between the fluctuations of the line-of-sight velocity and the magnetic field strengthϕv − ϕB, respectively. They suggested that the magnetic field fluctuation is caused by theopacity fluctuations that move upward and downward the region where the spectral lineprofiles are sensitive to magnetic fields. Norton et al. (2001) obtained the center-to-limbdependence of the phase angle between the magnetic and the velocity fluctuations withtheMichelson Doppler Imageraboard theSOHOsatellite. They reported that the phaseangle is near−90◦ at the disk center and near 0◦ at the limb, and made an importantcomment that the Alfven waves be more easily observed at the limb. They suggested thatthe phase relation reported in the paper is not due to the opacity effect. Khomenko et al.(2003) compared the analytical solution of the MHD equations including gravity, incli-nation of magnetic field, and effects of nonadiabaticity with the observations reported byBellot Rubio (2000), and concluded that the detected time variation in field strength couldbe partly due to magnetoacoustic waves. Ruedi & Cally (2003) suggested that most ofthe observed magnetic field oscillations is due to the opacity effect caused by temperatureand density fluctuations associated with magnetoacoustic waves.Recently apparent transverse oscillations, which are clear evidence of the Alfven waves,are detected in prominence (Okamoto et al. 2007), in spicules (de Pontieu et al. 2007,He et al. 2009), and in Ca jet (Nishizuka et al., 2008) with the Solar Optical Telescope

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(SOT; Tsuneta et al. 2008a; Suematsu et al. 2008; Ichimoto et al. 2008; Shimizu et al.2008) aboard theHinodesatellite (Kosugi et al., 2007). These Alfven waves have enoughPoynting flux to potentially heat the corona. We, however, cannot rule out the possibilitythat these waves are the standing Alfven waves. Transverse oscillations of coronal loopsare detected by Taroyan et al. (2008), Mariska et al. (2008), and Van Doorsselaere etal. (2008) using theEUV Imaging Spectrometer(Calhane et al., 2007) aboard theHinodesatellite as well. Ubiquitous upward Alfven waves in the corona are detected by Tom-czyk et al. (2007) using theCoronal Multi-Channel Polarimeterwithout magnetic fieldinformation. We stress that the observations of the magnetic field fluctuation with thesimultaneous velocity and photometric measurement allows us to identify propagatinghydromagnetic waves.

1.3 The Hinode Satellite

In this thesis, we analyze the data obtained with theHinodesatellite.Hinodeis the newJapanese satellite and was launched on 2006 Sep 23.Hinode has three observationalinstruments to investigate dynamics of the plasma related with the magnetic field, andits effect on the photosphere, the chromosphere, and the corona. The three instrumentsare the Solar Optical Telescope (SOT), the X-Ray Telescope (XRT), and the EUV Imag-ing Spectrometer (EIS). The SOT perform spectro-polarimetric observation and measurethree-dimensional vector magnetic components in the photosphere as well as take pho-tospheric and choromospheric images. The observation from space achieve very highspatial resolution (∼ 0.”32).SOT/SP is ideally suited to detect the magneto-hydrodynamic waves propagated along theflux tubes due to its high spatial and time resolution and its high polarimetric and photo-metric precision (e.g. Ploner& Solanki, 1997). SOT/SP obtains two spectra of iron lines(Fe I) with wavelengths of 630.15 nm and 630.25 nm, which are suitable for observinglower photosphere (del Toro Iniesta, 2003). Earlier studies about magnetic fluctuationswere done in sunspot umbra, since small-scale flux tube (∼ ”1) fluctuations might bedifficult to detect. The high spatial resolution ofHinode(∼ 0”.32) allows us to detect thefluctuations in such small-scale flux tubes. Furthermore, stable observations from spaceallow us to detect clear intensity fluctuations for the first time, and to obtain the phaserelations among the fluctuations in the magnetic flux, the velocity, and the intensity. Thisallows us to examine the opacity effect more in detail.

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Figure 1.1: Outward velocity of the plasma from the corona observed by ULYSEES. Thesolid line indicates that the outward velocity is∼ 400 km/s at lower latitude and∼ 800km/s at higher latitude.

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Figure 1.2: Schematic images of coronal heating.Top: Heating by nanoflares. Convectivemotion at the footpoint of flux tubes changes the topology of the magnetic field lines,causing many small-scale magnetic reconnections and releasing free magnetic energy inthe corona.Bottom: Heating by propagation of waves. Waves generated by the convectivemotion at the photosphere propagate upward along the magnetic field, and dissipate in thecorona.

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2 Measurement by Spectro-Polarimeter

2.1 Method to Calculate Parameters

The SP aboard SOT measure the three-dimensional magnetic field, line-of-sight velocity,and intensity obtaining the Stokes vectors I, Q, U, and V simultaneously. (Fig. 2.1) TheStokes I corresponds to intensity, the Stokes Q and U magnetic field perpendicular to theline-of-sight, and Stokes V line-of-sight magnetic field. We use the Fe I 630.25 nm lineto derive the line-of-sight velocity, the magnetic flux, and the intensity. The line-of-sightvelocity fluctuation(δvlos) is derived by measuring the Stokes V zero cross positionλc

performing linear fitting from peak to valley of Stokes V profiles. The Stokes V profilesreflect the motion of the magnetic atmosphere better than the Stokes I profiles, which alsocontain the information of the non-magnetic atmosphere. Intensity fluctuations in the linecore(δIcore) and in the continuum(δIcont) are derived from the line core intensityIcore

and continuum intensityIcont defined by,

Icont ≡ 4

(∫ λc−d2

λc−d3

I(λ)dλ +∫ λc+d3

λc+d2

I(λ)dλ

), (2.1)

Icore ≡ 4

(∫ λc

λc−d1

I(λ)dλ +∫ λc+d1

λc

I(λ)dλ

), (2.2)

whereI(λ) is the Stokes I profile observed with the SOT/SP,d1, d2, andd3 are 10.8 pm,43.2 pm, and 54.0 pm, respectively, The factor of 4 is to adjust the difference in the inte-gration range of Stokes profiles IQUV, as we will explain later.The line-of-sight magnetic flux fluctuation (δΦlos) and magnetic flux fluctuation perpen-dicular to the line-of-sight (δΦpar) are derived with the help of weak field approxima-tion (Landi degl’Innocenti & Landolfi 2004) rather than relying on the standard Milne-Eddington inversion (e.g. del Toro Iniesta 2003). The Milne-Eddington least-squares fit isperformed to the observed Stokes profiles of the Fe I 630.15 nm and Fe I 630.25 nm with12 parameters, which may be subject to noise that impedes the detection of fluctuationwith amplitude as small asδB/B0 ∼ 1.0 %. The intrinsic magnetic field strengthB0 andthe filling factorf are derived from the Milne-Eddington inversion to accurately deter-mine the Alfven speed. The intrinsic magnetic field strength is used only for this purpose.The filling factorf is defined as the fraction of area occupied with the magnetic field ina pixel (Orozco Suarez et al., 2007). The 12 free parameters are intrinsic field strength(B0), inclination(γ) and azimuth for magnetic field vector, line strength, Doppler width,damping factor, Doppler velocity, source function, source gradient, macro turbulence, fill-ing factor (stray-light factor), and the Doppler shift of the stray-light profile.For the weak field approximation, we first calculate the degree of the circular polarizationCP and the linear polarizationLP as defined by,

CP ≡ V

Icont

, (2.3)

LP ≡√

Q2 + U2

Icont

, (2.4)

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where

Icont ≡ 4

(∫ λc−d2

λc−d3

I(λ)dλ +∫ λc+d3

λc+d2

I(λ)dλ

), (2.5)

Q ≡∫ λc+d2

λc−d2

|Q(λ)|dλ, (2.6)

U ≡∫ λc+d2

λc−d2

|U(λ)|dλ, (2.7)

V ≡∫ λc

λc−d2

V (λ)dλ −∫ λc+d2

λc

V (λ)dλ, (2.8)

whereI(λ), Q(λ), U(λ), andV (λ) are the Stokes profiles observed with the SOT/SP,λc

is the measured zero cross position of the observed Stokes-V profiles as described above,andd1 andd2 are 10.8 pm and 43.2 pm, respectively. Since the integrations for CP and LPare done with respect toλc, the integral should not have any cross-talk with the Dopplervelocity. We take the integration ranges for CP and LP intentionally large so that theyare sure to cover the whole signals of Stokes Q, U, and V. The observed Stokes I, Q, U,and V profiles are shown in Fig. 2.1 as an example. TheCP andLP are related to theparameters derived from Milne-Eddington inversion as follows (e.g. Jefferies and Mickey,1991),

CP ∝ fB0 cos γ, (2.9)

LP ∝ f(B0 sin γ)2. (2.10)

We make the scatter plots indicating the relation betweenfB0 cos γ and CP (the toppanel of Fig. 2.2) and between

√fB0 sin γ andLP (the bottom panel of Fig. 2.2). Both

graphs indicate that CP and LP derived from weak field approximation and magnetic fluxderived from the Milne-Eddington inversion follows Eqs. 2.9 and 2.10. Performing thelinear fitting to the top panel of Fig. 2.2, we have the relation,

CP = klosΦlos, (2.11)

where conversion coefficientklos is (3.67±0.03)×10−5G−1 andΦlos = fB0 cos γ is line-of-sight magnetic flux derived from Milne-Eddington inversion, and the cross correlationis 0.97. Performing the quadratic fitting to the bottom panel of Fig. 2.2, we have therelation,

LP = apar + kparΦ2par, (2.12)

whereΦpar =√

fB0 sin γ is magnetic flux perpendicular to the line-of-sight derived fromMilne-Eddington inversion,apar = 0.00182 ± 0.00008, andkpar = (1.565 ± 0.072) ×10−8G−2. Thus we can measure the line-of-sight magnetic flux fluctuationδΦlos andmagnetic flux fluctuation perpendicular to the line-of-sightδΦpar from Φlos andΦpar.

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2.2 Detection Limit of Φlos

First we calculate the standard deviation of photon noise for each pixel in continuumrange of Stokes V normalized by the average intensity∆σ(V/I) as defined by,

∆σ(V/I) =1

< I >

√Σn

i V2i

n, (2.13)

whereVi are the photon counts of Stokes V in the continuum range (as shown in Fig. 2.1),< I > the average intensity in the continuum range, and n the number of pixels in thecontinuum range of Stokes V. We obtained∆σ(V/I) = 0.014.Next we calculate the standard deviation of V/Iσ(V/I). When the integration is per-formed over N pixels,σ(V/I) is given by,

σ(V/I) =

√N(∆σ(V/I)2

N=

∆σ(V/I)√N

. (2.14)

We integrate N=41 pixels to calculateCP = V/I in Eq. (2.8), soσ(CP ) = σ(V/I) isestimated to be2.0 × 10−4. Substituting3σ(CP ) = 5.8 × 10−4 (99% confidence level)into Eq. (2.11), the detection limit ofΦlos is estimated to be 16 G.In Eq. (2.8), we take the integration range of Stokes V intentionally large so that it is sureto contain the whole signal. Since the integration ranges for Stokes I and V are adjustedto be the same as mentioned above, the calculated CP is proportional toN−1 as long asthe integration range contains outside of signal,

CP ∝ 1

N. (2.15)

The conversion coefficient fromCP to Φlos (klos) is also proportional toN−1,

klos =CP

Φlos

∝ 1

N. (2.16)

From Eq. (2.14) we have,

σ(V/I) ∝ 1√N

. (2.17)

Thus the standard deviation of the line-of-sight magnetic fluxσ(Φlos) is proportional to√N ,

σ(Φlos) =σ(V/I)

klos

∝√

N. (2.18)

We change the N from 41 to 21 as is justified by Fig. 2.1, and obtained the detection limitof 12 G.

