Velocity and Magnetic Field Fluctuations in the Photosphere of a ...

THE ASTROPHYSICAL JOURNAL, 497 : 464È482, 1998 April 10 ( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

VELOCITY AND MAGNETIC FIELD FLUCTUATIONS IN THE PHOTOSPHERE OF A SUNSPOT BRUCE W. LITES,1 JOHN H. THOMAS,2 THOMAS J. BOGDAN,1 AND PAUL S. CALLY3 Received 1997 June 13 ; accepted 1997 November 14

ABSTRACT We use a data set of exceptionally high quality to measure oscillations of Doppler velocity, intensity, and the vector magnetic Ðeld at photospheric heights in a sunspot. Based on the full Stokes inversion of the line proÐles of Fe I 630.15 and 630.25 nm, in the sunspot umbra we Ðnd upper limits of 4 G (root mean square [rms]) for the amplitude of 5 minute oscillations in magnetic Ðeld strength and 0¡.09 (rms) for the corresponding oscillations of the inclination of the magnetic Ðeld to the line of sight. Our measured magnitude of the oscillation in magnetic Ðeld strength is considerably lower than that found in 1997 by Horn, Staude, & Landgraf. Moreover, we Ðnd it likely that our measured magnetic Ðeld oscillation is at least partly due to instrumental and inversion cross talk between the velocity and magnetic signals, so that the actual magnetic Ðeld strength Ñuctuations are even weaker than 4 G. In support of this we show, on the basis of the eigenmodes of oscillation in a theoretical model of the sunspot umbra, that magnetic Ðeld variations of at most 0.5 G are all that is to be expected. The theoretical model also provides an explanation of the shift of power peaks in Doppler velocity to the 3 minute band in chromospheric umbral oscillations, as a natural consequence of the drastic change in character of the eigenmodes of oscillation between frequencies of about 4.5 and 5.0 mHz due to increased tunneling through the acoustic cuto†Èfrequency barrier. Using measurements of the phase of velocity oscillations above the acoustic cuto† frequency, we determine the relative velocity response height in the umbra of four di†erent photospheric spectral lines from the phase di†erences between velocities in these lines, assuming that the oscillations propagate vertically at the local sound speed. In spacetime maps of Ñuctuations in continuum intensity, Doppler velocity, magnetic Ðeld strength, and Ðeld inclination, we see distinct features that migrate radially inward from the inner penumbra all the way to the center of the umbra, at speeds of a few tenths of a kilometer per second. These moving features are probably a signature of the convective interchange of magnetic Ñux tubes in the sunspot, although we failed to Ðnd any strong correlation among the features in the di†erent quantities, indicating that these features have not been fully resolved. Subject headings : Sun : atmosphere È Sun : magnetic Ðelds È Sun : oscillations È sunspots 1.

INTRODUCTION

magnetic Ðeld in the sunspot imposes a spatial order on the sunspot atmosphere that alters the 5 minute p-modes and produces new modes of oscillation not present in the quiet Sun. Sunspots also serve as local absorbers and scatterers of the acoustic power in the p-mode oscillations (Braun, Duvall, & LaBonte 1987, 1988 ; Braun, LaBonte, & Duvall 1990 ; Bogdan et al. 1993 ; Braun 1995 ; Bogdan et al. 1998). The data set we examine here is particularly well suited to detecting oscillations of the magnetic Ðeld within the sunspot, should they exist. Several previous attempts have been made to measure oscillations of umbral photospheric magnetic Ðeld strength, with conÑicting results. Some observers have reported signiÐcant oscillatory power in the umbral magnetic Ðeld strength, at various frequencies in the 5 and 3 minute bands (Mogilevskii, Obridko, & ShelÏting 1973 ; Gurman & House 1981 ; Horn, Staude, & Landgraf 1997) or at higher frequencies (Efremov & ParÐnenko 1996). Others have found no signiÐcant oscillations of the umbral magnetic Ðeld at their limit of sensitivity (Schultz & White 1974 ; Thomas, Cram, & Nye 1984). The observations reported here show that magnetic Ðeld Ñuctuations in the sunspot umbra are indeed very weak ; the root mean square (rms) amplitude of Ðeld strength variations in the 5 minute band are at most about 4 G, and probably less, and there are no signiÐcant power peaks outside the 5 minute band. We also show that such small variations in the magnetic Ðeld are all that should be expected on theoretical grounds. The high quality of our data set allows us to extract phase and coherence information about velocity and magnetic oscillations. In particular, the measured phase di†erences of

