Hinode and IRIS Observations of. the Solar MHD Waves Propagating from. the Photosphere to the Chromosphere

Transcription

1 Master Thesis Hinode and IRIS Observations of the Solar MHD Waves Propagating from the Photosphere to the Chromosphere Ryuichi Kanoh Department of Earth and Planetary Science Graduate School of Science, The University of Tokyo March, 2016

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3 Abstract The formation mechanisms of the high temperature plasmas in the chromosphere and the corona against the photospheric plasma have been still one of the major problems in solar physics. To heat the solar upper atmosphere, MHD waves excited by the photospheric turbulent motions are thought to be one of important candidates. Although there are many observational studies about MHD waves, quantitative estimation of the energy dissipated in the chromosphere and the corona has not yet been made. In this study, Hinode and IRIS satellites were utilized to investigate the dissipated energy flux, by comparing the energy flux measured at the different heights. We observed periodic oscillations of the physical parameters in a sunspot umbra at both the photosphere and the chromosphere. The period of the dominant oscillations is around 5 min at the photosphere and 3 min at the chromosphere. We found that the photospheric phase difference is about π radian for the intensity and magnetic flux, - π 2 for the doppler velocity and magnetic flux, and π 2 for the doppler velocity and intensity. The observed phase relation implies the superposition of the upward and downward slowmode waves at the photosphere. Since the identified mode is a longitudinal wave, the dominance of 3 min power rather than the 5 min power in the chromosphere can be explained by the acoustic cutoff in the stratified atmosphere. Sawtooth patterns in the temporal profiles of the chromospheric doppler velocity indicate that MHD waves are steepened in the chromosphere. In addition, a clear phase delay of the doppler velocity between the transition region Si IV line and the chromospheric Mg II suggests shock waves propagating from the chromosphere to the transition region. Based on the physical i

4 quantities of the propagating waves identified above, we derived that erg cm 2 s 1 as an upward energy flux at the photosphere and erg cm 2 s 1 as an upward energy flux at the chromosphere. The difference between the photospheric and the chromospheric energy fluxes is larger than the energy required to maintain the chromosphere in the sunspot umbrae, suggesting that the observed MHD waves can make a crucial contribution to the heating of the chromosphere above the sunspot umbrae. However, it is worth noting that there is a possibility that the energy dissipated at the chromosphere might be overestimated because of the opacity effect. The energy flux derived at the chromospheric level is smaller than the energy flux required for heating the corona. It implies that another heating mechanisms such as nanoflares might be important for the coronal heating above the sunspot umbrae. ii

5 MHD MHD Hinode IRIS 5 3 ϕ I ϕ B π, ϕ v ϕ B π, 2 ϕ v ϕ I π Slow-mode 2 acoustic cutoff Mg II Si IV erg cm 2 s erg cm 2 s 1 MHD iii

6 Opacity effect iv

7 Contents Abstract i iii 1 Introduction The chromospheric and coronal heating problem Heating mechanisms Nanoflares MHD Waves Aim of this study Observations and Analysis Basic diagnostics of the Stokes parameters LOS velocity measurement Magnetic field measurement Density measurement Hinode observations Hinode data analysis IRIS observations IRIS data analysis Co-alignment of each observation v

8 3 Results Oscillations at the photosphere observed with Hinode Oscillations at the chromosphere observed with IRIS Discussions Mode identification of the waves The wave mode at the photosphere The wave mode at the chromosphere Energy estimation Energy flux at the photosphere Energy flux at the chromosphere Implications to the solar atmospheric heating Summary and Future works 47 A Error Estimation 50 A.1 Hinode data A.2 IRIS data B Theory of MHD waves 55 B.1 MHD equations B.2 Wave modes B.3 Cutoff frequency C Manuals for IRIS data users 60 C.1 How to use C.2 Characteristics of the spectral lines C.2.1 O I C.2.2 Mg II C.2.3 C II vi

9 C.2.4 Si IV C.2.5 O IV C.2.6 Fe XXI C.2.7 Mg II triplet C.3 Some tips in IRIS data analysis C.3.1 Co-alignment between the SJIs and spectra C.3.2 Dark calibration C.3.3 Fe XXI emission seen in C II SJIs C.4 IRIS operations Acknowledgments 76 References 79 vii

10 Chapter 1 Introduction 1.1 The chromospheric and coronal heating problem Figure 1.1 shows the temperature and the total hydrogen density distribution as a function of the height in the solar atmosphere. From the solar surface to the height where the temperature is at minimum, the layer is defined as the photosphere. From the temperature minimum, the temperature rises toward the million kelvin (MK) corona. The intermediate region between the photosphere and the corona is called the chromosphere. At the top of the chromosphere, there is a narrow layer where the temperature increases drastically from K to a few MK. The layer is called the transition region (TR). The energy source of the sun is at the solar interior. Therefore, the temperature is expected to fall as moving outward from the solar interior. However, the temperature of the solar atmosphere actually increases as moving outward beyond the temperature minimum, instead of dropping. The reason for the temperature rise from the chromosphere has been still unclear and one of the major problems in solar physics. This is called the chromospheric and coronal heating problem. The energy fluxes required at the chromosphere and the corona above the areas of strong magnetic field concentrations, so called active regions, are thought to be around 10 7 erg cm 2 s 1 and 10 6 erg cm 2 s 1, respectively (Withbroe & Noyes, 1977; Anderson & Athay, 1989). Note that because of the coronal 1

11 Figure 1.1: Temperature (solid line) and density (dashed line) distribution along the height in the solar atmosphere. Taken from Withbroe & Noyes (1977) 2

12 low density, the energy flux required for heating the corona is smaller than that of the chromosphere. Understanding the chromospheric and coronal heating mechanisms is important because the heated atmosphere is thought to drive the solar-wind (Parker, 1958), which permeates the heliosphere and influences the near-earth environment. Recent researches with The Mars Atmosphere and Volatile Evolution (MAVEN: Jakosky et al., 2015b) such as Jakosky et al. (2015a) suggest that a major contributor to the long-term evolution of the Mars atmosphere is the solar activities. On the other hand, applicability of the solar physics to the other stars is another interesting motivation to investigate the chromospheric and coronal heating mechanisms. The sun is known as a fairy typical star in the universe. Schmitt & Wichmann (2001) reported an emission from highly ionized iron (Fe XIII at Å) in the corona of the red-dwarf star CN Leonis, using a ground-based telescope. The non-thermal line width of 18.4 km/s indicates great similarities between solar and stellar coronal heating mechanisms. Since it is difficult to investigate the stellar atmosphere in detail because of the lack of spatial resolution, studies on the solar atmosphere become an effective approach to understand the stellar atmosphere. For these reasons, understanding the chromospheric and coronal heating mechanisms is important not only for the solar physics but also for stellar and planetary physics. 1.2 Heating mechanisms Since we cannot understand the chromospheric and coronal heating with thermal conduction from the solar surface, non-thermal energy transportation mechanism is needed to heat the solar atmosphere. The solar atmospheric models such as Gary (2001) indicate that the plasma-beta, which is the ratio of gas pressure to magnetic pressure, becomes less than unity in the chromosphere and the corona. Therefore, magnetic field is believed to play an important role for heating the atmosphere. Actually, the areas of strong magnetic field concentrations have the hot atmosphere (see Figure 1.2). There are two candidates 3

13 Figure 1.2: The solar full disk images taken on 2014 December 16. Left panel shows a magnetogram taken by SDO. White and black colors imply the positive and negative magnetic polarity, respectively. Middle panel shows a broadband Hα image corresponding to the chromosphere taken at the Kanzelhoehe Solar Observatory. Right panel shows a soft X-ray image taken by Hinode X-ray telescope corresponding to the corona. Images are taken from the website that would transport the energy from the lower to upper atmosphere: (1) photospheric motions change the topology of the magnetic field lines, causing many small-scale magnetic reconnections so called nanoflares (Parker, 1972), and releasing free magnetic energy in the upper atmosphere ; (2) magnetohydrodynamic (MHD) waves, which propagate upward along the magnetic field lines and dissipate the energy at the upper atmosphere. In any case, the source of the energy for heating is believed to be in the photospheric fluctuations Nanoflares Solar flares are known to follow a simple scaling law between the number N and total energy E, dn(e) de ( ) α E = A, (1.1) E 0 where A is constant. This scaling law is universal in a broad energy range from large flares and stellar superflares (Maehara et al., 2012; Shibayama et al., 2013) to small transient brightenings (Shimizu et al., 1992). The total thermal energy in range E 1 to E 2 can be 4