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2.3 Detection Limit of Φpar

First we calculate the standard deviations of photon noises for each pixels in continuumrange of Stokes Q and U normalized by the average intensity∆σ(Q/I) and∆σ(U/I) asdefined by,

∆σ(Q/I) =1

< I >

√Σn

i Q2i

n, (2.19)

∆σ(U/I) =1

< I >

√Σn

i U2i

n, (2.20)

whereQi andUi the photon counts of Stokes Q and U in the continuum range (as shown inFig. 2.1),< I > the average intensity in the continuum range, and n the number of pixelsin the continuum range of Stokes Q and U. We obtained∆σ ≡ ∆σ(Q/I) = ∆σ(U/I) =0.014.The photon noise normalized by the average intensityq ≡ Qi/ < I > andu ≡ Ui/ < I >follow the Gaussian distribution. The probability density functions of q and u are givenby,

f(q) =1√

2π∆σexp(− q2

2(∆σ)2), (2.21)

f(u) =1√

2π∆σexp(− u2

2(∆σ)2), (2.22)

so that the probabilityp(q, u) is given by,

p(q, u) = f(q)f(u)dqdu =1

2π(∆σ)2exp(−q2 + u2

2(∆σ)2)dqdu. (2.23)

Now we change the variables from(q, u) to (l, θ) (l ≡√

q2 + u2 = Li/ < I >, q ≡l cos θ, u = l sin θ, 0 ≤ θ ≤ 2π). Thus we have,

dqdu =∂q/∂l ∂q/∂θ∂u/∂l ∂u/∂θ

dldθ = ldldθ, (2.24)

p(l, θ) =1

2π(∆σ)2exp(− l2

2(∆σ)2)ldldθ, (2.25)

f(l) =l

(∆σ)2exp(− l2

2(∆σ)2). (2.26)

We obtained the probability density functionf(l), so we can calculate the average ofl El

and standard deviation ofl for a pixel∆σl as follows,

El =∫ ∞

0lf(l)dl =

√π

2∆σ, (2.27)

∆σl =

√∫ ∞

0l2f(l)dl − [E(l)]2 =

√2 − π

2∆σ. (2.28)

9

When the integration is performed over N pixels,σ(L/I) is given by,

σ(L/I) =

√N(∆σl)2

N=

√2 − π

2

∆σ√N

. (2.29)

Substituting∆σ = 0.014 to Eq. (2.27), we obtainedEl = 0.00175, which is consistentwith the linear polarization withΦpar = 0 from the result of the fitting0.00182± 0.00008(Eq. (2.12)). Substituting∆σl = 0.014 andN = 41 into Eq. (2.29), we obtain3σ(L/I) =4.29 × 10−4. Thus from Eq. (2.12), the detection limit ofΦpar is ≥ 165G depending onthe real value ofΦpar.In Eqs. (2.6) and (2.7), we take the integration ranges of Stokes Q and U intentionallylarge so that they are sure to contain the whole signals. Since the integration ranges forStokes I, Q, and U are adjusted to be the same as mentioned above, the calculated LP isproportional toN−1 as long as the integration range contains outside of signals,

LP ∝ 1

N. (2.30)

The conversion coefficient fromLP to Φpar (kpar) is also proportional toN−1,

kapar =LP

Φ2par

∝ 1

N. (2.31)

From Eq. (2.29) we have,

σ(L/I) ∝ 1√N

. (2.32)

Thus the standard deviation of the magnetic flux perpendicular to the line-of-sightσ(Φlos)is proportional toN1/4,

σ(Φpar) =

√√√√σ(L/I)

kpar

∝ N1/4. (2.33)

We changed N from 41 to 11 as is justified by Fig. 2.1, and obtained the detection limitof ≥ 117 G.

2.4 Detection Limit of δvlos

As we have mentioned in Section 2.1, we determine the line-of-sight velocityδvlos fromzero cross position of Stokes V by performing linear fitting from peak to valley of theStokes V profiles. An example of a Stokes V profile overplotted by the regression line isshown in Fig. 2.3. The correlation coefficient of the linear regression is−0.99.In order to determine the detection limit ofδvlos, we statistically estimate the standarddeviation of the zero cross position considering photon noise following Gaussian distri-bution. A histogram indicating the probability density function of the photon noise isshown in Fig. 2.4. We take the photon noises from 32 Stokes V profiles in the continuumrange as indicated by filled circles in Fig. 2.1. The total number of wavelength point taken

10

for the histogram is 1600. The average and the standard deviation of the photon count is−0.829 and 33.7, respectively. Fig. 2.4 confirms that the histogram of the photon countis consistent with Gaussian curve with the calculated average and standard deviation.Then we randomly add the photon noise following the derived Gaussian function to eachpixel of the observed Stokes V profile, recalculate the zero cross position by linear fit-ting, and obtain a residual from the original zero cross position. We perform 100 suchcalculations for each Stokes V profile. We take 143 Stokes V profiles corresponding towide range of circular polarization (0.01− 0.06) determined by Eq. (2.3). A histogramindicating the probability density function of the residual from the original zero cross po-sition due to the added photon noise is shown in Fig. 2.5. Total number of the residuals is14300. The average and the standard deviation of the residuals are−6.63×10−4 pixel and0.0110 pixel, respectively. Fig. 2.5 confirms that the histogram of the residuals is consis-tent with Gaussian curve with the calculated average and standard deviation. The lattercan be considered to be a standard deviation of the zero cross position (σzc). The detectionlimit of line-of-sight velocityδvlos is given by3c(∆λ)σzc

λ0, wherec = 3.0×105 km s−1 is the

speed of light,∆λ = 2.15 pm pixel−1 the spectral resolution of SP, andλ0 = 630.25 nmthe center wavelength of the spectral line used in the analysis. Therefore, the detectionlimit of δvlos is estimated to be 34 m s−1.

2.5 Summery and Choice of Stokes V

The detection limits ofΦlos, Φpar, andvlos are estimated to be 12 G,>117 G, and 34 m s−1,respectively. We consider peak-to-valley amplitudes of the fluctuations in the magneticflux and the velocity above the estimated detection limits as significant fluctuations. Assummarized in section 4.1, the peak-to-valley amplitudes of the fluctuations in the line-of-sight magnetic flux and the velocity are 13− 59 G and 85− 340 ms−1, both of whichsatisfy the condition. On the other hand, the former is much smaller than the detectionlimit of Φpar with factor of>2. Thus we give up using Stokes Q and U, and concentrateon Stokes V to detect fluctuations inΦlos andvlos.

11

ab cde fg b f

b f

wavelengthwavelength

wavelength wavelength b d f

b fwavelengthwavelength

b f

Figure 2.1:Top: Profiles of Stokes I (top,left), Q (top,right), U (bottom,left), and V (bot-tom,right). Wavelength positions froma throughg define the integration ranges specifiedby λc, d1, d2, d3 in Eqs. (2.1)− (2.8). a throughg indicatea : λc − d3, b : λc − d2,c : λc − d1, d : λc, e : λc + d1, f : λc + d2, andg : λc + d3, respectively. The filled boxesin Stokes Q, U, and V indicate the wavelength ranges for calculation of CP and LP (Eqs.(2.6)− 2.8), while the filled circles for calculation of photon noises of Stokes Q, U, andV in continuum ranges (Eqs. (2.14), (2.19), and (2.20))Bottom: Profiles ofV/I (left) and√

Q2 + U2/I (right), whereI is the average intensity in continuum range.

12

Cir

cula

r P

ola

riza

tion

Φ los

Φ_par

Lin

ea

r P

ola

riza

tion

Figure 2.2: Top: Scatter plot indicating the relation between the l.o.s. magnetic fluxΦlos = fB0 cos γ from the Milne-Eddington inversion and the circular polarizationCPfrom the weak field approximation as defined by Eq. (2.3). Solid line indicates a linearregression line.Bottom: Scatter plot indicating the relation between the magnetic fluxperpendicular to the line-of-sightΦpar =

√fB0 sin γ from the Milne-Eddington inversion

and the linear polarizationLP from the weak field approximation as defined by Eq. (2.4).Solid line indicates the result of quadratic fitting.

13

Wavelength (pixel)

Ph

oton

Cou

nt

Figure 2.3: An example of a Stokes V profile overplotted by the regression line from peakto valley of the profile. The correlation coefficient of the linear regression is−0.99. Thearrow indicates the zero cross position.

14

Photon Noise

Pro

ba

bil

ity D

en

sit

y F

un

ctio

n

Figure 2.4: A histogram indicating the probability density function of the photon noise,which is consistent with Gaussian curve with the calculated average and standard devia-tion (solid curve).

15

Residual from the original zero cross (pixel)

Pro

ba

bil

ity D

ensi

ty F

un

ctio

n

Figure 2.5: A histogram indicating the probability density function of residual from theoriginal zero cross position due to the added photon noise, which is consistent with Gaus-sian curve with the calculated average and standard deviation (solid curve).

16

3 Observations and Data Analysis

3.1 Hinode Observation

The data for analysis was taken on 2007 February 3− 5. The region that we observedwas NOAA 10940, which moved from west 25.2 to 49.0 degrees in longitude during thecourse of the observation. The region consists of pores and plages. The integration time is4.8 s, and the field of view is 1”.92 (EW) by 81”.92 (NS). The pixel size is 0”.16. Periodicscanning was done by the SOT/SP for 1 hour or 3 hours depending on the flux tubes withcadence of 67 s. This time resolution allows us to detect magnetohydrodynamic waveswith a period longer than 134 s according to the Nyquist criteria. We analyzed 14 mag-netic flux concentrations as tabulated in Table 3.1. All these magnetic flux concentrationsare of positive polarity (magnetic field vector toward the observer). The region #05 isshown in Fig. 3.1 as an example of the data. The region #05 contains a pore in a plageregion.

Region Date Time Pore or Plage x1 y1 θ2

ID (UT) (”) (”) (deg)# 01 2007 Feb 03 13:18 - 14:28 pore 410 45 25# 02 2007 Feb 03 14:28 - 15:38 pore 410 -11 25# 03 2007 Feb 03 12:08 - 13:18 plage 410 -5 25# 04 2007 Feb 03 19:15 - 20:27 pore 460 46 29# 05 2007 Feb 03 19:15 - 20:27 pore 460 41 29# 06 2007 Feb 03 19:15 - 20:27 plage 460 -1 29# 07 2007 Feb 03 19:15 - 20:27 pore 460 -7 29# 08 2007 Feb 04 01:28 - 02:42 plage 510 42 32# 09 2007 Feb 04 01:28 - 02:42 plage 510 38 32# 10 2007 Feb 04 07:56 - 09:10 plage 560 -21 36# 11 2007 Feb 04 14:28 - 15:37 plage 602 35 39# 12 2007 Feb 04 12:45 - 13:54 plage 602 -5 39# 13 2007 Feb 04 13:31 - 14:40 plage 602 -27 39# 14 2007 Feb 05 06:56 - 08:08 plage 725 -10 49

Table 3.1: List of observed magnetic flux concentrations.

1: X-Y coordinate of the target region. X is to the West, and Y is to the North.2: Helio-longitudinal angle from the meridional line.

3.2 Time-Profile Data Analysis

We should track the region of interest (ROI), for which the wave analysis is performed,in a Lagrangian way for an extended period of time. In the case of pores, the overallmagnetic structure is maintained over 1 hour as shown is Fig. 3.1. In this case, we set theROI to cover a portion of a pore. The size of the ROIs for pores is typically 1”× 1”. The

17

physical parameters are averaged inside the ROI. On the other hand, magnetic structuresin plages are generally not maintained for 1 hour: magnetic elements may combine, split,or decay within a time period of several tens of minutes. Thus, we set the ROI largeenough to encompass the entire magnetic flux concentration in the spatial and temporaldomain, and average the physical parameters of the pixels withCP larger than 0.01 insidethe ROI. The size of the ROIs for plages is typically 1”× 1” to 2” × 4”. Examples of SPimages for a pore (ID #05) and a plage (ID #10) are shown in Fig. 3.2.We make the time profiles of line-of-sight magnetic fluxΦlos(t), line-of-sight velocityvlos(t), core intensityIcore(t), and continuum intensityIcont(t) as defined by,

Φlos(t) =

∑i CPi(t)

Nklos

, (3.1)

vlos(t) = −∑

i(λc,i(t) − λ)c

Nλ0

, (3.2)

Icore(t) =

∑i Icore,i(t)

N, (3.3)

Icont(t) =

∑i Icont,i(t)

N, (3.4)

whereIcore,i(t), Icont,i(t), CPi(t), are defined by Eqs. (2.1), (2.2), and (2.3), respectively,λc,i(t) the zero cross position,λ the zero cross position averaged over spatial and temporaldomain which indicates the central wavelength of the spectral line,c the speed of light,λ0 = 630.25 nm,klos = 3.7× 10−4 G−1 the conversion coefficient, subscripti each pixel,andN the number of pixel used to make the profiles as defined above. The line-of-sightmagnetic flux and velocity towards the observer is defined to be positive. The minus signin Eq. (3.2) is added for the purpose.