Many studies of oscillations in sunspots have been carried out over the past three decades, motivated by a desire not only to understand the intrinsic nature of these oscillations but also to extract the information these oscillations might contain about the basic structure of a sunspot above and below the solar surface. There are substantial di†erences between the oscillations observed in sunspots and those observed in the quiet Sun, and within a sunspot there are substantial di†erences between oscillations in the umbra and in the penumbra. Understanding the relations among these various oscillations, and the causes of the differences among them, is an important goal. A comprehensive review of the observations of sunspot oscillations was given by Lites (1992), and discussions of the theoretical interpretations of these oscillations may be found in Thomas & Weiss (1992), Chitre (1992), and Bogdan (1992, 1994). In this paper we are concerned with oscillations at photospheric levels in a sunspot. Oscillations in the 5 minute p-mode band have reduced amplitude in a sunspot compared to the surrounding quiet photosphere. The strong 1 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307-3000. The National Center for Atmospheric Research is sponsored by the National Science Foundation. 2 Department of Mechanical Engineering, Department of Physics and Astronomy, and C. E. K. Mees Observatory, University of Rochester, Rochester, NY 14627. Also Affiliate Scientist, HAO/NCAR. 3 Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.

464

VELOCITIES AND MAGNETIC FIELDS IN A SUNSPOT oscillations in di†erent spectral lines at frequencies above the acoustic cuto† frequency in the umbra allow us to estimate the relative velocity response heights of these spectral lines. We also study oscillations in the umbra-penumbra transition region and in the penumbra. Finally, we examine the more persistent, low-frequency behavior of Ñuctuations throughout the sunspot and Ðnd a pattern of inward radial motion of features in the inner penumbra and umbra that is indicative of the convective interchange process within the sunspot. 2.

DATA ACQUISITION AND ANALYSIS

The observations reported here were made with the HAO/National Solar Observatory (NSO) Advanced Stokes Polarimeter (ASP) at the Vacuum Tower Telescope at NSO/Sacramento Peak Observatory on 1995 October 15. The ASP, which measures the full Stokes vector of polarization proÐles of several spectral lines at each spatial point sampled, has been described in detail by Elmore et al. (1992) and has proved to be a very e†ective instrument for accurate measurements of vector magnetic Ðelds and Doppler velocities in sunspots and active regions (Lites & Skumanich 1990 ; Lites 1996 ; Lites et al. 1993, 1995 ; Lites, Marti nez Pillet, & Skumanich 1994 ; Skumanich, Lites, & Marti nez Pillet 1994 ; Marti nez Pillet et al. 1994 ; Keppens & Marti nez Pillet 1996 ; Marti nez Pillet, Lites, & Skumanich 1997 ; StanchÐeld, Thomas, & Lites 1997). 2.1. T he Observations Using the ASP, we observed a large, symmetric sunspot (Fig. 1 [Pl. 4]) in active region NOAA 7912, located near disk center (solar latitude 9S, longitude 5E). The spectrograph slit of the ASP was placed across the center of the sunspot, with the slit crossing the entire sunspot and including short regions of ““ quiet Sun ÏÏ on either side of the spot. The width of the slit was 0A. 375, and there were 229 pixels along its 84A. 7 length. The spectral measurements were made in a narrow wavelength range centered at 630.25 nm. An example of the instantaneous Stokes proÐles of the spectrum in this wavelength range may be seen in Figure 2 (Plate 5). The observations span a wavelength range of 0.302 nm with a sampling of 1.259 pm and a spectral resolution of about 3 pm. This wavelength range includes the strong magnetically sensitive lines Fe I 630.15 and 630.25 nm in addition to the weaker (high-excitation) magnetically sensitive lines Fe I 630.35 nm and Ti I 630.4 nm. In the cooler sunspot umbra, the high excitation Fe I 630.35 nm line is absent, and the region between Fe I 630.25 nm and Ti I 630.4 nm is populated by bands of molecular lines, most of them due to TiO and perhaps CaH (Boyer, Sotirovsky, & Harvey 1975). These molecular lines are apparent in the Stokes I and V images in the bottom panels of Figure 2, which have been scaled to reveal features in the umbral spectrum. A pair of narrow telluric absorption lines due to O are also present and provide a stable wavelength refer2 for determining Doppler velocities. ence The observations consist of four consecutive runs of 32.5 minute duration, separated by gaps of about 3 minutes (the time needed to restart an ASP sequence). Within each 32.5 minute segment, data were recorded at a nearly uniform cadence having a mean time step of 9.91 s. The ASP cameras acquire data at a 60 Hz frame rate. For the present observations, actual integration of the Stokes vector occurred during the Ðrst 2.13 s of each measurement, the