14 Figure 1.3: Frequency distribution of the transient brightenings as a function of the total energy derived from three different methods. Each fitted dashed-dotted line s index is Image is taken from Shimizu (1995). calculated as E2 E 1 N(E)EdE = A E α 0 = A E α 0 E2 E α E 1 α de ( ) E 2 α 2 E1 2 α. (1.2) If α < 2, large flares have major contribution to the chromospheric and coronal heating. On the other hand, small flares become dominant heating mechanism if α > 2. Since the corona has hot temperature even without large flares, small-scale brightenings might be important. Therefore, the determination of the index α is quite important when considering the possibility of nanoflare heating. To investigate the statistical properties of the solar small-scale brightenings, automatic recognition techniques for small brightenigs have been developed. By estimating the energy of brightenings from the observed parameters such as the emission measure, Shimizu (1995), Shimojo & Shibata (1999) and Aschwanden & Parnell (2002) found α < 2, suggesting that the total energy released by nanoflares 5

15 is not enough for heating the corona. On the other hand, Krucker & Benz (1998) and Parnell & Jupp (2000) found α > 2. Recently, a statistical study not only for coronal brightenings but also for chromospheric brightenings such as Ellerman bomb (Ellerman, 1917) was done by Nelson et al. (2013). Note that for estimating the energy, previous studies assumed isothermality and uniformity of the plasma for the sake of simplicity. However, we need to be careful about that the results highly depend on these assumptions. Hot plasma and its hard X-ray emission can be other indicators of nanoflares. Nanoflare models such as Klimchuk et al. (2008) predict that the small amounts of very hot (>5 MK) plasma, which may radiates hard X-ray, should be present. Multi-filter X-ray telescope (XRT: Golub et al., 2007) observations suggest that the temperature distributions may have hot distinct components (Schmelz et al., 2009; Reale et al., 2009). In addition, Brosius et al. (2014) resolved the lines of Fe XIX at 592.2Å formed at temperature 8.9 MK with Extreme Ultraviolet Normal Incidence Spectrograph (EUNIS-13) sounding rocket instrument. However, Ishikawa et al. (2014) reported that there is not enough hard X-ray emission at the active region corona by using Focusing Optics X-ray Solar Imager (FOXSI) sounding rocket. It is notable that direct observation of the nanoflares has not been done because of their tiny scale. Therefore, observations with high spatial and temporal resolution are needed for future studies to identify the nanoflares directly MHD Waves Compressible magnetoacoustic waves are steepened to form shock waves and its dissipation might contribute to the heating of the atmosphere. Magnetoacoustic waves generated at the photosphere are thought to be insufficient driver to heat the solar corona because of rapid dissipation before reaching the corona (Mein & Schmieder, 1981; Anderson & Athay, 1989). Therefore, such waves are currently considered as a possible candidate for heating the chromosphere and its discussions are ongoing (Carlsson et al., 2007; Bello González et al., 2010). 6

16 Figure 1.4: An example of the transverse displacement of spicules. (A) is a time-slice image along the white line in (B) to (F). Taken from De Pontieu et al. (2007) Figure 1.5: Time profiles and their Fourier power of the LOS magnetic flux, the LOS velocity and the intensity observed in a sunspot pore. These datasets are analyzed in Fujimura & Tsuneta (2009). The circles indicate the strong common Fourier peaks. Image is taken from Fujimura & Tsuneta (2009). 7

17 Alfvén waves are waves in incompressible wave modes and difficult to form the shock waves. Therefore, it is thought to carry much energy to the corona. Coronal Multi- Channel Polarimeter (CoMP), Hinode (Kosugi et al., 2007) and Atmospheric Imaging Assembly (AIA: Pesnell et al., 2012) onboard Solar Dynamics Observatory (SDO: Pesnell et al., 2012) found that the solar atmosphere is filled with Alfvén waves. Tomczyk et al. (2007) detected oscillations in coronal loop structures observed in the time series of the line-of-sight (LOS) velocity, the intensity, and the linear polarization images. Okamoto et al. (2007) and De Pontieu et al. (2007) found the transverse oscillations of the chromospheric prominences and spicules, that are frozen in the magnetic field lines, suggesting the existence of Alfvén waves (Figure 1.4). Although Alfvén waves are difficult to dissipate the energy compared to the compressible waves, there are some studies that report on the energy dissipation of Alfvén waves in the solar corona (Hahn et al., 2012; Hahn & Savin, 2013). In addition, the Interface Region Imaging Spectrograph (IRIS: De Pontieu et al., 2014) with the Solar Optical Telescope (SOT: Tsuneta et al., 2008b; Suematsu et al., 2008; Shimizu et al., 2008; Ichimoto et al., 2008) onboard the Hinode provide a spectroscopic measurement of oscillations in chromospheric prominence threads, suggesting the resonance absorption of Alfvén waves and its subsequent heating (Okamoto et al., 2015; Antolin et al., 2015). It should be noted that almost all of the previous studies on MHD waves do not evaluate the energy flux quantitatively because of the lack of magnetic field and density measurements. The energy flux F is written by F = ρδv 2 v g + (δv B) δb, (1.3) where ρ, B, v and v g are the mass density, the magnetic field strength, the velocity amplitude and the group velocity, respectively. The first and second terms on the right hand side are thermal-kinetic energy flux and poynting flux, respectively. From this equation, we can confirm the importance of magnetic field and density measurements. However, 8

18 because of the lack of physical parameter measurements, previous studies such as Bello González et al. (2010) and Kitagawa et al. (2010) assumed some physical parameters such as the density and the magnetic field strength for estimating the energy flux. Observations of the magnetic field fluctuation are attempted with ground-based telescopes (Landgraf, 1997; Lites et al., 1998; Bellot Rubio et al., 2000). However, they do not have confidence that the observed magnetic fluctuations are intrinsic because of the instrumental crosstalk, which is unwanted signal. Studies with space observatories have also been carried out so far. Rueedi et al. (1998) and Norton et al. (1999) analyzed the velocity and magnetic field oscillations observed in sunspots using the Michelson Doppler Imager (MDI) on NASA s Solar and Heliospheric Observatory (SOHO). As the latest achievements, Fujimura & Tsuneta (2009) reported that the spectropolarimeter onboard the Hinode satellite can catch the weak magnetic field fluctuations accurately at the photosphere. An example of their observed dataset is shown in Figure 1.5. Figure 1.5 shows time profiles and their Fourier powers of the LOS magnetic flux, the LOS velocity and the intensity observed in a sunspot pore. By using these observed amplitudes and phase relations, as we will see in later discussions, they identified a wave mode and derived photospheric energy flux. Note that the mode identification of MHD waves is crucial to estimate the energy flux. For example, the poynting flux term in equation 1.3 depends on whether the waves are transverse (v B) or longitudinal (v B). For studying the propagating processes of MHD waves, Okamoto & De Pontieu (2011) investigated the propagating processes of MHD waves along the spicules in the coronal hole with image processing technique and found that there are the mix of upward and downward propagating waves along the spicules. However, in their study, there is no quantitative information about magnetic field. For quantitative study, the seismology is one of the important techniques. It uses the properties in the waves to infer physical parameters of the solar atmosphere that are hard to measure directly. For example, Van Doorsselaere et al. (2008) estimated magnetic field strength of the coronal loop without polarimetric observation. Not only the coronal seismology but also the photospheric 9

19 Figure 1.6: Schematic drawing of the logic for estimating the dissipated energy flux seismology is developed by Fujimura & Tsuneta (2009), which estimates photospheric density even in optically thick region. Recently, Moreels & Van Doorsselaere (2013) improved a theoretical modeling of Fujimura & Tsuneta (2009). The modeling is the photospheric seismology, which is also a strong tool to identify the wave mode. 1.3 Aim of this study The fundamental motivation of this study is to estimate the dissipated energy of MHD waves at the solar atmosphere with mode identification. We emphasize that there are no study with identifying wave mode and estimating dissipated energy simultaneously. For identifying the mode of MHD waves, as we will see later, the temporal evolution of the physical parameters is important. Therefore, the data obtained with ground-based telescopes is not suitable to investigate the temporal evolution because of seeing effect. In addition, spectroscopic observations, not imaging, are more suited to detect the fluctuations caused by MHD waves quantitatively. For these reasons, Hinode and IRIS satellites are ideally suited to detect the fluctuations. Hinode can reveal the photospheric tiny fluctuations of the physical parameters including the magnetic flux accurately with spectropolarimetric observation. Moreover, recent launched IRIS performs spectroscopic observation to derive the physical quantities of the chromospheric and the TR plasma that have not yet been discussed with high-cadence and continuous observations. The combination of these satellites makes us possible to understand the behaviors of MHD waves 10