18

217”.2

10

8”.

5

FG image (FeI)

19:15 (UT)

2”.1

9”.

6

I VII I VV V

19:36 19:57 20:18

Figure 3.1: Top: The SOT filtergraph (FG) image taken in the FeI 630.25 nm line at19:45(UT) on 2007 February 3. The field of view is 217”.1 (EW)× 108”.5 (NS). Thepixel size is 0”.108. Exposure time is 90 ms. The black rectangular box indicates theregion #05 (Table 3.1).Bottom: Zoomed SP images (Stokes I and V) for the region #05taken at 19:15-20:18 (UT) on 2007 February 3. Periodic scanning was done by SP forabout 1 hour with a cadence of 67 s. The integration time is 4.8 s. The field of view is2”.08 (EW)× 81”.92 (NS), part of which is shown here. The pixel size is 0”.16. Theblack region in the Stokes I map, which corresponds to the white region in the Stokes Vmap, is a pore. These images show that the magnetic structure in a pore is maintained forat least one hour.

19

Figure 3.2: The boxes in each image indicate the region-of-interest (ROI) for the waveanalysis.Left: SOT/SP images for the region #05 (Table 3.1). The ROI with size of 1”.28(EW) × 0”.64 (NS) is located inside the pore.Right: SOT/SP images for the region #10.The ROI contains a plage. The images show that the plage is not maintained for 1 hour.The size of the ROI is 2”.08 (EW)× 3”.52 (NS).

20

4 Power Spectrum and Phase Relation

4.1 Statistical Analysis

The top panels of Fig. 4.1 show the time profiles of the line-of-sight magnetic flux, theline-of-sight velocity, and the line core intensity for the region #04 (Table 3.1). We ap-plied the Fast Fourier Transform to all time profiles. The result for the region #04 is shownin the bottom panels of Fig. 4.1. The power spectra generally show one or two isolatedsharp peaks in the shorter periods, while broader peaks are found in the longer periods,corresponding to a gradual rise and fall in the time profiles. Some of the peaks have thesame period in the magnetic and velocity field, and the photometric intensity. We found20 such common peaks, which are all tabulated in Table 4.1. We analyzed 29 flux tubes,and such common peaks are found in 14 (48%) flux tubes, which are all tabulated in Table3.1.We derive the root-mean-square (r.m.s.) amplitudes of the line-of-sight fluctuation inmagnetic flux(δΦlos,rms) and velocity(δvlos,rms), the line core intensity fluctuations(δIcore,rms),and the continuum intensity fluctuations(δIcont,rms) at the peak periods in the power spec-tra. We also obtain phase difference between the fluctuations in the magnetic flux(ϕB),the velocity(ϕv), the line core intensity(ϕI,core), and the continuum intensity(ϕI,cont);ϕB − ϕv, ϕv − ϕI,core, ϕI,core − ϕB, andϕI,core − ϕI,cont, all for the peak periods. Thephase relations between the fluctuations in the magnetic flux, the velocity, and the inten-sity fluctuations are of critical importance to identify modes and properties of magneto-hydrodynamic waves as we will see later.Whenxn is the raw time profile(0 ≤ n ≤ N − 1) (N is the number of data points), thenthe complex amplitudeXk at the frequencyk in the frequency domain is converted to ther.m.s. value of the wave amplitudeAk,rms and the phaseθk as follows:

Xk =1

N

N−1∑n=0

xn exp(−2πikn

N), (4.1)

Ak,rms =√

2|Xk|, (4.2)

θk = arctan[Im(Xk)

Re(Xk)]. (4.3)

We calculate these values for all the peaks, and Table 4.1 lists the l.o.s. magnetic fluxΦ0,los = B0,losf , whereB0,los is the line-of-sight magnetic field andf the average fillingfactor, both of which are derived from Milne-Eddington inversion, the r.m.s. line-of-sightmagnetic flux fluctuations(δΦlos,rms),

δΦlos,rms

Φ0,los, the r.m.s. line-of-sight velocity fluctua-

tions (δvlos,rms), the r.m.s. line core and continuum intensity fluctuations normalized bythe average intensity,δIcore,rms

Icoreand δIcont,rms

Icont, and the phase difference among magnetic,

velocity, and intensity fluctuations;ϕB −ϕv, ϕv −ϕI,core, ϕI,core−ϕB, andϕI,core−ϕI,cont

derived from Eq. (4.3). There are 8 cases for pores and 12 cases for plages where mag-netic and velocity fluctuations have strong power at the same periods. The histograms ofthe phase difference and period for 20 such common peaks are shown in Fig. 4.2. The his-tograms for the phase difference show striking concentrations at around−90◦ for ϕB −ϕv

andϕv − ϕI,core, at around180◦ for ϕI,core − ϕB, and at around10◦ for ϕI,core − ϕI,cont.Here, for instance,ϕB − ϕv ∼ −90◦ means that the velocity leads the magnetic field by a

21

quarter of cycle.The periods are around 3−5 min for pores, while the periods are around 4−9 min forplages. There is no power between 134s (the detection limit due to the Nyquist crite-ria, see section 3.1) and 204s (region #04 in Table 4.1). As pointed out in Sect. 2.1, nocross-talk should be expected in the l.o.s. magnetic signal from the velocity fluctuations.Furthermore, the phase difference between the magnetic flux and the velocity fluctuationϕB − ϕv, if caused by the cross-talk, should be0◦ or 180◦, while the observed phase dif-ference shows a strong concentration at around−90◦. A similar phase relation is obtainedby Bellot Rubio et al. (2000) for sunspot umbrae. On the other hand, Ruedi et al. (1999)and Norton et al. (1999) came to an opposite conclusion that the magnetic field leads thevelocity by about a quarter of a cycle.

4.2 Dependence of Fluctuations on the Position of a Pore

We now take the ROIs in region #05 as shown in Fig. 4.3 in order to know the dependenceof fluctuations on the position of a pore. We take 10 ROIs with box sizes of 0”.96× 0”.96,which are located inside of the pore (ROIs 6 and 8), at the edge of the pore (ROIs 3, 4,5, 7, and 10), and outside of the pore (ROIs 1, 2, and 10) as identified from Stokes I map(the top panel of Fig. 4.3). Stokes I and V maps in Fig. 4.3 indicate that the magneticstructure of the pore is stable over 1 hour.We make the time profiles of the l.o.s. magnetic flux, l.o.s. velocity, and intensity for eachROI averaging the parameters over whole pixels, and perform Fast Fourier Transform tothe profiles just the same as sections 3.2 and 4.1. The power spectra of the profiles areshown in Fig. 4.4. This graphs indicate that the strong fluctuations in the magnetic fluxand the velocity with common period are seen in ROIs 2, 3, 4, 9, and 10, all of which arelocated at the edge of the pore or outside of the pore. On the other hand, no significantfluctuations in the magnetic flux are found in ROIs 6 and 8, both of which are locatedinside of the pore.This difference may be due to the plasmaβ in the ROIs; Flux tubes in higherβ region maybe sensitive to the motion of plasma surrounding the flux tube that generates the MHDwaves, while those in lowerβ region may not since the plasma is strongly frozen in bythe magnetic field. Statistical analysis for pores with smaller choice of ROIs may give usthe information about dependence of the fluctuations in the magnetic flux, the velocity,and the intensity on magnetic field strength.

22

Reg

ion

δΦ

1 los,

rms

Φ2 0,los

δΦ

los,

rms

Φ0

,los

f3

δv4 lo

s,rm

s

δIcore

,rm

sIcont

5δIrm

s,cont

Icont

5P

6ϕ

B−

ϕ7 v

ϕv−

ϕ7 I,c

ore

ϕI,c

ore

−ϕ

7 Bϕ

I,c

ore

−ϕ

7 I,c

ont

ID(G

)(1

03G

)(%

)(m

/s)

(%)

(%)

(min

)(d

eg)

(deg

)(d

eg)

(deg

)#0

117

.11.

161.

480.

7770

0.58

0.38

4.0

−67

−103

170

−3

#02

8.8

1.05

0.84

0.75

600.

320.

175.

2−

57

−74

131

43-

8.3

-0.

79-

680.

550.

254.

9−

58

−123

−179

16-

9.4

-0.

90-

570.

590.

294.

0−

54

−110

164

15#0

38.

90.

781.

140.

6586

0.47

0.36

5.2

−74

−70

145

−5

#04

10.0

1.02

0.98

0.73

360.

570.

283.

4−

94

−107

−159

−12

#05

13.8

0.97

1.42

0.81

120

0.97

0.89

4.9

−57

−73

130

14#0

64.

40.

670.

660.

5676

0.27

0.11

5.2

−71

−47

118

12#0

79.

81.

080.

910.

7267

0.36

0.34

7.6

−67

−41

108

19-

7.7

-0.

71-

590.

350.

254.

3−

96

−76

172

11#0

83.

90.

540.

720.

4977

0.37

0.30

7.6

−58

−61

119

14-

3.5

-0.

65-

620.

280.

126.

8−

58

−120

178

22#0

94.

80.

461.

040.

4698

0.74

0.53

5.7

−60

−85

145

14#1

04.

50.

510.

880.

5734

0.19

0.12

7.6

−105

−156

−99

−21

#11

7.4

0.47

1.57

0.52

350.

920.

587.

6−

102

−87

179

16#1

24.

50.

580.

780.

5344

0.25

0.15

5.7

−38

−46

8416

-3.

5-

0.60

-82

0.41

0.30

5.2

−48

−100

148

8#1

35.

10.

730.

700.

6173

0.39

0.18

5.2

−48

−71

118

4-

6.4

-0.

88-

470.

470.

264.

5−

55

−93

138

21#1

46.

40.

391.

640.

4340

0.20

0.11

8.5

−77

−88

165

-7

Tabl

e4.

1:P

hysi

calp

aram

eter

sco

rres

pond

ing

toth

epr

inci

palp

eak

inth

epo

wer

spec

tra

ofal

lreg

ion

ofin

tere

sts

(sho

wn

inTa

ble.

3.1)

with

com

mon

peak

sin

the

mag

netic

flux,

the

velo

city

,and

the

phot

omet

ricin

tens

ity.

1:r.m

.s.

(roo

tmea

nsq

uare

)l.o

.s.

(line

ofsi

ght)

mag

netic

flux

ampl

itude

.2:

l.o.s

.m

agne

ticflu

xfr

omM

ilne-

Edd

ingt

onin

vers

ion.

3:av

erag

efil

ling

fact

or.

4:r.m

.s.

l.o.s

.ve

loci

tyam

plitu

de.

5:r.m

.s.

inte

nsity

fluct

uatio

nno

rmal

ized

byth

eav

erag

ein

tens

ityfo

rlin

eco

rean

dco

ntin

uum

.6:

Flu

ctua

tion

perio

d.7:

phas

edi

ffere

nce

betw

een

the

fluct

uatio

nsin

the

mag

netic

flux

,the

velo

city

,the

core

inte

nsity

,and

the

cont

inuu

min

tens

ity.