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remaining time being needed to process and record data from the two other ASP cameras, which were used with the universal birefringent Ðlter (UBF) to obtain simultaneous Doppler images in Fe I 557.6 nm for a region consisting of the sunspot and its immediate surroundings.4These runs were carried out in the interval UT 14 : 30È16 : 49, during which the seeing quality was uniform and exceptionally good (\0A. 5), except for a very few brief episodes of mild image disturbance. These data represent perhaps the highest image quality of any obtained to date for ASP observations. For this reason alone, these data are unique. Prior to the time series observations, a spatial map of the sunspot and its surroundings was performed with a coarse (0A. 75) stepping of the slit perpendicular to its length. The inferred vector magnetic Ðeld for this map (UT 13 : 58È 14 : 13) is shown in Figure 1 as a color display with the directions of the Ðeld vectors shown in perspective. Also shown in color in this Ðgure are spacetime maps of the measured magnetic Ðeld strength, azimuth angle, and zenith angle for the Ðrst of the four 32.5 minute runs that make up the whole time series. Tracking of the sunspot was accomplished with the NSO ““ quad-cell ÏÏ sunspot tracker, which is extremely e†ective at compensating for both bulk image motion and the small (subarcsecond) periodic wobble of the image arising from the rotating polarization modulator. Because the ASP observing procedure recalibrates the position of the spectrograph slit at the start of each observing run, there are small (>1A) shifts in the location of the slit relative to the sunspot between the Ðnish of one run and the start of the next. These displacements are evident in the time series only under very close scrutiny. 2.2. Data Processing The data in the sequential time series were interpolated onto a uniform time grid, and the short gaps between the four individual time series were Ðlled by linearly interpolating between the end points of the adjacent time series. The result is a continuous series of data equally spaced in time (9.91414 s), of total duration 139.3 minutes. The corresponding temporal Nyquist frequency for the computed power spectra and cross-correlations is therefore 50.433 mHz. The raw polarimetric data are calibrated and compensated for the polarizing properties of the Vacuum Tower Telescope and the intrinsic response of the polarimeter. The details of this procedure are set forth in Skumanich et al. (1997) and involve a global least-squares solution that simultaneously determines the telescope parameters and the response matrix of the polarimeter and leads to an assessment of the instrument-induced cross talk between Stokes I, Q, U, and V . By applying these postprocessing algorithms, the ASP is routinely able to measure peak linear polarization amplitudes of 0.1% of the continuum intensity against an instrumentally produced background polarization that is larger by a factor of 20. The calibrated data are then subjected to a least-squares inversion based on the Ðt of the Milne-Eddington solution of the Unno-Rachkovsky equations (Unno 1956 ; Rachkovsky 1962) of plane-parallel magnetized radiative 4 Analysis of data from the UBF measurements during this observing run will be presented in a future paper.