20 at the lower and the upper atmosphere at the same time. We estimate the dissipated energy flux by comparing the energy flux both in the photosphere and the chromosphere as shown in Figure 1.6. Even though Fujimura & Tsuneta (2009) is a very good research for quantitative energy estimation, it is worth noting that their dataset is observed only in the photosphere, limiting quantitative understanding of propagation and dissipation mechanisms of waves. In addition, their temporal resolution (67 sec) is not enough to resolve the waveform as seen in Figure 1.5. They focused only on one strong Fourier power peak for quantitative investigation such as mode identification and energy estimation. However, it is risky for quantitative study, since it might depend on the window of the Fourier transformation. For these reasons, a high-cadence study with Hinode and IRIS is needed to study the energy balance of the MHD waves. This paper presents Hinode and IRIS highcadence simultaneous observations and their results, that are used for discussing energy dissipation mechanisms of MHD waves at the solar atmosphere. We describe observational methodologies in Chapter 2. Their results and interpretations are in Chapter 3 and 4, respectively. A summary of this thesis and future works are given in Chapter 5. 11

21 Chapter 2 Observations and Analysis 2.1 Basic diagnostics of the Stokes parameters Stokes parameters can be strong tools for plasma diagnostics. There are four kinds of Stokes parameters, Stokes I, Q, U, and V. Stokes I is an intensity of the light. Stokes Q and U are linear polarization signal, which corresponds to transverse magnetic field against the LOS. Stokes V is a circular polarization signal corresponding to the LOS magnetic fields. The Stokes parameters are defined with amplitude of electric field by I Ex 2 + E 2 y, (2.1) Q Ex 2 E 2 y, (2.2) U Ea 2 E 2 b, (2.3) V El 2 E 2 r, (2.4) where the subscripts refer to three different bases : the standard Cartesian basis (x, y), a Cartesian basis rotated by 45 (a, b), and a circular basis (l, r). 12

22 Figure 2.1: A schematic picture of Stokes I (left) and V (right). Dashed line and solid line are not shifted and redshifted, respectively LOS velocity measurement LOS velocity is derived by using the Doppler effect. The Doppler effect changes the frequency of a wave when an observer moves relative to its source. The doppler velocity is calculated as v = c δλ λ 0, (2.5) where c is the light speed, λ 0 is the reference wavelength and δλ is wavelength deviation from λ 0. If there is a spectral line, the Doppler effect shifts the spectral line to the wavelength direction as shown in Figure 2.1. Note that not only Stokes I, but also Stokes Q, U and V are also shifted with doppler effect. In this study, Stokes V is used for measuring the doppler velocity Magnetic field measurement Magnetic field is measured by using the Zeeman effect. The Zeeman effect splits degenerate states and its spectral broadening depends on magnetic field strength. The Zeeman 13

23 Figure 2.2: A schematic picture of Stokes V with strong (red) and weak (black) LOS magnetic flux. Dashed and solid lines imply the before and after subtracting each polarization signal (l, r), respectively. wavelength splitting is known to be λ B = (m l g l m u g u )λ B, (2.6) where subscription means two levels (l is for lower and u is for upper). m and g are the magnetic quantum number and the Landé factor, respectively (del Toro Iniesta, 2007). Here λ B λ 0ν L, (2.7) c where ν L e 0B 4πm e c (2.8) is the Larmor frequency, e 0 is the charge, m e is the mass of the electron, and c is the speed of the light. The Milne-Eddington fitting (del Toro Iniesta, 2007) is one of the famous inversions of polarization measurement to derive three components of magnetic field strength. Assuming that the source function is a linear function and other physical parameters are constant along the LOS, we can obtain 12 physical parameters such as the magnetic field strength by fitting Unno-Rachkovsky solution (Unno, 1956; Rachkovsky, 14

24 Figure 2.3: Grotrian diagram of the O IV levels in the atomic model, where the black arrows represent allowed transitions and the gray arrows represent forbidden transitions. Taken from Olluri et al. (2013). 1962, 1963, 1967) to observed Stokes parameters. As the LOS magnetic field becomes stronger, the amplitude of Stokes V signal arises and then the absolute area of Stokes V becomes bigger because of the reducing the cancelled area as shown in Figure 2.2. Therefore, by using the area of Stokes V, we can estimate LOS magnetic field strength. This is called as weak-field approximation Density measurement Measuring electron densities using the flux ratio of two spectral lines is a popular diagnostic technique. In this section, we summarize basic mechanisms for diagnostics of the electron density, based on Phillips et al. (2012). We consider the equilibrium between excitation and de-excitation to and from levels i and g, N g N e C e gi = N i (N e C d ig + A ig ), (2.9) 15

25 Figure 2.4: Line ratio of O IV Å/ Å as a function of electron density, calculated by using CHIANTI database. 16

26 where A ig is the Einstein coefficient for the spontaneous emission in the transition from i to g and N e is the electron density. N g and N i are the number of ions in level g and level i, respectively. Cig e and Cgi d are collisional excitation rate and collisional de-excitation rate, respectively. The left hand side and right hand side of the equation (2.9) mean the excitation and de-excitation, respectively. Below is a variation of equation (2.9), N i N g = N e Cgi e N e Cig d + A. (2.10) ig When we use two forbidden lines, in the cases where radiative cascades from higher levels can be neglected, the population of these metastable levels is given by N i N g = N e Cgi e j<i A ij + N e j<i C, (2.11) ij where g is the ground level. In case of N e j<i C ij << j<i A ij, we have N i N g = Cgi e j<i A N e. (2.12) ij On the other hand, in case of N e j<i C ij >> j<i A ij, we have N i N g = Cgi e j<i C. (2.13) ij These equations imply that the level populations depend linearly on density when N e j<i C ij is enough smaller than j<i A ij, and are constant otherwise. In an optically thin atmosphere, the intensity only depends on the density and temperature of the medium. The emissivity ϵ ν = hν 4π N ia ij (2.14) is now electron density dependent through the population density of the upper level. Assuming that the intensity behaves like the emissivity, the ratio produces a value for the electron density. The ratio of intensity from two emission lines O IV Å/1401.2Å 17

27 as a function of electron density derived from CHIANTI v8.0 (Del Zanna et al., 2015) is shown in Figure Hinode observations The SOT onboard the Hinode satellite observed a well-developed leading sunspot of NOAA Active Region In this paper, we study the energy balance in the sunspot umbrae. Although the sunspot is only a part of the solar surface, investigations of MHD waves in the sunspot umbrae can give a better opportunity to understand the fundamental physical mechanisms because of its simple structures. We focus on the observation taken from 15:39 to 16:31 UT on 4 September The sunspot was located at (x,y)=(510,75 ). The SOT s spectropolarimeter (SP: Lites et al., 2013) of SOT recorded the four Stokes profiles of the Fe I lines at Å and Å with a spectral sampling of må (Figure2.5). Sparse raster scanning was used to achieve higher temporal cadence for a wider field of view. A 3.8 arcsec range was mapped with measurements at 12 slit positions; one spectral measurement with the slit width of 0.15 and then the next measurement after moving 0.30 in the west direction. The exposure time and average cadence are 1.6 sec and 27 sec, respectively. In the slit direction, the two pixels were summed, providing the spectral data with the pixel size of We used the calibrated Stokes data (Level 1 data), calibrated by the SOT-SP calibration software (Lites & Ichimoto, 2013) and available via CSAC at HAO/NCAR. 2.3 Hinode data analysis For the detection of weak magnetic fluctuations, Stokes V is preferable instead of Stokes Q and U because of its much higher sensitivity. The detection limits of the longitudinal and transverse magnetic fields observed by the SOT-SP are known to be 1 5 G and G, respectively (Tsuneta et al., 2008b). We used the Stokes I and V profiles of 18

28 Figure 2.5: Typical averaged Stokes I, Q, U and V profiles in the sunspot umbra observed with Hinode/SOT-SP. The profiles are normalized to the continuum intensity. 19

29 the Fe I Å line to derive the LOS velocity, the LOS magnetic flux and the intensity. The LOS velocity is derived by using the Stokes V zero cross-position instead of the line center of the Stokes I. It is because that the Stokes I contains the information of the non-magnetic atmosphere. The LOS magnetic flux is derived with the area of Stokes V (so called weak-field approximation) instead of Milne-Eddington fitting, which may be subject to noise that impedes the detection of fluctuation caused by MHD waves because of many free parameters. The validity of the weak-field approximation inside sunspot umbrae is guaranteed by Felipe et al. (2014) for Fe I Å line with their synthetic observation. For doing the weak-field approximation, we first calculate the degree of the circular polarization CP as defined by CP = V Icont, (2.15) where and Å V V (λ)dλ (2.16) Å Å Icont I(λ)dλ. (2.17) Å The CP is related to the parameters derived from Milne-Eddington inversion as follows, CP fb LOS, (2.18) where f and B LOS are filling factor and LOS magnetic intrinsic field strength, respectively. The coefficient of proportionality is derived from a linear regression to scatter plots between the CP and the LOS magnetic flux derived by the Milne-Eddington fitting (Figure 2.6). 20