23

magnetic flux velocity intensity

time(min)l.o.

s. m

agn

etic

flu

x (

G)

time(min)

l.o.

s. v

eloc

ity (

km

/s)

time(min)nor

ma

lize

d i

nte

nsi

ty

period(min)

pow

er o

f m

agn

etic

flu

x (

G2

/min

)

period(min)

pow

er o

f vel

ocit

y (

(km

/s)2

/min

)

period(min)pow

er o

f in

ten

sity

(1

/min

)

Figure 4.1:Top: Time profiles for the region #04 of Table 3.1: the line-of-sight (l.o.s.)magnetic flux(left), the l.o.s. velocity(center), and the line core intensity(right) asdefined by Eqs. (3.1)− (3.3). The intensity profile is normalized to the peak value of thetime profile. Images of the region #04 are shown in Fig. 3.1.Bottom: The power spectraof the l.o.s. magnetic flux(left), the l.o.s. velocity(center), and the normalized line coreintensity(right). The circles indicate the common, isolated peaks.

24

phase difference (deg) phase difference (deg)

phase difference (deg) phase difference (deg)

nu

mb

er o

f ev

ents

nu

mb

er o

f ev

ents

nu

mb

er o

f ev

ents

nu

mb

er o

f ev

ents

nu

mb

er o

f ev

ents

period (min)

Period

Figure 4.2:Left: Histograms of the phase difference between fluctuations in the magneticflux, the velocity, the line-core intensity, and the continuum intensity,ϕB − ϕv (top, left),ϕv − ϕI,core (top right), ϕI,core − ϕB (bottom left), andϕI,core − ϕI,cont (bottom right).Solid lines indicate the phase difference for pores and plages, while dashed lines thephase difference for only pores. The histograms show striking concentrations at around−90◦ for ϕB − ϕv andϕv − ϕI,core, at around180◦ for ϕI,core − ϕB, and at around10◦

for ϕI,core − ϕI,cont. Right: Histogram of the periods of the common peaks in the powerspectra. The peak periods are around 3−5 minutes for pores, while the peak periods forplages are around 4−9 minutes.

25

I

V

19:15 19:39 20:03 20:27

2007 Feb 3

1 2

3 4

5 6

7 8

9 10

Figure 4.3: Stokes I and V maps of the region #05 in Table. 3.1. 10 ROIs with the sizeof 0.”96 × 0”.96 are indicated in the white boxes. The ROIs are numbered as indicatedin the right panel. The black region in Stokes I, which corresponds to the white region inStokes V, indicate a pore, whose magnetic field is maintained over an hour. The wholeFOV of the maps is 1”.92× 9”.60.

26

magnetic field

(G^2/min)

magnetic field

(G^2/min)

velocity

((km/s)^2/min)

velocity

((km/s)^2/min)

intensity

(1/min)

intensity

(1/min)

1 2

3 4

5 6

7 8

9 10

Figure 4.4: Power spectra of l.o.s. magnetic flux, l.o.s. velocity, and intensity for the ROIsindicated in Fig. 4.3. The numbers for the spectra correspond to those defined in the rightpanel of Fig. 4.3. The vertical axes indicating the power spectra of the magnetic flux, thevelocity, and the intensity are all scaled to0−200 G2 min−1, 0−0.01 (km/s)2 min−1, and0 − 1.0 × 10−4 min−1. The circles indicate the strong fluctuations in the magnetic fluxwith the velocity or the intensity with same periods.

27

5 Intensity Fluctuation

Previous authors (e.g. Bellot Rubio et al. 2000) detected fluctuations in the magneticfield strength and the velocity for a sunspot umbra, and obtained a phase difference of∼ 90◦. They concluded that the observed fluctuations in magnetic field strength is mainlycaused by the opacity effect. Temperature and density fluctuations associated with thepropagation of a hydrodynamic (acoustic) or magneto-hydrodynamic (magneto-acoustic)wave may cause the opacity fluctuation that moves the line formation layer upward ordownward, resulting in an apparent magnetic field fluctuation, if the magnetic field has agradient with geometrical height(dB/dz). This is called the opacity effect.In this section, we consider whether the observed fluctuation is due to the opacity effect.The photometric intensity that we observe is given by

I =∫ τ

0

σT (τ)4

πe−τdτ , (5.1)

whereT is the local temperature at the optical depthτ . The intensity modulation can takeplace due either to change in the temperature or to change in the optical depth, whichdepends on the density and the temperature in the optical path. The opacity effect in-volves the second term (e−τ ). Fluctuation in intensity indicates a compressive nature ofthe fluctuation due to the first term (σT (τ)4

π) and/or to the second term (e−τ ) in eq. (5.1).

Thus, waves with low intensity fluctuation, especially those with an intensity fluctuationclose to zero, can be considered to be a incompressible mode (such as the kink mode),while those with high intensity fluctuation can be considered to be a compressible mode(such as the sausage mode). The schematic images of the kink-mode MHD wave and thesausage-mode MHD wave are shown in Fig. 5.2.The top panels of Fig. 5.1 show the histograms of the line core(δIcore,rms) and the contin-uum(δIcont,rms) intensity fluctuations normalized by the average intensityIcore andIcont;δIcore,rms

Icore(core fluctuation) andδIcont,rms

Icont(continuum fluctuation) for all the peaks. The re-

lation between the core and the continuum fluctuations is shown in the bottom panel ofFig. 5.1. The scatter plot indicates that the fluctuation at the line core is larger than thecontinuum fluctuation for all the peaks, and that the line-core and the continuum fluctu-ations are linearly correlated. A linear fitting between the line-core and the continuumfluctuations is given by,

δIcont,rms

Icont

= 0.79δIcore,rms

Icore

− 0.00066. (5.2)

The cross correlation coefficient is 0.91. Fig. 4.2 shows that phase difference between theintensity fluctuation in the core and in the continuum,(ϕI,core − ϕI,cont), has a concentra-tion at around10◦ ± 14◦.We here consider the opacity effect due either to the density or to the temperature fluc-tuations. First we assume only the density fluctuation (without the temperature fluctua-tion). Magnetic field strength is smaller with height (dB/dz < 0) because of the canopystructure of magnetic flux tubes. Since the observations are carried out with 25.2 to 49degrees away from the normal, we simply assume here that the magnetic field strengthalong the line of sight decreases with height in the following discussion. The temperature

28

is lower with height below the temperature minimum. When the atmosphere in the lineformation layer is compressed (or decompressed), the line formation layer moves upward(downward), because the opacity along the line of sight in the flux tube increases (de-creases). When the line formation layer moves upward (or downward), both the magneticfield strength and the intensity decrease (increase). Therefore, the observed magneticfield strength and the constant-temperature intensity fluctuation caused by the opacity ef-fect should have had the phase difference of0◦, while the observed phase differencesϕI,core − ϕB have a concentration at around180◦. Thus, the observed phase difference isnot consistent with that caused by the opacity effect, if the opacity effect is caused onlyby the density fluctuation without temperature fluctuation.On the other hand, the line formation layer may be compressed (or decompressed) underthe adiabatic condition. We here consider the opacity effect due to temperature, assumingthat the optical depthτ depends only on the temperature. The dominant absorber in thevisible wavelengths is the H− ion (e.g. Stix, 2002). The populations of H− and HI arerelated with the Saha equation (Rutten 1995, eq. (8.2))

logN(HI)

N(H−)= −0.1761 − log Pe + log

U(HI)

U(H−)+ 2.5 log Te −

5040χ

Te

, (5.3)

wherePe is the electron pressure,Te the electron temperature,χ the ionization energyfrom H− to H, N(H−) andN(HI) the population densities of H− and HI,U(H−) andU(HI) the partition function of H− and HI. Equation (5.3) indicates that the populationof H− depends highly on the temperature, and decreases with the temperature in the caseof the adiabatic compression, while the population depends weakly on the pressure, andincreases with the pressure in the constant temperature case. Thus, we cannot determinethe population of H− in the actual situation without employing a model taken into accountthe radiation exchange between the inside and the outside of the flux tubes.We point out that regardless of mechanism to change the opacity, the phase differencebetween the fluctuations in the magnetic field and the intensity (ϕI −ϕB) depends only onthe sign of magnetic gradient along the line of sight when the line formation height movesupward or downward due to the lateral expansion and the contraction of the tube. The fluxtubes that we observed were located 25.2 to 49.0 degrees away from the sun center. If themagnetic field strength decreases with height along the oblique line of sight, the phasedifference between the fluctuations in the magnetic field and the intensityϕI −ϕB shouldhave been0◦, whereas we obtainedϕI−ϕB ∼ 180◦. Therefore, the phase relation betweenthe fluctuations in the magnetic field and the intensity from the observation would not beconsistent with that caused by the opacity effect under the assumption of the decreasingfield strength with height along the line of sight.If the effect of the adiabatic compression (or decompression; first term in equation (5.1))is larger than the opacity effect due to the density and/or temperature fluctuation (secondterm in equation (5.1)), the phase difference between the magnetic field strength and theintensity fluctuation is0◦ for the case of the fast-mode wave, while that is180◦ for the caseof the slow-mode wave. Thus, we can rule out the fast-mode wave, since the observedphase difference is close to180◦.

29

continuum line core

intensity fluctuation (%)intensity fluctuation (%)

nu

mb

er o

f ev

ents

nu

mb

er o

f ev

ents

line core

con

tin

uu

m

intensity fluctuation (%)

Figure 5.1:Top: Histograms of continuumδIcont and line coreδIcore intensity fluctuationsnormalized by the average intensityIcont and Icore,

δIcont,rms

Icont(left) and δIcore,rms

Icore(right)

(solid lines). Dashed lines indicate histograms for pores.Bottom: Scatter plot between theintensity fluctuationsδIcont,rms

Icontand δIcore,rms

Icore. The solid line indicates the linear regression

line.

30

z

l.o.s.l.o.s.

Figure 5.2:Left:kink mode MHD waveRight: sausage mode MHD wave

31

6 Kink Mode MHD Waves

In this section, we examine whether the observed properties of waves are consistent withthe kink mode MHD waves (the left panel of Fig. 5.2). Though the magnetic and velocityfluctuations that we observe could be either parallel or perpendicular to the flux tubes, wehere consider the possibility that the observed fluctuations are transverse to the magneticfield. As discussed in section 5, Fig. 5.1 shows that some of the fluctuations has verysmall intensity fluctuation. Since the kink mode is essentially of non-compressive nature,those fluctuations with little intensity fluctuation may dominantly have properties of thekind mode.

6.1 Phase Relation of Propagating Kink Waves

The dispersion relation of the kink mode neglecting gravitational stratification is given by(e.g. Spruit, 1981; Edwin and Roberts, 1983, Moreno-Insertis, Schussler,& Ferriz-Mas,1996, Ryutova & Khijakadze, 1990)

ck =ω

k= vA

√ρi

ρi + ρe

, (6.1)

whereck is the phase speed of the kink mode,ω the frequency,k the wave number,vA theAlfv en speed,ρi the density inside the flux tube, andρe the density outside the flux tube.The transverse displacement of the flux tubeδx with geometrical heightz and timet canbe expressed asδx(z, t) = x0 cos(ωt ± kz), wherex0 is the amplitude of the transversedisplacement. The transverse magnetic field and velocity component are given by,

δB = B0∂(δx)

∂z= ∓B0k sin(ωt ± kz), (6.2)

δv =∂(δx)

∂t= −ω sin(ωt ± kz), (6.3)

whereB0 is the vertical magnetic field strength. From Eqs. (6.1)− (6.3), we obtain

δB

B0

= ± δv

ω/k, (6.4)

δB = ±√

4π(ρi + ρe)δv. (6.5)

The phase relation of the kink mode is the same as that of the Alfven mode. Magnetic fieldis directed away from the Sun in our case. If the kink wave propagates to the directionsame as that of magnetic field vector, minus sign should be taken, and vice versa. If apure ascending or descending kink wave propagates toward the observer along magneticfield, phase difference between the magnetic field and the velocity fluctuations(ϕB − ϕv)should, therefore, have been 180◦ or 0◦, respectively, as shown in Fig. 6.1. Fig. 4.2 showsthat this is not the case.