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transfer to the full Stokes I, Q, U, and V line proÐles of Fe I 630.15 nm and Fe I 630.25 nm. The inversion procedure determines a magnetic and a nonmagnetic contribution to the emergent intensity that may have di†ering line-of-sight velocities, but the procedure takes no account of velocity gradients and therefore cannot reproduce asymmetric Stokes Q, U, and V proÐles (e.g., Skumanich & Lites 1987 ; Je†eries, Lites, & Skumanich 1989 ; Westendorp Plaza et al. 1998). From the full least-squares inversion, Doppler velocities were determined for the Fe I lines, denoted here as Fe V1 (for Fe I 630.15 nm) and Fe V2 (for Fe I 630.25 nm). Two additional Doppler velocities, denoted by Fe V (for Fe I zc 630.15 nm) and Ti V (for Ti I 630.4 nm), were determined zc from the shift of the zero crossing of the Stokes V proÐles for these two lines. A Ðfth Doppler velocity, denoted by Mol V, was constructed by cross-correlating successive Stokes V spectra across the wavelength band spanned by the TiO and CaH molecular lines. Spectral regions around the strong Fe I and Ti I lines (see the bottom, right-hand image of Fig. 2) were excluded from this cross-correlation. Our computation and display of power, phase, and coherence spectra follows closely the procedure used by Lites, Rutten, & Kalkofen (1993, hereafter LRK), which is based on the method of Edmonds & Webb (1972). When computing a spatial-average power spectrum for Ñuctuations in an observable quantity, we Ðrst compute the discrete Fourier transform of the time series at each spatial pixel along the slit and then average the power spectra. Unlike LRK, phase di†erences at each frequency are computed from the vector addition of the phase di†erences for the individual spatial-frequency samples, with each vector represented by a length (P P )1@2, where P and P are the b a the respective b powers at that frequency acorresponding to variables a and b involved in the selected phase di†erence. Thus, only one phase di†erence is displayed at each frequency, rather than a gray scale as presented by LRK. Coherence spectra are displayed as the quantity C2 deÐned by LRK so as to indicate more clearly the range of observed coherence, but here we have adopted a smoothing width in frequency of Ðve rather than nine frequency points. 2.3. Quality of the Inferred Doppler V elocities We can estimate the level of systematic instrumental error in our Doppler velocity measurements by examining the time series of the spatially averaged wavelength positions of the two narrow O telluric lines, which are plotted 2 a linear trend, representing a in Figure 3. After removing velocity drift (from redshift to blueshift) with time, the residual Doppler signal from the telluric lines is found to have a total rms Ñuctuation of 13 m s~1. The upper panel of Figure 4 shows that the power spectrum of these telluric velocity Ñuctuations appears to be a featureless l~1 noise spectrum. Within the 5 minute (2 \ l \ 4.5 mHz) and 3 minute (4.5 \ l \ 7 mHz) bands, the integrated rms telluric velocity Ñuctuations are 2.7 m s~1 and 2.2 m s~1, respectively. Examination of the two time series in Figure 3 shows that most of these Ñuctuations are real, and possible sources for these small Ñuctuations include seeing Ñuctuations and vibration in the spectrograph and winds in the EarthÏs atmosphere. Between 70 and 100 min from the start of the time series the two curves of Figure 3 di†er by 10È20 m s~1, but their variations on short timescales remain very similar. The relative departure of the curves during this epoch of the time series could be due to slight di†erences in the accuracy

Vol. 497

FIG. 3.ÈPosition of each of the two telluric O spectral lines as a function of time for the entire time series. The rms 2Ñuctuations in these positions are used to determine the precision of the spectrograph for measurements of Doppler velocities on the Sun.

of the Ñat-Ðeld correction at the location of the two narrow telluric lines. Because we reference our solar Doppler measurements to these inferred O line positions, our wavelength calibration procedure 2will compensate for small Ñuctuations in wavelength arising from the spectrograph, but not for Ñuctuations arising from terrestrial winds. Hence, we estimate that the precision of our measurements of solar Doppler velocities in the 5 and 3 minute bands may be no better than about 3 m s~1. On the other hand, the random noise level present for individual spatial locations along the slit is much larger than this precision. Therefore, we expect that most of the Doppler velocity results for oscillatory phenomena presented in this paper are limited fundamentally by random noise of the measurement process or by solar perturbations of the line proÐle shapes, but not by systematic errors of instrumental origin. 2.4. Quality of the Inferred V ector Magnetic Field Measurements A discussion of the accuracy of vector magnetic Ðelds as inferred from ASP data may be found in Lites et al. (1994) and Lites (1996). Because the photosphere of a sunspot umbra is perhaps the ““ calmest ÏÏ location in the visible atmosphere of the Sun, we have found that the present time series measurements provide us with another means of verifying the accuracy and reproducibility of inversion of ASP data. Figure 5 presents the results of the inversion of the Ðrst 32.5 minute time series run of the observations presented here. The panels on the left in Figure 5 show the timeaveraged vector Ðeld quantities (Ðeld strength, zenith angle, and azimuth angle in the local solar frame) as solid lines, showing their variations along the slit. The rms Ñuctuation of each quantity, computed from the time series and multiplied by 10, is shown as a shaded region around each curve.