30 Figure 2.6: Scatter plots of the LOS magnetic flux and the circular polarization as defined in equation (2.15). The red dashed line indicates a linear regression line. 21

31 2.4 IRIS observations IRIS observed the same sunspot at the almost same periods (16:00-17:54 UT on 4 September 2013). The IRIS performs spectroscopic observations of the chromosphere and the TR. The UV spectral data was acquired with sit-and-stare mode; the slit (0.33 width) was pointed at one solar location and its position was in the SOT-SP s FOV. The spectral lines identified as Si IV at 1403Å ( K), O IV at 1400Å, 1401Å, 1405Å ( K), Mg II at 2796Å and 2803Å ( K) were measured as shown in Figure 2.7. The pixel size along the slit direction is 0.17 and the cadence is 3 seconds. At the same time, the series of the slitjaw images (SJI) for Si IV ( K), C II ( K), Mg II ( K) and Mg II wing ( K) were obtained every 12 seconds. Their field of view is 35 x40 (Figure 2.8) and they are used to identify the exact location of the slit on the solar features. We used the level 2 data created with the instrumental calibration including the dark current subtraction, flat field, and geometrical corrections (De Pontieu et al., 2014). 2.5 IRIS data analysis We applied a single gaussian fit to the Mg II 2796Å and Si IV 1403Å spectra to derive the LOS velocity at two different temperatures. Here the center position of the spectral lines averaged over the field of view are used as the reference wavelength. Since the Mg II spectra, which have the large opacity, is formed in a non-local thermodynamic equilibrium condition, central reversals are typically observed in the line core (Leenaarts et al., 2013a,b; Pereira et al., 2013). However, note that the Mg II lines have no central reversed profiles in sunspots as reported by Morrill et al. (2001). The inter-combination multiplet of O IV lines at Å, Å, Å, Å and Å provides a well known set of density-sensitive pairs. Since the O IV lines at Å and Å are outside the spectral window of our dataset, we can use Å, Å and Å lines. However, the line at Å is blended with a S IV line as reported in Young 22

32 Figure 2.7: Typical averaged line profiles in the sunspot umbra observed with IRIS. The spectral sampling of top and bottom panels is må and må, respectively. Figure 2.8: IRIS SJI images in different filters. 23

33 (2015). We thus used the ratio between the Å and Å. As seen in Figure 2.4, the line ratio is monotonically increased as a function of the electron density. Therefore, we used the line ratio for specifying electron density. 2.6 Co-alignment of each observation The data co-alignment between the Hinode and IRIS is done by using the Mg II wing SJI image and the Fe I continuum image. Photospheric sunspot features such as umbra and penumbra worked as fiducial features in the co-alignment. We applied the IDL procedure get correl offsets.pro to get a rigid displacement of the SJI image from the Fe I continuum image in reference. Note that the co-aligned SP maps were stretched in the X direction because of the sparse raster mapping. In addition, the pixel scale is aligned to the IRIS SJI images by using the IDL procedure congrid.pro. The accuracy of the co-alignment is better than 0.5 according to the visual inspection of the co-aligned data. The aligned FOV is shown in Figure

34 Figure 2.9: (a)(b): Co-aligned IRIS SJI image in Mg II wing and Hinode/SOT-SP continuum map. Yellow dotted line is at the IRIS slit. (c): Ca II H image of the sunspot in NOAA Active Region at 16:00 UT on 4 September, observed with SOT s filtergraph. The red square and yellow dashed line give the field of view of Hinode/SOT-SP and IRIS raster, respectively. The purple square gives the field of view used in the subsequent chapters. 25

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36 Chapter 3 Results 3.1 Oscillations at the photosphere observed with Hinode Figure 3.1 shows the temporal evolution of the magnetic flux, the doppler velocity and the line core intensity derived from the SP data averaged in the 6 x 6 pixels (1.92x1.92) area inside the sunspot umbra. Periodic oscillating features are visible in the profiles. The dominant periods are around 5 min. We subtracted the 12 points (324 sec) running average from the original time series data to remove the long-term gradual change in the profiles. We derive the amplitude of the fluctuations by multiplying 2 to the root-meansquare values. Since the observed sunspot is not at the disk center, we divided them by cos θ, where θ is a heliolongitudinal angle 31 degree from the meridional line. Here it is assumed that the observed fluctuations are mainly in the direction of the umbral magnetic field, which is normal to the solar surface. This assumption will be supported by the mode identification later. The results are tabulated in Table 3.1. The noise levels are estimated in Appendix A.1. To check the phase relations between the LOS magnetic flux, the LOS velocity and the intensity, we checked cross-correlations among the physical parameters. Figure 3.2 shows the cross-correlations as a function of time lag. The maximum correlation coefficient 27

37 Figure 3.1: Time series of the doppler velocity, the magnetic flux and the intensity, respectively. Lower figures are residuals after subtracting 12 points running average from the original time series data. 28

38 Figure 3.2: Correlation coefficient for physical parameters observed in the sunspot umbra as a function of time lag. Each symbol implies the pair of physical parameters. Blackdiamond, blue-asterisk, red-triangle and orange-square show correlation coefficients between δi core δb, δv δi core, δv δb and δi core δi cont respectively. 29

39 δbz δvz δicore δicont δbz/b 0 δicont/icont δicore/icont (G) (km/s) (DN/s) (DN/s) (%) (%) (%) Table 3.1: Physical parameters obtained in the sunspot umbral photosphere with Hinode/SOT-SP between δi core and δi cont without time lag means that there is no phase shift between I core and I cont. The minimum correlation coefficient between δi and δb without time lag implies their out-phase relation. On the other hand, other correlation coefficients are close to zero when the time lag is zero and they gradually increase with time lag. Figure 3.2 shows that the phase differences ϕ v ϕ B and ϕ v ϕ I are π 2 radian. A similar phase relation is obtained by Fujimura & Tsuneta (2009) for pores and plages. The phase relations between the fluctuations in the LOS magnetic flux, the LOS velocity, and the intensity are important to identify wave modes and properties of MHD waves as we will see later. Note that the phase relations discussed above are common in the sunspot umbra. It can be checked in Figure 3.3, showing the spatial distribution of the cross-correlation coefficients of the physical parameters at the three time lags. 3.2 Oscillations at the chromosphere observed with IRIS Figure 3.4 shows a comparison between the doppler shift of the photospheric Fe I Å line observed with Hinode and that of the TR Si IV line observed with IRIS. Note that the LOS velocity derived from IRIS spectra is averaged in 3 pixels along the slit, which overlaps to the region of interest in the photosphere. Compared to the Hinode data, the sawtooth pattern - slow evolution of the velocity toward redshift followed by a sudden blueshift - is clearly seen in the temporal evolution of the chromospheric LOS velocity. High frequency waves stand out in IRIS data, which can be seen in the power spectra shown in Figure 3.5. We note that before calculating the Fourier transform, we subtracted the 324 sec running average from the original data obtained by both Hinode and 30

40 Figure 3.3: Spatial distribution of the cross-correlation coefficients of the physical parameters with the time lag (-54 sec, 0 sec and +54 sec). Field of view is in the sunspot umbra and penumbra, shown in the left intensity image. Top, middle and bottom rows show the distribution of the cross-correlation coefficient between δi core δb, δv δb, δv δi core, respectively. 31

41 Figure 3.4: Temporal evolution of the doppler velocities derived from Fe I (photospheric) line obtained by Hinode and Si IV (TR) line obtained by IRIS. The observed region is in the sunspot umbra. Positive and negative values are blueshift and redshift, respectively. An enlarged view of one period waveform is plotted in right side. Red and blue dashed lines are manually fitted to the waveforms as an indicator of the velocity gradient. IRIS in order to reduce the influence of the long trend, which is not well expressed by Fourier transform. Therefore, the power spectra in < 3 mhz is not reliable. Similar high-frequency enhancements are reported by Centeno et al. (2006, 2009) in the sunspot umbra with the photospheric Si I line and the chromospheric He I line obtained by the Tenerife Infrared Polarimeter (TIP) aboard Vacuum Tower Telescope (VTT). Figure 3.6 shows doppler velocities of the chromospheric Mg II line and the TR Si IV line as a function of time. The oscillation in the Si IV time profile is about 20 sec delayed from the oscillation in the Mg II profile. The amplitude of the Mg II and Si IV oscillations is 2.0 km/s and 6.2 km/s, respectively. The electron density (N e ) derived by using an emission line pair (O IV Å/1401.2Å) is /cm 3, which is contained in the good density sensitivity range for O IV Å and Å pairs (see Figure 2.4). We should note that our density diagnostics is based on the assumption of the equilibrium ionization. However, the actual solar atmosphere is not in equilibrium ionization. Considering the non-equilibrium ionization, Olluri et al. (2013) and Young (2015) reported that the elec- 32