32

6.2 Phase Relation of Standing Kink Waves

We then consider a superposition of the ascending kink wave and the descending waves,which is the reflected ascending wave at the photosphere-chromosphere boundary. Whenthe ascending and the descending kink waves coexist in the line formation layer, the super-posed wave form is determined by six variablesδBu, δvu, ϕu, δBd, δvd, ϕd, which indicatethe amplitude of the magnetic field fluctuation (δB), the amplitude of the velocity fieldfluctuation (δv), and the initial phase (ϕ) of upward (subscriptu) and downward (subscriptd) waves. When magnetic field vector is toward the observer, the transverse magnetic fieldand velocity displacement of the superposed kink wave are given by

δB = −δBu cos(ωt + ϕu) + δBd cos(−(ωt + ϕd)), (6.6)

δv = δvu cos(ωt + ϕu) + δvd cos(−(ωt + ϕd)). (6.7)

Note that the phase difference between magnetic and velocity fluctuation in the ascendingkink wave is 180◦, while that in the descending kink wave is 0◦. This fact is reflected inthe sign of each term in Eqs. (6.6) and (6.7). We can rewrite these equations as follows:

δB = δBs cos(ωt + ϕB), (6.8)

δv = δvs cos(ωt + ϕv), (6.9)

whereδBs andδvs are the magnetic and the velocity amplitudes of the superposed kinkwave, andϕB andϕv are phases of the magnetic field and the velocity of the superposedkink wave. In Eqs. (6.8) and (6.9),δBs, δvs, ϕB, andϕv are given by:

δBs =√

δB2u + δB2

d − 2δBuδBv cos(ϕu − ϕd), (6.10)

cos ϕB =δBu sin ϕu − δBd sin ϕd

δBs

, (6.11)

sin ϕB =−δBu cos ϕu + δBd cos ϕd

δBs

, (6.12)

δvs =√

δv2u + δv2

d + 2δvuδvv cos(ϕu − ϕd), (6.13)

cos ϕv =−δvu sin ϕu − δvd sin ϕd

δvs

, (6.14)

sin ϕv =δvu cos ϕu + δvd cos ϕd

δvs

. (6.15)

From Eq. (6.5), we obtainδvδB

= 1√4π(ρi+ρe)

. Therefore, we obtain the following relation

among the quantities in Eqs. (6.6) and (6.7):

δvu

δBu

=δvd

δBd

. (6.16)

Using Eqs. (6.10)−(6.16), the following phase difference between magnetic and velocityfluctuations is obtained:

cos(ϕB − ϕv) = cos ϕB cos ϕv + sin ϕB sin ϕv

33

=−δBuδvu + δBdδvd

δBsδvs

=δBu/δvu

δBsδvs

(−δv2u + δv2

d) =δvu/δBu

δBsδvs

(−δB2u + δB2

d). (6.17)

This equation shows that the phase difference between the magnetic and the velocityfluctuations (ϕB − ϕv) should be−90◦ or 90◦ when the amplitude of the reflected de-scending kink wave is exactly the same as that of ascending kink wave (i.e.δvu = δvd

andδBu = δBd). The observed phase relation is consistent with this prediction.The transverse displacement of magnetic field line in the presence of upward (δxu) anddownward (δxd) kink wave is written as a function of height(z) and time(t),

δxu(t, z) = xu0 cos(ωt + kz + ϕu), (6.18)

δxd(t, z) = xd0 cos(ωt − kz + ϕd), (6.19)

wherexu0, xd0, ϕu, ϕd are the transverse amplitude and the initial phase of the magneticfield line fluctuation in the presence of the upward (subscriptu) and the downward (sub-scriptd) kink wave. Whenxu0 = xd0 ≡ x0, which corresponds to the case for perfectreflection, the transverse displacementδxs of the magnetic field line in the presence ofthe superposed kink waves is given by

δxs(t, z) = δxu(t, z) + δxd(t, z) = 2x0 cos(ωt +ϕu + ϕd

2) cos(kz +

ϕu − ϕd

2). (6.20)

Equation (6.20) shows that the superposed kink wave, if with perfect reflector, is a stand-ing wave. Sketches of standing kink wave are shown in Fig. 6.2. Whether the phasedifference is90◦ or −90◦ depends on the distance from the reflection boundary (anti-node).

6.3 Interpretation for Standing Waves

We here give one interpretation for the concentration of the phase difference at around−90◦. Fig. 6.3 indicates that the electron densityne drops with factor of102 at thetransition region, so that the Alfven velocityvA = B√

4πnm, where n is the number density

and m the average particle mass, jumps with factor of 10 assuming constantB from thechromosphere to the corona (which can be true at magnetic structures in active regions.)Thus the transition region can play an role in a chromospheric-coronal boundary. Theascending kink wave reflect back at the boundary, even though residual upward wavemay go through the transition region and reach the corona as we will discuss in section6.4. When the ascending and the descending kink waves coexist in the line formationlayer beneath the reflector, the superposed wave has a property of a standing wave. As theAlfv en velocity in the corona is much larger than that in the chromosphere, the standingwave should have an anti-node at the boundary as shown in Fig.6.2.The phase difference between the magnetic and the velocity fluctuations should have beeneither90◦ or −90◦, while observed phase angle concentrates at around−90◦. Whetherthe phase angle is90◦ or −90◦ depends on the distance between the reflector and theline formation layer (Fig. 6.2). The concentration at−90◦ indicates that the separationbetween the reflecting boundary and the line formation layer is fixed for all the flux tubessuch that it corresponds to−90◦ phase difference. If we perform similar observations withdifferent absorption lines with different formation height, and the difference in height is

34

larger than the quarter of the wavelength (800km), this conjecture can be verified.

6.4 Leakage of Poynting Flux to the Corona

As shown in Fig. 6.3, some residual upwarding wave may reach to the corona, eventhough the superposed wave is almost a standing wave. The Poynting flux above thereflecting layer is the difference between the ascending one and the descending one in theline formation layer. We here estimate the effective or residual upward-directed Poyntingflux along a flux tube above the reflector. The Poynting flux of the kink wave is given byF = fB0

4π(δBrmsδvrms), so that the difference of the Poynting flux between the ascending

and the descending kink waves is given by

△F =fB0

4π(δBu,rmsδvu,rms − δBd,rmsδvd,rms), (6.21)

whereδBu,rms = δBu/√

2, δvu,rms = δvu/√

2, δBd,rms = δBd/√

2, δvd,rms = δvd/√

2.Using Eq. (6.17), we can rewrite the equation as follows:

△F = −fB0

4π(δBs,rmsδvs,rms) cos(ϕB − ϕv), (6.22)

whereδBs,rms = δBs/√

2, δvs,rms = δvs/√

2. It turns out that the effective upward-directed Poynting flux is proportional tocos(ϕB − ϕv). δBs,rms, andδvs,rms in Eq. (6.22)are related to the observables, assuming that the fluctuations are transverse (i.e. normal tothe flux tubes),

δBs,rms =δΦlos,rms

f sin θ, (6.23)

δvs,rms =δvlos,rms

sin θ, (6.24)

whereθ is helio-longitudinal angle from the meridional line. If the phase difference from−90◦ is just 6◦ as an exercise, i.e.ϕB − ϕv = −96◦, we obtain△F = 2.7 × 106 ergcm−2 s−1 by substitutingB0 = 1.7 × 103 G, δΦlos,rms = 7.7 G, δvlos,rms = 0.059 km/s,f = 0.73, andθ = 29◦ (region #07). (The energy flux required for the coronal heating(∼ 3 × 105 erg cm−2 s−1 for the quiet Sun; Withbroe & Noyes, 1977))

6.5 Seismology of Photospheric Flux Tubes

We show in this chapter that various physical parameters that characterize the magneticflux tubes are obtained simply from the amplitude and period of the magnetic and velocityfluctuations. We estimate the physical parameters for the region #02. The intensity fluc-tuation is 0.17% in continuum (Table 4.1), and we assume that the observed fluctuation isdue to the superposition of upward and downward kink waves.We define the coronal/chromospheric boundary, which is considered to be a reflector, tobe the origin of the z-axis, which is normal to the solar surface (away from the Sun).A schematic behavior of the standing kink wave is shown in the left panel of Fig. 6.4.Substitutingϕu+ϕd

2= 0 (without losing generality) andϕu−ϕd

2= 0 (to make the height at

35

z = 0 the anti-node) into Eq. (6.20), the transverse displacement of the flux tube is givenby,

δxs(t, z) = 2x0 cos(ωt) cos(kz). (6.25)

The transverse components of the magnetic field and the velocity are given by,

δBs(t, z) = B0∂δxs

∂z= −2B0x0k cos(ωt) sin(kz), (6.26)

δvs(t, z) =∂δxs

∂t= −2x0ω sin(ωt) cos(kz), (6.27)

Equations (6.26) and (6.27) indicate that the phase difference between the fluctuations inthe magnetic field and the velocityϕB − ϕv is,{

−90◦ for (n + 12)π ≤ kz ≤ (n + 1)π (sector (b) in Fig.6.4),

90◦ for nπ ≤ kz ≤ (n + 12)π (sector (a) in Fig.6.4),

(6.28)

wheren = −1,−2,−3, .... Equation (6.28) indicates that the observed phase differenceϕB − ϕv ∼ −90◦ is consistent with the situation that the line-formation height is locatedin the sector (b). From Eqs. (6.1), (6.26), (6.27), we have

|δvs||δBs|

=ω/k

B0

| tan(kz)| =| tan(kz)|√4π(ρi + ρe)

, (6.29)

ρi + ρe =( |δBs||δvs|

)2 | tan(kz)|2

4π, (6.30)

where|δBs| and|δvs| are the amplitude of the fluctuations in the magnetic field and thevelocity, and are the function of heightz. |δBs| and|δvs| in Eq. (6.30) are related to theobservables,

|δBs| =

√2δΦlos,rms

f sin θ, (6.31)

|δvs| =

√2δvlos,rms

sin θ. (6.32)

Assuming that the flux tubes that we observe here are in pressure equilibrium, and do nothave a helical structure (azimuthal component), the equation for the pressure equilibriumfor the flux tube is simply expressed as

B2i

8π+

ρi

mkBTi =

B2e

8π+

ρe

mkBTe, (6.33)

ρeTe − ρiTi =m

8πkB

(B2i − B2

e ), (6.34)

whereB, ρ, andT are the magnetic field strength, the mass density and the temperature,and the subscripti ande indicate the inside and the outside of the flux tube, respectively,m average particle mass, andkB the Boltzmann constant. From Eqs. (6.30) and (6.34),we can determineρi andρe assuming that outside the flux tube is field-free (Be = 0 G),

36

as is inferred by the observations.The line formation height in the umbra is deeper than that in the quiet Sun, because of thelower temperature and density (e.g. Stix, 2002). The Wilson depression for the flux tubewith B ∼ 2000 G reaches about 300− 400 km (Deinzer, 1965; Mathew et al., 2004). Thetemperature and the average molecular weight at the height∼ −350 km isTe = 1.0×104

K andµ = 1.2 (from Table 2.4, Stix, 2002). Since the temperatures inside the flux tubeis lower than that outside the flux tube in the subsurface region (Maltby et al., 1986), weassumeTi = 7.0 × 103K. We choosekz = −585◦ (see section 7.3 for justification tochoose the value). Substitutingm = µmp = 1.9× 10−24 g, wheremp is the proton mass,kB = 1.4 × 10−16 erg K−1, Bi = B0 = 1.9 × 103 G, δΦlos,rms = 8.8 G, δvlos,rms = 0.060km s−1, f = 0.75, andθ = 29◦, we obtain mass densitiesρi = 6.3 × 10−8 g cm−3

andρe = 2.4 × 10−7 g cm−3. The number densities inside and outside the flux tube areni = ρi

m= 3.3 × 1016 cm−3 andne = ρe

m= 1.2 × 1017 cm−3, respectively. The mass

density for the height of−300 −−400 km isρe = 3.5 − 4.5 × 10−7 g cm−3 (from Table6.1, Stix, 2002). This is consistent with our estimation within a factor of 2.We also estimate other physical parameters associated with the flux tube: (1) Alfvenspeed inside the flux tubevA,i = Bi√