42 Figure 3.5: Normalized LOS velocity power spectra in the sunspot umbra. Black and red colors show the photospheric (black) and chromospheric (red) power spectra. Blue dotted line is a noise level for the photospheric power, which is calculated from the average in >10 mhz. tron density derived by the line ratio method with O IV lines might be overestimated by up to several factors. 33

43 Figure 3.6: Temporal evolution of the LOS velocity at the sunspot umbra. The LOS velocities derived from Mg II and Si IV are plotted by red and black line, respectively. 34

44

45 Chapter 4 Discussions 4.1 Mode identification of the waves The wave mode at the photosphere First, we will show that the intensity fluctuations can be considered as perturbation in the electron density and thus in gas pressure. The continuum intensity is given by Icont = 2hν c 2 1 exp (hν/k B T ) 1, (4.1) where h is the Planck constant, k B is the Boltzman constant, T is the temperature, ν is the frequency, and c is the speed of light. Considering density perturbation ρ = ρ 0 + δρ, Moreels & Van Doorsselaere (2013) calculated that the intensity can be approximated by Icont I hν δρ I 0, (4.2) 3 k B T 0 ρ 0 where I 0 is given by I 0 = 2hν3 c 2 ( exp hν ), (4.3) k B T 0 where Moreels & Van Doorsselaere (2013) assumed that the specific heat ratio γ is 5 3 and that the gas is ideal and adiabatic. From equation (4.2), we can say that I is positively 36

46 correlated to δρ. In adiabatic condition (pρ γ = C), the gas pressure is positively correlated to ρ, where p, γ and C are gas pressure, specific heat ratio and constant, respectively. Therefore, we can interpret the increase of intensity as an increase of gas pressure. For the incompressible mode, density fluctuation and the resultant intensity fluctuation are zero. However, our observational results suggest that the intensity also oscillates with the doppler velocity and the magnetic flux. Therefore, we can exclude the incompressible mode for the observed MHD waves. In the MHD theory, there are two compressible wave modes, i.e., fast-mode and slow-mode. The difference between the fast-mode and the slow-mode waves is the phase relation of restoring forces. For the fast-mode waves, the phase relation between the gas pressure and magnetic pressure is in-phase. It becomes opposite for the slow-mode waves, i.e., the out-phase relation between the gas pressure and magnetic pressure. Since the gas pressure and magnetic pressure are proportional to the intensity and magnetic flux, respectively, the phase difference between the magnetic flux and the intensity fluctuations is 0 for the case of the fast-mode wave, while it is π radian for the case of the slow-mode wave. Thus, we can rule out the fast-mode wave, since the observed phase difference is close to π radian. Next, we will show that using other phase relations (ϕ v ϕ B, ϕ v ϕ I ), we can distinguish the propagating and standing wave modes. We model a flux tube as a straight cylinder. The plasma is uniform both inside and outside the cylinder with a possible jump at the boundary. We point out that this model does not consider the gravity and the Coriolis force. By using the MHD equations in a straight cylinder shown in Appendix B, Moreels & Van Doorsselaere (2013) calculated the z-component of the velocity and the magnetic flux perturbations. For propagating (δρ = ρ 0 ρ cos(ωt kz)) slow-mode waves, 37

47 Wave mode ϕ I ϕ B ϕ v ϕ B ϕ v ϕ I slow propagating π π 0 slow standing π ±π/2 ±π/2 Table 4.1: Phase differences between the LOS velocity, the LOS magnetic flux and the intensity fluctuations for different wave modes. the results are δb = B 0 ω 2 k 2 c 2 s ω 2 ρ cos(ωt kz), (4.4) δv = c2 s ρ sin(ωt kz), ω/k (4.5) δi = 2 hν 3 k B T I 0 ρ cos(ωt kz), (4.6) where ω, t and k are angular frequency, time and wave number, respectively. Thus, the equations (4.4), (4.5) and (4.6) tell that ϕ I ϕ B = π radian, ϕ v ϕ B = π radian and ϕ v ϕ I = 0 radian. On the other hands, For standing (δρ = ρ 0 ρ [cos(ωt kz) + cos(ωt + kz)] = ρ 0 ρ cos(ωt) sin(kz)) slow-mode waves, the results are δb = B 0 ω 2 k 2 c 2 s ω 2 ρ cos(ωt) sin(kz), (4.7) δv = c2 s ρ sin(ωt) cos(kz), ω/k (4.8) δi = 2 hν 3 k B T I 0 ρ cos(ωt) sin(kz). (4.9) The equations (4.7), (4.8) and (4.9) give that ϕ I ϕ B = π radian, ϕ v ϕ B = ± π 2 radian and ϕ v ϕ I = ± π 2 radian. These results are summarized in Table 4.1. Our observed phase relations (ϕ I ϕ B = π radian, ϕ v ϕ B = π 2 radian and ϕ v ϕ I = π 2 radian) are consistent with those of the slow-standing wave. Periodic fluctuations imply that the excitation source of the observed waves is a global eigenmode oscillation (Goldreich & Keeley, 1977). The observed period ( 5 min) is consistent with the typical frequency of global eigenmode oscillations. Since the wave 38

48 Figure 4.1: The pressure profiles of a finite-amplitude wave at three times in the formation of a shock wave. front of the global oscillations is likely to be perpendicular to the solar surface because of the gravitational stratification, the observed longitudinal waves are easy to interpret The wave mode at the chromosphere Since IRIS cannot perform spectropolarimetric observations, we cannot identify the wave mode by using the phase relations of observed parameters. However, IRIS observes not only one line but several lines. Considering the different formation heights of the chromospheric Mg II line and the TR Si IV line, the clear phase difference in these lines shown in Figure 3.6 implies that the chromospheric waves propagate upward. A time lag between Mg II and Si IV is around 20 second. The difference in the line formation heights of Mg II and Si IV is about 0.5 Mm (Rathore et al., 2015a), and thus their propagating speed is roughly 25km/s, which is close to the sound speed in the atmosphere where Mg II (T K and c s 15 km/s) and Si IV (T K and c s 40 km/s) are formed. The velocity amplitude observed with IRIS is larger than that with Hinode because of the density stratification caused by the gravity. As a result, steepening is observed with IRIS as a consequence of shock formation and energy dissipation. Since longitudinal waves are easily steepened compared to transverse waves, the observed steepening signature also supports the identified slow-mode waves at the photosphere. The dominant frequency of the chromospheric waves is 7 mhz, whereas the observed dominant frequency is

49 mhz at the photosphere. The change of the dominant power to higher frequency can be explained with the acoustic cutoff. The oscillations below the cutoff frequency do not propagate upward. On the other hand, above the cutoff value, waves propagate upward freely into the chromosphere. Photospheric standing mode is a consequence of cut and reflected waves, because the frequencies of almost all the photospheric waves are below the cutoff frequency, which is roughly 6 mhz, i.e. the lower edge of the strong IRIS power (Figure 3.5). 4.2 Energy estimation The energy flux of the slow-mode wave is described as F = ρδv 2 v g, (4.10) where ρ is the density, δv is the amplitude of doppler velocity and v g is the group velocity. Note that since the direction of δv is same as B in case of slow-mode waves, poynting flux, S = (δv B) δb, (4.11) becomes zero Energy flux at the photosphere For estimating the energy flux, we need to estimate the mass density at the photospheric height. By using equations (4.7) and (4.9), the photospheric phase speed can be written by ω k = c δi/i 0 s δb/b 0 [ 2 hν 3 k B T + δi/i ] 1/2 0 (4.12) δb/b 0 The phase speed of slow (sausage) mode waves is close to c T (Edwin & Roberts, 1982), where the tube speed c T = c sv A, the sound speed c c 2 s = s +va 2 γk B T m and the Alfvén 40

50 speed v A = B 0 4πρ. Therefore, the comparison between equation (4.12) and c T gives ρ = g/cm 3 by substituting the observed parameters. Figure 3.5 suggests that the waves with the frequency above 6 mhz can penetrate to the chromosphere. Since Centeno et al. (2009) suggest that the waves reachable to the chromosphere come directly from the photosphere by means of linear wave propagation rather than from nonlinear interaction of modes. Thus, the upward energy flux at the photosphere (F Hinode ) is estimated by using the doppler velocity amplitude δv = km/s, which is derived from the 6 to 10 mhz data. Note that the strong IRIS power exists in the 6-10 mhz range. The waves in the 6 10 mhz may propagate to the chromosphere because of the frequency higher than the cut off frequency. With ρ = g/cm 3 and v g = c s = 5.4 km/s, we derived F Hinode = erg cm 2 s Energy flux at the chromosphere The energy flux of the waves at the formation height of the Si IV line is estimated by using the observed velocity amplitude δv = 6.2 km/s, the sound speed of 40 km/s calculated using the formation temperature of Si IV and the mass density of g/cm 3. The mass density ρ is estimated by ρ = N e µm p with the estimated N e, the proton mass m p = g and µ = 1.25 assuming the solar atomic abundance H : He = 3 : 1. From these parameters, we obtained the energy flux of erg cm 2 s 1. Nindos et al. (2000) reported that sunspot temperatures and emission measures are still lower than the average active region parameters but higher than the quiet region plasma parameters. Since the coronal energy loss at the quiet region is erg cm 2 s 1, which is larger than our estimated chromospheric energy flux, we can say that our estimated chromospheric energy flux is not enough for the requirement of the coronal heating. In addition, note that our estimated density might be overestimated by up to several factors because of the non-equilibrium ionization effect (see Section 3.2). Since the density is proportional to the energy flux, the energy flux might also be overestimated by up to 41

51 Figure 4.2: Schematic summary of the observational results. several factors. As a supplement, it is notable that if there is no energy dissipation from the photospheric energy flux, chromospheric δv should be around 100 km/s, which is quite larger than the observed amplitude. 4.3 Implications to the solar atmospheric heating The observational results are summarized in Figure 4.2. The comparison between F Hinode and F IRIS suggests that the amount of dissipated energy flux is enough to heat the umbral chromosphere ( erg cm 2 s 1, estimated by Avrett (1981) and Lee & Yun (1985)). Here we neglect the effect of spreading magnetic field because it is not large above the sunspot umbra at the chromospheric level (several factors). Note that the energy flux required to heat the umbral atmosphere is lower than that of active region. Since the sunspot umbral magnetic field is highly bundled, we guess that the discontinuity of magnetic field is not likely to be made inside umbral fields. Therefore, small energy releases such as nanoflares might not contribute to the sunspot umbral atmospheric heating. On the other hand, the energy flux observed in the chromosphere is much smaller than the required energy input for the umbral coronal heating. It implies that other heating 42

52 Figure 4.3: Schematic of the opacity effect. Note that magnetic field becomes weak in the upper atmosphere. mechanisms should be important in the corona. A possible candidate is small-scale transverse oscillation, which has amplitude relatively smaller than the longitudinal waves. The magnetic field above the sunspots is strong even in the solar corona, providing rather large poynting flux even with small fluctuations. We should note that our estimated photospheric density is larger than that in standard empherical models such as Maltby et al. (1986) and Fontenla et al. (2006). As an example, in Maltby et al. (1986) s model, ρ < 10 7 g/cm 3 at z = 300 km corresponding to the formation height of Fe I Å line (Felipe et al., 2014). In the following, we will discuss the reason why the estimated photospheric density with the seismology becomes large. We consider that the opacity effect, as discussed in Lites et al. (1998), Bellot Rubio et al. (2000), Khomenko et al. (2003) and Felipe et al. (2014), might be one possible reason. The opacity effect means that the region where the spectral line profiles are sensitive to magnetic fields moves upward and downward. Temperature and density fluctuations associated with the propagation of a compressible waves may cause the opacity fluctuations that move the line formation layer upward or downward, resulting in an apparent magnetic field fluctuation. For estimating the photospheric density, we assume here that the observed magnetic fluctuation is fully intrinsic (δb = δb intrinsic ). However, there is 43

53 Figure 4.4: Photospheric mass density estimated by the seismology discussed in Chapter as a function of intrinsic magnetic field fluctuation. The red dashed line shows the observed amplitude. Note that there is a possibility that a part of observed fluctuation is not an intrinsic fluctuation of the magnetic field. a possibility that opacity change may cause a false oscillation in observed magnetic fluctuation (δb = δb intrinsic + δb opacity ). Once the density increases, the intensity also increases according to equation (4.2). At the same time, the increase of the density moves the line formation layer upward. Since the magnetic field at the upper atmosphere is weaker than that in the lower atmosphere as shown in Figure 4.3, when we only consider the opacity effect caused by density fluctuation, ϕ I ϕ B should be π radian and it is consistent to our observational result. It means that the observed δb gives the maximum value of δb intrinsic. Figure 4.4 is a photospheric mass density derived by the seismology as a function of δb intrinsic. It shows that if δb intrinsic becomes small, the density also becomes small. Since the density is positively proportional to the energy flux, our estimated photospheric energy flux is the maximum value. The previous numerical simulation by Rüedi & Cally (2003) suggested that most of the expected fluctuations in the magnetic flux is actually due to a cross-talk from the temperature and density oscillations associated with MHD waves, implying the opacity effect. However, synthetic observations of Fe I Å line by Felipe et al. (2014) suggested that the photospheric magnetic field 44

54 retrieved from the weak field approximation provides the intrinsic magnetic field oscillations associated with the wave propagation because of the low magnetic field gradient. It implies the vertical structure might be important. Collados et al. (1994) reported the differences in vertical gradient db dz between the umbra of large and small sunspots, and found an order of magnitude difference. In the case of big sunspot, db dz 0.25 G/km, which is close to the condition of the numerical calculations in Felipe et al. (2014). It should be noted that if we consider that the temperature decrease along the height is dominant compared to the effect of density fluctuation, it works in the opposite sense as discussed in Fujimura & Tsuneta (2009). For these reasons, we point out that we cannot conclude whether our observed magnetic field fluctuations come from the opacity effect or not. Another possible reason for the density discrepancy is due to the low temperature in the sunspot umbra. The cool atmosphere at the umbral photosphere may reduce the amount of H ion, which is a dominant absorber in the visible wavelength (e.g., Stix, 2002). As a sequence, the line formation layer moves downward and may create our estimated high-density because of the gravity stratification. Previous studies, such as Mathew et al. (2004) and Martinez Pillet & Vazquez (1993), obtained that the magnitude of the Wilson depression is km in the umbra, which is enough longer than the photospheric scale height ( 150 km). Note that not only the temperature but also the magnetic field affects the Wilson depression. 45

55

56 Chapter 5 Summary and Future works We investigated energy dissipation mechanisms of MHD waves above the sunspot umbra. This work is valuable because it is still unclear whether MHD waves are important or not quantitatively for the chromospheric and the coronal heating. We estimated upward energy fluxes both at the photosphere and the chromosphere with Hinode and IRIS satellites, providing the difference in these energy fluxes, which may be considered as the dissipated energy in the region between the two atmospheric layers. We detected periodic fluctuations at both the photosphere and the chromosphere. The period of the fluctuations is around 5 min at the photosphere and 3 min at the chromosphere. We identified the wave mode of the observed fluctuations for formulating the energy flux. By using the phase relations between the LOS magnetic flux, the LOS velocity and the intensity obtained by Hinode, we suggested that the nature of the observed fluctuations are the standing slow-mode wave at the photosphere. In addition, chromospheric counterparts observed with IRIS suggest that high-frequency parts of photospheric waves penetrate upward to the chromosphere and form shock waves because of amplitude enhancement with decreasing density. The photospheric and chromospheric mass densities are estimated with the seismology and the line ratio method respectively. We calculated the ascending energy flux with an assumption that photospheric waves with the frequency >6 mhz penetrate to the chromosphere. As a result, we derived that erg cm 2 47