4πρi, (2) plasmaβ = ρikBT/m

B2i /8π

inside the flux tube in

the line formation layer, (3) wavelength of the kink modeL = vA,i

√ρi

ρi+ρeP , whereP is

the fluctuation period, (4) propagation time of fast magneto-acoustic wave across the fluxtubeτ = R√

v2A,i+c2s

, whereR is the tube radius, and (5) distance between the boundary

and the line formation layerd = L |kz|360

. Other obvious useful parameters are the pres-sure scale heightH = kBT

mg, whereg is the gravity in the solar surface, and the sound

speed in the photospherecs =√

γkBTm

, whereγ is the adiabatic coefficient. SubstitutingBi = 1.7 × 103 G, ρi = 6.3 × 10−8 g cm−3, ρe = 2.4 × 10−7 g cm−3, g = 2.7 × 104 cms−2, P = 312 s, γ = 5/3, andR = 2000 km (case # 02), we obtainvA,i = 21 km s−1,β = 0.22, L = 3.0× 103 km, τ = 8.6× 101 s,d = 4.9× 103 km, H = 3.9× 102 km, andcs = 11 km s−1. The propagation time of the fast magneto-acoustic wave across the fluxtubeτ is less than the oscillation periodP , and this is consistent with the assumption ofthe kink wave.Mathew et al. (2004) calculated the physical parameters (magnetic pressure, gas pres-sure, Wilson depression, and plasmaβ) for a sunspot by performing an inversion to in-frared spectro-polarimetric profiles, and the derived plasma beta for the umbraβ ∼ 0.5 isconsistent with our estimation. Ruedi (1992) also performed an inversion to the infraredlines, and obtained the plasmaβ ∼ 0.25 at z = 0 km in the plage region. The plasmabeta is generally higher atz = −350 km, following the increase in the mass density (Stix,2002).As demonstrated here, we are potentially able to obtain all the physical parameters of theflux tube from the information on the MHD fluctuations. This indicates that seismologyof magnetic flux tubes is possible with multiple lines corresponding to different height(photosphere and chromosphere) of the solar atmosphere.

37

Figure 6.1: Schematic images indicating the phase difference between the fluctuationsin the magnetic field and the velocityϕB − ϕv with kink waves propagating to the same(left) or opposite (right) direction of the magnetic field.ϕB − ϕv should have been 180◦

or 0◦ degrees respectively, neither of which is consistent with result from the observationϕB − ϕv ∼ −90◦.

38

Figure 6.2:Top: A standing kink wave along magnetic field lineB0 is divided into fourparts (1) through (4) each separated by nodes and anti-nodes. The standing kink wavesshould have an anti-node (see text for detail).Bottom: Time evolution of the standing kinkwaves. The wave evolves from (a) to (h), and goes back to (a). The arrows with filledbox indicate velocity vector, while the arrows with circle indicate perturbed component ofmagnetic field vector. The length of the arrows indicates the magnitudes of the vector atcertain space and time points. Schematic representation of the standing kink wave showsthat the phase difference between magnetic and velocity fluctuations (ϕB − ϕv) is −90◦

at the portions (1) and (3), and90◦ at the portions (2) and (4).

39

Figure 6.3: (top) Electron densityne and electron temperatureTe as function of height.(The origin of height is defined by the optical depth at the wavelength of 500 nmτ500 beingunity.) The graph indicates thatne drops at the transition region, so that the transitionregion can play a role as a chromospheric-coronal boundary. (bottom) Schematic imageof a standing kink wave. Although some residual upward wave may propagate throughthe transition region and reach to the corona, the ascending wave reflects at the boundary,the superposed wave has a property of a standing wave.

40

boundary (node)z

line formation layer

(b)

(a)

(b)

(a)

O(a)

(b)

(a)

(b)

(a)

Figure 6.4: The standing kink wave (left) and the standing slow sausage wave (right). Thephase difference between the fluctuations in the magnetic field and the velocity (ϕB −ϕv)is 90◦ in the sector (a) and−90◦ in the sector (b). The arrows indicate the transversemotion of the magnetic fields at the anti-nodes.

41

7 Sausage Mode MHD Waves

We here consider the alternative possibility that the observed magnetic and velocity fluc-tuations are due to the longitudinal MHD waves or the slow sausage mode oscillation (theright panel of Fig. 5.2: Ryutova, 2009; Defouw, 1976; Roberts and Webb, 1978; Ryutova1981).

7.1 Phase Relation of Propagating Slow Sausage Wave

We consider a slow mode perturbation propagating along a cylindrical flux tube, neglect-ing gravitational stratification, following Ryutova (2009). We assume that the magneticand velocity fluctuations with higher intensity fluctuation (Fig. 5.1) may have the sausage-mode nature. The momentum equation perpendicular to the flux tube is given by

B0∥δB∥

4π+ δp = 0, (7.1)

where the subscript0 means these values in unperturbed state, andδ means perturbationof these values. We have the relation under the adiabatic condition

δp = c2s0δρ, (7.2)

and the flux conservation is given by

B0∥δS + δB∥S0 = 0. (7.3)

The momentum equation parallel to the flux tube is (substituting eq. (7.2))

ρ0

∂δv∥∂t

= −∂δp

∂z= −c2

s0

∂δρ

∂z, (7.4)

and the continuity equation is

∂

∂t(δρS0 + δSρ0) + S0ρ0

∂δv∥∂z

= 0, (7.5)

whereS = πR2, B∥, ρ, p, v∥ are the cross section of the flux tube, the longitudinalmagnetic field, the density, the pressure, and the longitudinal velocity, respectively, andcs,0 is the sound speed. From Eqs. (7.1)− (7.3), we have

δB∥ = −4πδp

B0∥= −4πc2

s0

B0∥δρ, (7.6)

δS = −S0

δB∥

B0∥= S0

4πc2s0

B20∥

δρ. (7.7)

The continuity equation (Eq. 7.5) becomes1 +4πc2

s0ρ0

B20∥

∂δρ

∂t+ ρ0

∂δv∥∂z

= 0. (7.8)

42

Taking the time derivative, and substituting Eq. (7.4), we have the dispersion relation,1 +4πc2

s0ρ0

B20∥

∂2δρ

∂t2− c2

s0

∂2δρ

∂z2= 0. (7.9)

We therefore obtain the phase velocity of the slow sausage modecT (c.f. Edwin andRoberts, 1983),

c2T =

c2s0v

2A

c2s0 + v2

A

, (7.10)

wherevA is the Alfven velocity.Hereafter we define positive as away from the solar surface. We consider a simple sinu-soidal wave propagating upward (k > 0) or downward (k < 0) along the flux tube ofpositive (B∥ > 0) or negative (B∥ < 0) polarity,

δρ = δρ cos(ωt − kz) (ω = kcT ), (7.11)

whereδρ is the amplitude of the density fluctuation. Substituting Eq. (7.6), we have

δB∥ = −4πc2s0δρ

B∥= −4πc2

s0

B∥δρ cos(ωt − kz), (7.12)

and we have from Eq. (7.4)

ρ0

∂δv∥∂t

= −c2s0kδρ sin(ωt − kz). (7.13)

Taking the integration with time (neglecting integration constant), we have

δv∥ =c2s0

cT

δρ

ρ0

cos(ωt − kz), (7.14)

Assuming that the flux tube has an axis-symmetric sausage oscillation, the transversevelocity averaged over the whole pixels within the flux tube should be canceled out. Thus,what we detect as a clear strong peak in the l.o.s. velocity must be longitudinal, if thefluctuation is due to the propagating slow sausage mode.From Eqs. (7.11), (7.12), and (7.14), we have the phase relations between the fluctuationsin the magnetic field, the velocity, and the density,

δρ

δB∥= −

B∥

4πc2s,0

, (7.15)

δB∥

δv∥= −4πcT ρ0

B∥, (7.16)

δv∥δρ

=c2s,0

cT ρ0

, (7.17)

Although we observedϕB − ϕv ∼ −90◦, Eq. (7.16) indicates that the phase differencebetween the fluctuations in magnetic field and the velocityϕB − ϕv with slow sausage

43

wave propagating to the same or the opposite direction to the magnetic field should havebeen180◦ or 0◦, respectively, as shown in Fig. 7.1. Thus we can rule out the possibilitythat the observed fluctuations are due to the propagating wave with slow sausage mode.

7.2 Phase Relation of Standing Slow Sausage Wave

We here consider the superposition of ascending and the descending slow sausage waveswith the same amplitude of the density fluctuation which has a property of a standingwave, assumingB∥ > 0 from our observation,

δρ = δρ[cos(kcT t − kz + ϕu) + cos(kcT t + kz + ϕd)] =

δρ cos(kcT t +ϕu + ϕd

2) cos(kz +

ϕu − ϕd

2), (7.18)

whereϕu andϕd are the initial phases of the upward and downward propagating waveswith slow sausage mode, andk > 0 without losing generality. From Eqs. (7.6) and (7.18)we have

δB∥ = −4πc2s0

B∥δρ cos(kcT t +

ϕu + ϕd

2) cos(kz +

ϕu − ϕd

2), (7.19)

and from Eqs. (7.4) and (7.18) we have

δv∥ = − c2s0

ρcT

δρ sin(kcT t +ϕu + ϕd

2) sin(kz +

ϕu − ϕd

2). (7.20)

Equations (7.18) and (7.19) indicate that the phase difference between the fluctuations inthe magnetic field and the density is180◦.Equations (7.19) and (7.20) indicate that the phase difference between the fluctuations inthe magnetic field and the velocityϕB − ϕv is 90◦ or −90◦, depending on the distancebetween the boundary and the line formation layer, as shown in Fig. 7.2. It is consistentwith he observed phase relation between the fluctuations in the magnetic field and thevelocityϕB − ϕv ∼ −90◦.In the previous section, we discussed that waves with low intensity fluctuation be consid-ered to be a incompressible mode (such as the kink mode), while those with high intensityfluctuation is considered to be a compressible mode (such as the sausage mode). How-ever, Eq. (7.18) indicates that the density fluctuation and the resultant intensity fluctuationare zero at the nodal points for the standing sausage wave. Thus, there may be cases thatthe standing sausage wave may not show intensity fluctuation with large amplitude.

7.3 Seismology of Photospheric Flux Tubes

We here show that the seismology of magnetic flux tubes is also possible for the sausageMHD oscillation. We assume that the observed fluctuation is due to the superposition ofupward and downward compressible sausage waves for the region #05. This is justifiedby the fact that the region #05 has very high intensity fluctuation (Table 4.1).A schematic behavior of the standing sausage wave is shown in the right panel of Fig.6.4. Substitutingϕu+ϕd

2= 0 (without losing generality) andϕu−ϕd

2= 0 (to make the

44

height atz = 0 the anti-node) into Eqs. (7.18)− (7.20), the variations of the density, thelongitudinal magnetic field, and the longitudinal velocity are given by,

δρ = δρ cos(kcT t) cos(kz), (7.21)

δB∥ = −4πc2s0

B∥δρ cos(kcT t) cos(kz), (7.22)

δv∥ = − c2s0

ρcT

δρ sin(kcT t) sin(kz), (7.23)

Equations (7.22) and (7.23) indicate that the phase difference between the fluctuations inthe magnetic field and the velocityϕB − ϕv is given by,{

90◦ for (n + 12)π ≤ kz ≤ (n + 1)π (sector (a) inFig. 6.4),

−90◦ for nπ ≤ kz ≤ (n + 12)π (sector (b) inFig. 6.4),

(7.24)

wheren = −1,−2,−3, ... . Equation (7.24) indicates that the observed phase differenceϕB − ϕv ∼ −90◦ is consistent with the situation that the line-forming layer is located inthe sector (b).Eqs. (7.22) and (7.23) are reduced to,

|δB∥||δv∥|

=4πc2

s,0δρ| sin(kz)|/B∥

c2s,0δρ| cos(kz)|/ρcT

=4πρcT | tan(kz)|

B∥, (7.25)

where|δB∥| and|δv∥| are amplitudes of longitudinal fluctuations in the magnetic field andthe velocity.B∥, |δB∥|, and|δv∥| are related to the observables,

B∥ = B0, (7.26)

δB∥ =

√2δΦlos,rms

f cos θ, (7.27)

δv∥ =

√2δvlos,rms

cos θ. (7.28)

Sincecs =√

γkBTm

andvA =B∥√4πρ

,

c2T =

c2sv

2A

c2s + v2

A

=γkBTB2

∥

4πργkBT + B2∥m

. (7.29)

From Eqs. (7.25) and (7.29), we have

( |δB∥||δv∥|

)2=

(4πρ)2γkBT | tan(kz)|2

4πργkBT + B2∥m

. (7.30)

Equation (7.30) leads to a second order equation forρ,

a1ρ2 − a2ρ − a3 = 0, (7.31)

a1 = (4π|δv∥|)2γkBT | tan(kz)|2, (7.32)

45

a2 = 4πγkBT |δB∥|2 (7.33)

a3 = (B∥|δB∥|)2m (7.34)

Sinceρ > 0, we can take onlyρ =a2+

√a22+4a1a3

2a1. This indicates that we can determine the

mass density inside the flux tube with the additional knowledge oftan(kz) for the line-forming height. However, there are multiple solutions due to ambiguity intan(kz). Theregion that we chose for the photospheric seismology (region #02 with the assumption ofthe kink wave and #05 with the assumption of the sausage wave) are both pores, whosemagnetic field strength is almost the same. We assume that the parameters of the flux tube(the mass density, plasma beta, and Alfven velocity) and distance between the boundaryand the line formation layer derived from the analysis of the region #05 (sausage-wavedominant) should be consistent with those derived from the analysis of the region #02(kink-wave dominant, section 6.5). The choice ofkz = −585◦ for Eq. (6.30) andkz =−707◦ for Eq. (7.32) in the following exercise is based on the assumption.We first calculate mass density inside the flux tubes for kink waveρi (from Eqs. (6.30 and(6.34)) and slow sausage waveρ (from Eq. (7.31)) assuming kz satisfyingϕB−ϕv ∼ −90◦

for each mode. Once we obtained the mass density, plasmaβ, Alfv en speedvA, phasespeed for each modeck andcT , wavelengthL, and distance between the boundary and theline formation layerd can be calculated in course. A diagram indicatingβ andd for boththe kink wave and the slow sausage wave is shown in Fig. 7.3. We found a solution thatthe both waves have the same value ofβ andd with d less than5.0× 103 km, which is thereasonable value for distance between the line-formation-layer and the transition region.SubstitutingB0 = 1.7 × 103 G, δΦlos,rms = 13.8 G, δvlos,rms = 0.12 km s−1, f = 0.81,θ = 29◦, γ = 5/3, kB = 1.4 × 10−16 erg K−1, T = 1.0 × 104 K, andm = 1.9 × 10−24

g, we obtain the mass density inside the flux tubeρ = 5.2 × 10−8g cm−3. We then derivethe values associated with the flux tube; (1) Alfven speedvA = B0√

4πρ, (2) plasma beta

β = ρkBT/mB2

0/8π, (3) phase speed of the slow sausage modecT , (4) wavelength of the slow

sausage modeL = cT P , whereP is the observed oscillation period, and (5) distancebetween the boundary and the line formation layerd = L |kz|

360. SubstitutingP = 294 s,

we obtainvA = 21 km s−1, β = 0.23, cT = 8.4 km s−1 (sound speedcs = 11 km s−1),L = 2.5 × 103 km, andd = 4.9 × 103 km.The set of parameters derived here satisfy the condition thatρ (or ρi), β, vA, andd derivedhere are consistent with those derived in section 6.5. The distance between the boundaryand the line formation layerd = 4.9× 103 km is consistent with the distance between theline-formation height and the transition region. This indeed indicates that the transitionregion is the reflecting layer for such waves. Note that the wavelengthL is much largerthan the scale heightH = kBT

mg∼ 3.9 × 102 km, and the effect of the gravity has to be

taken into account for more rigorous treatment.

46

Figure 7.1: Schematic images indicating the phase difference between the fluctuationsin the magnetic field and the velocityϕB − ϕv with slow sausage waves propagating tothe same (left) or opposite (right) direction of the magnetic field.ϕB − ϕv should havebeen 180◦ or 0◦ degrees respectively, neither of which is consistent with result from theobservationϕB − ϕv ∼ 90◦.

47

(1) (2) (4)(3)

nodenode nodeanti-nodeanti-nodeboundary

B 0

(a) (e)

(b)

++

(f)

+ +(g)

++++

(c)

++ ++(d)

++

(h)

+ +

Figure 7.2:Top: A standing slow sausage wave along magnetic field lineB0 is dividedinto four parts (1) through (4) each separated by nodes and anti-nodes. The standing waveshould have an anti-node at the transition region as described in section 6.3.Bottom: Timeevolution of the standing slow sausage waves. The wave evolves from (a) to (h), and goesback to (a). The arrows with filled box indicate velocity vector, the arrows with circleperturbed component of magnetic field vector, and+ and− signs perturbed componentof density. The length of the arrows and the number of plus/minus sign indicate themagnitudes of the perturbation at certain space and time points. Schematic representationof the standing slow sausage wave shows that the phase difference between magnetic andvelocity fluctuations (ϕB − ϕv) is 90◦ at the portions (1) and (3), and−90◦ at the portions(2) and (4). Note that the phase difference between the fluctuations in the magnetic fieldand the densityϕB − ϕρ should be always180◦, because the total pressure inside the fluxtube due to magnetic field and plasma is constant in the slow sausage mode (Eq. (7.1))

48

Figure 7.3: A diagram indicating the plasmaβ and the distance between the boundaryand the line-formation layerd for the kink wave (without asterisks) and the slow sausagewave (with asterisks). Each line is derived from the equations in sections 6.5 and 7.3 forkz satisfying the observableϕB − ϕv ∼ −90◦ for each mode. The circle indicates one ofthe solution satisfying the condition that both waves have same value ofβ andd with dless than5×103 km, which is reasonable value for the distance between the line-formationlayer and the transition region. (The condition is based on the fact that the region that wechose for the photospheric seismology (region #02 with the assumption of the kink waveand #05 with the assumption of the slow sausage wave) are both pores, whose magneticfield strength is almost the same) The circle corresponds to kz=−585◦ for the kink waveand kz=−707◦ for the slow sausage wave, which are used to the estimation of physicalparameters for both modes.

49

8 Center-to-Limb Variation

In sections 6 and 7, we showed that the observed phase relations are consistent with bothstanding kink waves and standing slow sausage waves. In this section, we try to distin-guish between the two modes performing center-to-limb observation. Kink waves can beconsidered to be barely observed at the disk center, while sausage waves can be at thelimb. Thus we first estimate the limit ofµ angle(θ) for wave detection in kink mode andslow sausage mode, using the detection limit ofΦlos = 12 G andΦpar > 117 G estimatedin section 2.From Table. 4.1, the upper limit of l.o.s. r.m.s. magnetic flux fluctuationδΦlos,rms is 17.1G, which correspond to the l.o.s. magnetic flux variation of 48.3 G from peak-to-valley,where theθ is 25◦. If we assume pure kink wave propagates along the flux tube, thetransverse component of magnetic flux fluctuation isδBt = 48.4G

sin 25◦= 113 G. The magnetic

field fluctuation perpendicular to the l.o.s. is (113 Gcos 25◦) = 103 G, which is belowthe detection limit ofΦpar of 117 G. Thus we only analyzeΦlos. Assuming that the upperlimit of δBt = 113 G is independent ofθ, the boundary ofθ for wave detection in kinkmode isθk = arcsin 12G

113G= 6◦. Thus kink waves would not be observed atθ < 6◦.

If we assume pure slow sausage wave propagates along the flux tube, the longitudinalcomponent of magnetic flux fluctuation isδBl = 48.4G

cos 25◦= 53 G. The magnetic field fluc-

tuation perpendicular to the l.o.s. is (53 Gsin 25◦) = 22 G, which is below the detectionlimit of Φpar of 117 G. Assuming that the upper limit ofδBl = 53 G is independent ofθ, the boundary ofθ for wave detection in slow sausage mode isθt = arccos 12G

53G= 77◦.

Thus slow sausage waves would not be observed atθ > 77◦.We perform the center-to-limb observation withµ angle of1◦ < θ < 64◦ for 10 poresand 46 plages including the result of section 4. The method of analysis is just same assections 3.2 and 4.1. We counted the number of flux tubes with common strong peaks asfunction ofθ, and the result is tabulated in Table. 8.1.A histogram indicating the probability of plages with strong common peaks is shown inFig. 8.1. The probability should have had singularity of increase or decrease withθ, ifonly kink waves or slow sausage waves exist in the flux tubes, respectively. Fig. 8.1clearly indicate this is not the case, thus we say that both mode exist in the flux tubes.Further statistical analysis with more flux tubes withθ of 0−20◦ and40−60◦ would giveus the information for mode identification.

50

θ (◦) Pore Plage Total0− 10 0/0 1/3 1/310− 20 0/1 1/4 1/520− 30 8/8 6/9 14/1730− 40 1/1 6/16 7/1740− 50 0/0 6/16 7/1750− 60 0/0 1/4 1/460− 70 0/0 0/1 0/1

Total 9/10 16/46 25/56Table 8.1: number of the flux tubes with common strong peaks/number of analyzed fluxtubes

51

Figure 8.1: A histogram indicating the probability of plages with strong common peaks(Top) Bottom graphs suggest that the obtained probability as a function ofθ can be ex-plained by slow sausage waves and kink wave both existing in the observed flux tubes.

52

9 Comparison between Positive and Negative Polarity

In sections 6 and 7, we showed that the observed phase differences in magnetic structuresof positive polarity is concentrated aroundϕB − ϕv ∼ −90◦, ϕv − ϕI ∼ −90◦, andϕI − ϕB ∼ 180◦, which are consistent with both standing kink waves and standing slowsausage waves. If the standing waves also exist in magnetic structures of negative polarity,phase relation associated with the magnetic field should change by 180◦ (i.e. the observedphase difference should beϕB −ϕv ∼ 90◦, ϕv −ϕI ∼ −90◦, ϕI −ϕB ∼ 0◦). We analyzedpores and plages of negative polarity as tabulated in Table. 9.1.

Region Date Time Pore or Plage x1 y1 θ2 FOV cadenceID (UT) (”) (”) (deg) (”) (sec)

# 01 2007 Feb 15 13:18 - 14:28 plage -24 0 1 2.6x82 30# 02 2007 Feb 23 14:28 - 15:38 pore -356 56 22 9.6x82 120# 03 2009 Nov 21 11:00 - 13:00 plage 143 284 19 5.1x82 60# 04 2009 Nov 22 11:00 - 13:00 plage 311 288 26 5.1x82 60# 05 2009 Nov 23 11:00 - 13:00 plage 492 292 37 5.1x82 60# 06 2009 Nov 24 11:00 - 13:00 plage 731 231 53 5.1x82 60# 07 2009 Nov 24 11:00 - 13:00 plage 731 231 53 5.1x82 60# 08 2009 Nov 25 11:00 - 13:00 plage 830 236 64 5.1x82 60# 09 2009 May 27 09:34 - 10:32 plage -50 -542 35 9.6x82 120# 10 2009 May 27 09:34 - 10:32 plage -50 -542 35 9.6x82 120# 11 2009 Jun 02 10:03 - 11:00 pore -325 384 32 9.6x82 120# 12 2009 Jun 02 10:03 - 11:00 pore -325 384 32 9.6x82 120# 13 2009 Jun 04 09:40 - 10:36 pore 42 381 24 9.6x82 120

Table 9.1: List of observed magnetic flux concentrations.

1: X-Y coordinate of the target region. X is to the West, and Y is to the North.2: Helio-longitudinal angle from the meridional line.