57 s 1 for the photospheric upward energy flux and erg cm 2 s 1 for the chromospheric upward energy flux. Their difference is enough larger than the energy loss required at the chromosphere above the sunspot umbra. It means that the observed MHD waves can play an important role for heating the chromosphere. However, there is a possibility that the opacity effect can also cause the fluctuation in the magnetic field. Therefore, what we need to do next is to distinguish δb intrinsic and δb opacity. The proportion of δb intrinsic and δb opacity highly depends on the vertical structure. Inverting the observed Stokes profiles with the SIR code (Stokes Inversion based on Response Functions; Ruiz Cobo & del Toro Iniesta, 1992) gives us the height distribution of physical parameters. It can be a clue to distinguish δb intrinsic and δb opacity. The effect of the gravity and shape of flux tubes also should be considered for more rigorous treatment. For considering them, improvement of theoretical models for seismology becomes important. Data-driven simulations like Felipe et al. (2011) might also be an effective approach. It helps us to note some missing physics such as mode conversion (Cally et al., 1994), which is not considered in this paper. In this study, transverse waves were not focused. However, for understanding the coronal heating mechanism, transverse waves might play an important role. To make it easier to detect the transverse oscillations, observations closer to the solar limb might be interesting. In addition, the solar polar region is one interesting region for solar-wind acceleration as discussed in Tsuneta et al. (2008a). Studies on waves at polar regions might connect to the future study with Solar orbiter (Müller et al., 2013), a mission of collaboration between ESA and NASA that was selected as the first medium-class mission of ESA s Cosmic Vision programme. When we discuss dissipation mechanisms of transverse waves, the chromospheric poynting flux, which cannot be obtained with IRIS because of the lack of polarimetric observations, becomes crucial. Solar-C (Watanabe, 2014), which plans to observe chromospheric magnetic fields in the future, makes us possible to measure the chromospheric poynting flux. 48

58

59 Appendix A Error Estimation A.1 Hinode data For estimating the noise level of the LOS velocity and the LOS magnetic flux, we calculated the standard deviation of intensity fluctuations in a continuum range of Stokes V profiles observed in the sunspot umbra. The total number of the used Stokes V profiles is Its probability density function is shown in Figure A.1. Then we randomly added the photon noise to each pixel of the observed Stokes V profiles. With these profiles, we recalculated the LOS velocity and the LOS magnetic flux, which was compared to the original values. Their distributions of residuals can be well represented by gaussian functions (see Figure A.2 and A.3). We applied a single gaussian fit to the probability density functions and estimated error levels. For LOS velocity, σ v = km/s, km/s, km/s and km/s for 1 1 summing, 2 2 summing, 4 4 summing and 6 6 summing, respectively. For LOS magnetic flux, σ B = 17.3 G, 8.7 G, 4.2 G and 2.9 G for 1 1 summing, 2 2 summing, 4 4 summing and 6 6 summing, respectively. Since we used 6 6 summing dataset in this paper, the error level is much smaller than observed amplitudes tabulated in Table 3.1. The signal-to-noise ratio for Stokes I with 1.6 sec exposure and 2 pixel summing observations is known to be around 1000 (Ichimoto et al., 2008). Since both δi cont /I cont 50

60 Figure A.1: A histogram indicating the probability density function of the intensity fluctuations of Stokes V profiles observed with Hinode/SOT-SP (Black line). The distribution can be well represented by a gaussian function (Red line). Figure A.2: A histogram indicating the probability density function of residual from the original Hinode doppler velocity due to the added photon noise. Each color shows the pixel summing dependence. 51

61 Figure A.3: A histogram indicating the probability density function of residual from the original Hinode LOS magnetic flux due to the added photon noise. Each color shows the pixel summing dependence. 52

62 Figure A.4: Histograms indicating the probability density function of the noise count of Stokes I observed with IRIS Si IV and Mg II (Black line). They are consistent with gaussian function (Red line). and δi core /I cont is around 0.7%, the observed amplitude of the intensity is much larger than the noise level. A.2 IRIS data For estimating the noise level of the LOS velocity, we calculated the standard deviation of intensity fluctuations in a continuum range of the Si IV spectra and a line wing range of the Mg II spectra observed in the sunspot umbra. The total number of the used intensity profiles is Probability density functions for Si IV and Mg II are shown in Figure A.4. Note that the center of the probability density functions is set in the peak of the distribution. With the same method done for Hinode error estimation (Appendix B.1), we obtained probability density functions of residuals from the original values. The results are shown in Figure A.5. The standard deviation of doppler velocities with Si IV and Mg II are 0.24km/s and 0.12 km/s, respectively. These errors are much smaller than observed 53

63 Figure A.5: Histograms indicating the probability density function of residual from the original IRIS doppler velocity due to the added photon noise. amplitudes and does not strongly affect to the energy estimation done in Chapter 4. 54

64 Appendix B Theory of MHD waves B.1 MHD equations In this chapter, we summarize the basic formula of MHD waves, based on Priest (2014). The basic MHD equations are dρ + ρ v = 0, dt (B.1) ρ dv = p + ( B) B/µ ρgẑ 2ρΩ v, dt (B.2) ( ) d p = 0, (B.3) dt ρ γ B = (v B) and t (B.4) B = 0, (B.5) where ρ, t, v, p, B, µ, g and Ω are the density, the time, the velocity, the plasma pressure, the magnetic field strength, the magnetic permeability, the gravitational acceleration and the angular velocity, respectively. The vector and scalar are represented by bold type and small type, respectively. We consider small departures from the equilibrium ρ = ρ 0 + δρ, p = p 0 + δp, B = B 0 + δb, (B.6) 55

65 and linearize the basic equations as follows, (δρ) + (δv )ρ 0 + ρ 0 ( δv) = 0, (B.7) t (δv) ρ 0 = (δp) + ( δb) B 0 /µ δρgẑ 2ρ 0 Ω δv, (B.8) t ( ) (δp) (δρ0 ) + (δv )p 0 c s + (δv )ρ = 0, (B.9) t t (δb) = (δv B 0 ) and (B.10) t δb = 0, (B.11) where c 2 s = γp 0 = γk BT 0 ρ 0 m (B.12) By using these equations, a generalized wave equation can be written 2 δv t 2 = c 2 s ( δv) (γ 1)gẑ( δv) g (δv z ) 2Ω (δv) t + { [ (δv B 0 )]} B 0 µρ 0. (B.13) Substituting plane-wave solutions, we can obtain general dispersion relation ω 2 δv = c 2 sk(k δv) + i(γ 1)gẑ(k δv) + igkδv z 2iωΩ δv + {k [k (δv B 0 )]} B 0 µρ, (B.14) where ω and k are the angular frequency and the wave number, respectively. Linear perturbations of the variables are taken in the form A exp[i(kz + mϕ ωt)], where A and m are constants. As calculated in Sakurai et al. (1991) and Moreels & Van Doorsselaere (2013), when we consider straight cylinder of magnetic flux and g = Ω = 0, by introducing the displacement ξ such that v = ξ, we can integrate these equations to rewrite t MHD equations in terms of the plasma displacement ξ and the total pressure perturbation δp are written by D d dr (rξ r) = CrδP, (B.15) 56

66 d(δp ) dr = ρ 0 (ω 2 k 2 v 2 A)ξ r, (B.16) ρ 0 (ω 2 k 2 va)ξ 2 ϕ = i m δp and (B.17) r ρ 0 (ω 2 k 2 va)ξ 2 z = ikδp + ik B2 0 ξ, (B.18) µ where D = ρ 0 (c 2 s + v 2 A)(ω 2 k 2 v A )(ω 2 k 2 c 2 T ), (B.19) ( ) m C = ω 4 (c 2 s + va) 2 2 r + 2 k2 (ω 2 k 2 c 2 T ), (B.20) ξ = ω 2 δp ρ 0 (c 2 s + v 2 A )(ω2 k 2 c 2 T ) = ir, (B.21) and compression inside the flux tube can be written as ωr exp i(ωt + mϕ + kz), the multiplication by ω is to ensure the correct dimensions. Here v A and c s are the Alfvén speed and the sound speed, respectively. B.2 Wave modes When we consider g = Ω = 0, equation (B.14) becomes ω 2 δv/v 2 A = k 2 cos 2 θ B δv (k δv)k cos θ B ˆB 0 + [1 + c 2 s/v 2 A](k δv) k cos θ B (ˆB 0 δv)k, (B.22) where θ B is the angle between the direction of propagation (k) and B 0. Taking the scalar product with k and ˆB 0, we have ( ω 2 + k 2 c 2 s + k 2 v 2 A)(k δv) = k 3 v 2 A cos θ B (ˆB 0 δv) (B.23) and k cos θ B c 2 s(k δv) = ω 2 (ˆB 0 δv) (B.24) 57

67 Figure B.1: Sausage and kink mode disturbance travelling along the slab. If (k δv) vanish, these equations are satisfied. This mode is called Alfvén mode (transverse wave). On the other hand, vanishing (k δv)/(ˆb 0 δv) can also satisfy these equations. In that condition, we have the dispersion relation: ω 4 ω 2 k 2 (c 2 s + v 2 A) + c 2 sv 2 Ak 4 cos 2 θ B = 0 (B.25) This dispersion relation has two distinct solutions [ ω 1 k = 2 (c2 s + va) 2 ± 1 1/2 (c 2 4 s + va 4 2c2 sva B)] 2 cos 2θ (B.26) The modes that have larger and smaller value are known as fast-mode and slow-mode, respectively. These waves are longitudinal waves. For the slow-mode, as θ B approaches to π, the phase velocity along the field approaches a value 2 c T = c sv A c 2 s + v 2 A (B.27) When we consider non-uniform magnetic field, such as a flux-tube, we have kink-mode (transverse wave) and sausage-mode (longitudinal wave) shown in Figure B.1. The kink 58

68 mode and sausage mode are related to Alfvén-mode and magneto-acoustic mode, respectively. Magnetic tension force of the flux tube is the restoring force in the kink mode, and is essentially incompressible. On the other hand, magnetic pressure force of the flux tube is the restoring force in the sausage mode, and is essentially compressible. B.3 Cutoff frequency When B = Ω = 0, taking the scalar product of the equation (B.14) with k and z in turn yields a pair of equations for δv z and k δv, which give the dispersion relation ω 2 (ω 2 N 2 s ) = (ω 2 N 2 sin 2 θ g)k 2 c 2 s, (B.28) where N s = γg 2c s, (B.29) N = g γ 1 c s, (B.30) sin 2 θ g = 1 k 2 z k 2, k = k + i 2H ẑ. (B.31) (B.32) Vertical propagation (θ g = 0) gives ω 2 = N 2 s + k 2 c 2 s (B.33) This equation means that wave exists only if ω > N s. N s is called cutoff frequency. We point out that this model omits magnetic field, but the idea is same even with the magnetic field. 59

69 Appendix C Manuals for IRIS data users IRIS C.1 How to use IRIS Web Figure C.1 FITS Level-2 IRIS Technical Note IRIS (RASTER) (SJI) RASTER FITS IDL read_iris_l2,files,index,data,wave= Mg II k 2796 files FITS index data 60

70 Figure C.1: IRIS read iris l2 SJI IRIS FITS read iris l2 IRIS SJI IDL> help, data_sji DATA_SJI FLOAT = Array[409, 385, 210] IDL> help,index_sji INDEX_SJI STRUCT = -> <Anonymous> Array[210] 61

71 ( Data Flow Level 2 keywords ) RASTER IDL> help, data_raster DATA_RASTER FLOAT = Array[165, 385, 16, 105] sit-and-stare SJI IDL> help,index_raster INDEX_RASTER STRUCT = -> <Anonymous> Array[1680] IRIS Figure C.2 files_r= iris_l2_ _114030_ _raster_t000_r00000.fits read_iris_l2,files_r,in2796_r,da2796_r,wave= Mg II k 2796 x=indgen(n_elements(da2796_r[*,0,0]))*in2796_r[0].cdelt1+in2796_r[0].cdelt1+in2796_r[0].wavemin!x.margin=[6,2]!p.multi=[0,3,2] set_plot, ps device,filename= example.eps,/color,set_font= Helvetica,xoffset=3,yoffset=2,$ xsize=30,ysize=30,/encapsulated,bits=16 loadct,3 plot_image,rotate(reform(da2796_r[134,*,*]),12),xtitle= X (arcsecs),ytitle= Y (arcsecs),$ scale=[in2796_r[0].fovx/n_elements(da2796_r[0,0,*]),in2796_r[0].fovy/n_elements(da2796_r[0,*,0])],$ charsize=3,font=1,title= Core,max=5000,min=500 loadct,1 plot_image,rotate(reform(da2796_r[300,*,*]),12),xtitle= X (arcsecs),ytitle= Y (arcsecs),$ scale=[in2796_r[0].fovx/n_elements(da2796_r[0,0,*]),in2796_r[0].fovy/n_elements(da2796_r[0,*,0])],$ charsize=3,font=1,title= Wing,max=400,min=0 loadct,8 plot_image,rotate(reform(da2796_r[229,*,*]),12),xtitle= X (arcsecs),ytitle= Y (arcsecs),$ 62

72 Figure C.2: C.1 節に記載されているコードによる出力画像 63

73 Figure C.3: IRIS IRIS Technical Note 1,scale=[in2796_r[0].FOVX/n_elements(da2796_r[0,0,*]),in2796_r[0].FOVY/n_elements(da2796_r[0,*,0])],$ charsize=3,font=1,title= Triplet,max=300,min=0 loadct,0 plot,x,da2796_r[*,100,0],yr=[0,3000],position=[.1,.1,.9,.5],xr=[2794,2806],charsize=3,font=1,$ title= Mg II h \& k,ytitle= DN,xtitle= Wavelength (angstrom),thick=3 set_line_color plots,[in2796_r[0].wavemin+134*in2796_r[0].cdelt1,in2796_r[0].wavemin+134*in2796_r[0].cdelt1],[0,3000],$ col=3,linestyle=2,thick=3 plots,[in2796_r[0].wavemin+300*in2796_r[0].cdelt1,in2796_r[0].wavemin+300*in2796_r[0].cdelt1],[0,3000],$ col=5,linestyle=2,thick=3 plots,[in2796_r[0].wavemin+229*in2796_r[0].cdelt1,in2796_r[0].wavemin+229*in2796_r[0].cdelt1],[0,3000],$ col=4,linestyle=2,thick=3 device,/close 64

74 Figure C.4: Mg II Pereira et al. (2013) C.2 Characteristics of the spectral lines IRIS Figure C.3 C.2.1 O I O I 1356Å ( Mm) O I 1356Å Lin & Carlsson (2015) C.2.2 Mg II Mg II 2796Å Mg II 2803Å Figure C.4 (central reversal) central reversal (Morrill et al., 2001) ( Mm) Hα Ca II K Mg II k 3 τ = 1 k 2r k 2v 65

75 Figure C.5: 放射強度 (上列) ドップラー速度 (中列) ライン形成高度 (下列) を k2v k3 k2r の各々で計算した結果 Leenaarts et al. (2013b) より引用 66

76 Figure C.6: Mg II C II Si IV のライン中央における放射強度のマップ 異なる構造 が観測されているのがわかる Rathore et al. (2015b) より引用 して得られる構造は何を反映しているのかは解釈が容易ではない (Figure C.5) 詳細 は Leenaarts et al. (2013a,b) や Pereira et al. (2013) に記されている C.2.3 C II C II 1334A と C II 1335A のラインは central reversal を Mg II 2796A のように示す C II 1334A と C II 1335A の比は観測的に常に 1.8 よりも小さく 放射が光学的に厚いこ とを示している Mg II や Si IV のマップ (Figure C.6) と比較してみると C II のマッ プからは Mg II と共通する構造もあれば Si IV と共通する構造も確認でき 彩層上 部と遷移層の間で形成されていることが示唆されている 詳細は Rathore & Carlsson (2015) Rathore et al. (2015a) Rathore et al. (2015b) で数値的 観測的に検証されて いる C.2.4 Si IV Si IV 1394A と Si IV 1403A のラインは central reversal を示さない しかしながら Yan et al. (2015) により報告された self-absorption と呼ばれる現象には注意が必要で ある self-absorption は 低空で加熱が生じる Ellerman bomb などの現象の際に見ら れる これらのラインは基本的には光学的に薄い IRIS による観測で 非常に微弱 67

77 Si IV (Schmit et al., 2014) C.2.5 O IV O IV (Section ) log N e =10-13 O IV Young (2015) Olluri et al. (2013) C.2.6 Fe XXI 10 7 K Fe XXI Fe XXI (Young et al., 2015; Tian et al., 2015) C.2.7 Mg II triplet Mg II triplet (<1.0 Mm) Mg II triplet Ellerman bomb (δt ) Pereira et al. (2015) 68

78 Figure C.7: (a)sji (b)raster C.3 Some tips in IRIS data analysis C.3.1 Co-alignment between the SJIs and spectra Level 2 RASTER SJI Figure C.7 Figure C.8 RASTER SJI C.3.2 Dark calibration Level 2 Figure C.9 69

79 Figure C.8: IRIS の光学系の模式図 画像は IRIS Technical Note 1 より引用 Figure C.9: 左 IRIS により観測された Si IV のスペクトル線 右 連続光の範囲に おけるカウントの頻度分布 CCD の上半分 (黒) と下半分 (赤) に分けてプロットして ある 70

80 Figure C.10: C II SJI Fe XXI Mg II C II C II SJI C.3.3 Fe XXI emission seen in C II SJIs Fe XXI C II C II SJI Fe XXI Figure C.10 C.4 IRIS operations IRIS 1 3 IRIS OBS-ID OBS-ID (Figure C.11 and C.12) IRIS Technical Note 31 IRIS Technical Note 31 71

81 Figure C.11: OBS-ID IRIS Technical Note 31 72

82 73

83 Figure C.12: OBS-ID IRIS Technical Note 31 74