We make the time profiles of line-of-sight magnetic flux, line-of-sight velocity, andnormalized intensity, and performed the Fast Fourier Transform to the profiles just sameas Sections 3.2 and 4.1. We found common strong peaks in power spectra of magneticflux, velocity, and intensity in 8 flux tubes out of 13. The properties of the common strongpeaks are summarized in Table. 9.2.We make histograms comparing the phase relations and fluctuation period between posi-tive and negative polarities, as shown in Fig. 9.1. We clearly see that the phase relationsassociated with magnetic flux (i.e.ϕB − ϕv andϕI − ϕB) is differed by 180◦ betweenpositive and negative polarity. The result supports our consideration that the observedphase relation is due to standing kink waves and/or standing slow sausage waves.

53

Region δΦ1los,rms δv2

los,rms

δIcore,rmsIcont

3P 4 ϕB − ϕ5

v ϕv − ϕ5I,core ϕI,core − ϕ5

B

ID (G) (m/s) (%) (min) (deg) (deg) (deg)

#01 7.4 74 0.30 5.5 116 -30 -86- 9.0 34 0.27 5.0 166 139 55- 6.5 56 0.16 4.3 -128 -96 -138- 6.2 35 0.07 4.0 114 -35 -79

#02 9.4 54 0.12 5.8 111 -93 -18#04 12.3 63 0.27 4.4 116 -81 -35#06 5.2 29 0.13 8.8 -163 -151 -46#09 10.7 47 0.44 6.2 85 -71 -14#11 18.3 43 0.72 4.8 -163 159 4#12 21.8 62 1.42 5.8 127 -59 -68#13 15.4 84 0.29 5.2 152 -141 -11

- 15.5 88 0.44 4.7 82 -64 -18

Table 9.2: Physical parameters corresponding to the principal peak in the power spectraof all region of interests (shown in Table. 9.1) with common peaks in the magnetic flux,the velocity, and the photometric intensity.

1: r.m.s. (root mean square) l.o.s. (line of sight) magnetic flux amplitude.2: r.m.s. l.o.s. velocity amplitude.3: r.m.s. intensity fluctuation normalized by the average intensity for line core.4: Fluctuation period.5: phase difference between the fluctuations in the magnetic flux , the velocity, and theline core intensity.

54

Figure 9.1:Top: Histograms of the phase difference between fluctuations in the magneticflux, the velocity, the line-core intensity, and the continuum intensity,ϕB − ϕv (top, left),ϕv−ϕI,core (top right), ϕI,core−ϕB (bottom left), andϕI,core−ϕI,cont (bottom right). Solidlines indicate histograms for positive polarity, while dashed lines for negative polarity.The histograms clearly shows the phase differencesϕB − ϕv andϕI,core − ϕB differs by180◦ between positive and negative polarity, while there are no significant changes inϕv − ϕI,core andϕI,core − ϕI,cont. Bottom: Histogram of the periods of the common peaksin the power spectra. The period is mainly concentrated around4− 6 min, and there is nosignificant change between positive and negative polarity.

55

10 Conclusion and Future Work

We have detected clear signatures of the MHD waves propagating along the magnetic fluxtubes in a form of magnetic, velocity, and intensity sinusoidal waves with exactly the sameperiod. One or two strong and sharp peaks with common periods in the power spectra ofthe l.o.s. magnetic flux, the l.o.s. velocity, and the intensity time profiles are evident inthe pores and the plages. We note that 90% (9/10) of the pores and 35% (16/46) of plageshave such common peaks. Periods of the peaks concentrate at around 3−6 minutes forpores and 4−9 minutes for plages. Phase difference between the l.o.s. magnetic flux(ϕB), the l.o.s. velocity(ϕv), the line core intensity(ϕI,core), and the continuum intensity(ϕI,cont) have striking concentrations at around−90◦ for ϕB −ϕv andϕv −ϕI,core, around180◦ for ϕI,core − ϕB, and around10◦ for ϕI,core − ϕI,cont in case of positive polarity (Fig.4.2).ϕB −ϕv andϕI,core −ϕB differ by 180◦ (Fig. 9.1). These fluctuations are associatedwith the intensity fluctuationsδIcont,rms

Icontand δIcore,rms

Icore. The amplitude of the intensity fluc-

tuations amount to 0.1− 1.0 % of the average intensity level. Some flux tubes have a verysmall intensity fluctuation, and the wave mode for such flux tubes is considered to be theincompressible kink mode. On the other hand, flux tubes with higher intensity fluctuationmay have the compressible sausage mode.The phase relationϕI − ϕB ∼ 180◦ from the observation would not be consistent withthat caused by the opacity effect (e.g. Bellot Rubio et al., 2000), if the magnetic fieldstrength decreases along the line of sight toward the observer. We propose that the longi-tudinal and/or transverse MHD waves propagating along the flux tube are responsible forthe fluctuations. The observed phase differenceϕB − ϕv ∼ −90◦ is consistent with thephase relation of the superposition of the ascending and the descending kink wave. Thisindicates that the ascending kink wave is substantially reflected at the chromospheric-coronal boundary. The superposed waves have the property of the standing waves. Inaddition to the standing kink mode, the observed phase relation between the fluctuationsin the magnetic flux and the velocity is consistent with superposition of the ascending andthe reflected descending slow-mode sausage waves. We tried to distinguish between kinkwaves and slow sausage waves comparing the probability of wave detection for the fluxtubes at the desk center with that far away from disk center. The result suggest the possi-bility that the both waves exist in the flux tubes, although further statistical analysis mustbe required. We performed statistical analysis for magnetic structures of both positiveand negative polarities. The phase relations associated with magnetic field differ by 180◦,while others do not show any significant changes. This result supports that the observedfluctuation is due to the standing kink and/or slow sausage waves.So far our analysis is based on the assumption that the either the kink mode or the sausagemode is dominant in the flux tubes. Both the kink mode and the slow sausage mode maybe excited in the same flux tube. Torsional waves are not discussed in this paper. Theregion of interest encompasses the entire magnetic flux concentrations in the spatial andtemporal domain (i.e. in the case of plages), and we average the physical parameters in-side the ROI. Thus, we are probably unable to detect the torsional Alfven waves, even ifthey exist, because the perturbation of the magnetic flux and the velocity is averaged overthe whole flux tube, and are canceled out.We derive the various physical parameters of the flux tubes only from the observed pe-riod and the amplitudes of magnetic and velocity oscillations. Such parameters include

56

(1) mass density inside and outside the flux tube, (2) plasmaβ inside the flux tube, (3)Alfv en speed inside the flux tube, (4) phase speed, (5) wavelength, (6) distance betweenthe boundary and the line formation layer, and (7) propagation time of fast magneto-acoustic wave across the flux tube. In the examples presented in this paper, we choosesimilar sets ofkz as defined in sections 6.5 and 7.3 for both cases (the kink-wave domi-nant case and the sausage-wave dominant case) such that the derived physical parametersof the flux tubes coincide. The choice determines the distanced between the boundary(node) and the line formation layer. The derived mass density outside the flux tube isconsistent with that of the standard solar model in the case of the kink wave. Note that wecan not derive the mass density outside the flux tube in the case of slow sausage mode,because the flux tube is not in the pressure equilibrium. This exercise demonstrates thatthe seismology of magnetic flux tubes is possible with the observations of the oscillationperiod and amplitudes for various photospheric and chromosheric lines, and may open anew channel for the diagnostics of the magnetic flux tubes.Magnetohydrodymanic waves are believed to play a vital role in the acceleration andheating of the fast solar wind. However, it has been thought that the Alfven speed rapidlyincreases with height due to the rapid decrease in the plasma density, resulting in signif-icant reflection at the chromosphere-corona boundary. We indeed show that this may bethe case in this paper: the upward propagating kink and/or sausage waves must be signif-icantly reflected back above the line formation layer. Deviation in the phase differencebetween the magnetic and velocity fluctuations from−90◦ as seen in Fig. 4.2 may indi-cate residual waves propagating to the corona. Indeed, the upward Poynting flux abovethe reflecting layer is estimated to be2.7 × 106 erg cm−2 s−1 in one case (kink wave).

Tsuneta et al (2008b) conjectures that a rapid decrease in the magnetic field strengthassociated with the rapidly expanding flux tube near the chromosphere-corona boundaryfor the polar kG patches reduces the vertical change in Alfven speed, and the Alfveniccutoff frequency be lower in the polar flux tubes. Magnetohydrodymanic waves gener-ated in the photosphere may be more efficiently propagated to the corona through suchfanning-out flux tubes with large expansion factor observed in the polar coronal holes.On the other hand, the observations presented here suggest significant reflected waves. Itwould therefore be interesting to see whether the reflectivity of the magnetohydrodymanicwaves depends on the locations or environment e.g. coronal holes vs the quiet Sun.We are also interested in dynamics of MHD waves between the photosphere and thecorona. Velocity fluctuations in the coronal loop, which can be considered to be the kinkwaves, have been detected by Van Doorsselaere et al. (2008), Mariska et al. (2008),and so on using EIS. EIS can perform a repeated scan with the narrow slit to detect thefluctuations in the Doppler velocity and in the intensity in the transition region and thelower corona. SP can perform a repeated scan with high time cadence, which allows usto observe apparent fluctuations in the line-of-sight magnetic flux, in the velocity, and inthe intensity at lower photosphere, as we have explained in this thesis. The collaborationbetween SP and EIS allows us to study the properties of MHD waves, such as the type ofMHD wave (standing or propagating, kink or sausage, and so on) and physical parametersin a flux tubes.

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Acknowledgements

First of all, I would like to express my sincere appreciation to my supervisor ProfessorSaku Tsuneta (National Astronomical Observatory of Japan(NAOJ)), for his warm andstrong guidance, useful comments and discussions, and continuous encouragements toachieve my works. He taught me many important things for a scientist, such as methodsof data analysis, making slides, having a presentation, writing a paper, and so on. Healso gave me many chances to present my works such as international conferences, work-shops, and seminars, in which I was given many comments useful for improvement of myworks. I have been going through many fruitful and interesting experiences throughoutthe two years in his laboratory.I would also like to thank to Associate Professor Tamaki Yokoyama (Tokyo University,my formal supervisor) for kindly allowing me to study in NAOJ, giving me many usefulcomments in seminars, and kindly encouraging me to achieve my works. He stimulatedmy interest in solar physics through his classes.I would like to thank the members of the Hinode Science Center in NAOJ, especiallyH.Hara, R.Kano, Y.Katsukawa, S.Imada, T.Magara, N.Narukage, T.J.Okamoto, Y.Masada,D.Orozco, and R.Ishikawa for fruitful comments and discussions that improved my workvery much. I also thank Y.Shiozu, K.Okabe, K.Ueda, N.Sako (NAOJ), Y.Iida, N.Kitagawa(Tokyo University), H. Ito (Nagoya University), and H.Watanabe (Kyoto University) forgetting along with me very well. I thank all the members associated with Hinode ScienceProject for giving me opportunity to study using the interesting and great data obtainedwith Hinode.We gratefully thank M. Ryutova and O. Steiner for the fruitful discussions on the sausagemode and the opacity effect. M. Ryutova helped us to theoretically formulate the proper-ties of the sausage mode in section 7.1. We acknowledge encouragements from E. Priest,B. Roberts, R. Erdelyi, E. Khomenko, N.Yokoi and M. Velli.Finally, I would like to express my largest appreciation to my parents for their continuoussupports and encouragements.

Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborat-ing with NAOJ as a domestic partner, NASA and STFC (UK) as international partners.Scientific operation of the Hinode mission is conducted by the Hinode science team orga-nized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partnercountries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan),STFC (U.K.), NASA, ESA, and NSC (Norway). This work was carried out at the NAOJHinode Science Center, which is supported by the Grant-in-Aid for Creative ScientificResearch ”The Basic Study of Space Weather Prediction” from MEXT, Japan (Head In-vestigator: K. Shibata), generous donations from Sun Microsystems, and NAOJ internalfunding.

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