No. 1, 1998

VELOCITIES AND MAGNETIC FIELDS IN A SUNSPOT

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and the time-averaged uncertainties of these quantities resulting from the least-squares inversion (d o B o , dc , lsq dt ) on the horizontal axis. The thin straight linelsqin each lsq plot denotes where the two plotted quantities would be equal. We see that this line e†ectively represents a lower bound to the distribution of points in all three scatter plots. The fact that the distributions approach these lines from above shows that the estimates of errors from the inversion procedure are valid and that indeed some positions within the umbra are so quiet that the observed Ñuctuations in the vector Ðeld measurements are due mostly to errors in our ability to infer the Ðeld quantities (to a few gauss and a fraction of a degree) rather than to real Ñuctuations in the umbra itself. Conversely, points that fall signiÐcantly above these lines represent places in the sunspot where there are real time variations. We will demonstrate in the following that these variations are mostly secular and that oscillatory variations of the umbral magnetic Ðeld are so small that they are below our present ability to measure them. 3.

FIG. 4.ÈTop : Average log power spectrum of Doppler velocities measured in Fe I, Ti I, and the molecular lines, averaged over all spatial points in the umbra. Also shown is the average log power spectrum of shifts of the telluric lines used as a wavelength reference. Bottom : Average phase di†erence (points) and coherence (solid curve) between the Doppler velocities measured in the umbral core in the molecular lines and the Ti line, as functions of frequency.

Outside of the sunspot there are regions where no inversion was attempted because the observed net polarization in the Fe I lines was too small. The average vector Ðeld quantities are shown as null in these regions. The Ñuctuations can be large in these regions if the polarization exceeds the threshold for inversion at some of the time steps. Fluctuations in the penumbral regions are also large due to slow variations in the magnetic Ðeld, as will be demonstrated later in this paper. Furthermore, there are signiÐcant Ñuctuations in both zenith and azimuth angle close to the center of the umbra (near 37A). This is mostly an artifact of the ambiguity of the azimuth angle in the observerÏs frame near the point where the vector magnetic Ðeld is aligned with the line of sight. Otherwise, it is clear that the inferred vector magnetic Ðeld orientation in the umbra is quite stable during this 32.5 minute span, to within 1¡È2¡, and that the Ðeld strength is stable to within a few tens of gauss. These observations demonstrate not only that the sunspot umbra is very quiescent from the standpoint of plasma motions, but also that the umbral magnetic Ðeld is not changing very much on short timescales. In ° 5 below we o†er theoretical arguments why this should be true. The panels on the right in Figure 5 demonstrate that error estimates arising solely from the least-squares inversion of Stokes proÐles do have some validity as measures of the uncertainties in the inferred physical quantities. These scatter plots, shown for Ðeld strength and orientation angles, present the rms Ñuctuation from the observed vector Ðeld (d o B o , dc , dt ) time series on the vertical axis rms rms rms

OSCILLATIONS IN THE UMBRA

We use the time-averaged continuum intensity I to distinguish among sets of contiguous pixels along thec Ðeld of view of the slit corresponding to regions of quiet Sun, penumbra (center-side and limb-side), umbra-penumbra transition (center-side and limb-side), and umbra, arranged in order of decreasing mean continuum intensity. (The locations of these regions are indicated in the right-hand panels of Fig. 8.) In our discussions below we will refer to average power, phase, and coherence spectra of various quantities computed for some of these distinct regions. In this section we discuss the measured oscillations in velocity, intensity, and magnetic Ðeld in the umbra of the sunspot. As deÐned here, the umbra consists only of the dark core, symmetric about the point of minimum intensity, where the molecular lines are clearly visible in the spectrum. We note that along the length of the path traversed by the slit through the umbra, there are subtle di†erences in power spectra. SpeciÐcally, the peak power of the 5 minute band for Fe V1 increases slightly with distance from the center of the sunspot. This e†ect is not obvious in Mol V. Furthermore, many measures experience higher noise levels (apparent at high frequencies) near the very center of the umbra, where the intensity and the signal-to-noise ratio are low, and hence the Doppler shifts are less well determined. However, these variations are not so large as to invalidate our averages over the regions of the sunspot. 3.1. Umbral V elocity Oscillations The upper panel of Figure 4 displays the average umbral power spectra for three of our Ðve measures of Doppler velocity (deÐned in ° 2.2) out to the Nyquist frequency of 50 mHz. Also shown is the power spectrum for the telluric O lines, discussed earlier in relation to systematic instrumental2 errors of measures of the Doppler velocity. The three solar Doppler velocities all yield comparable power levels in the 5 minute p-mode band (see Table 1). Above 10 mHz the spectra are all featureless, with the Ñuctuations in the Mol V Doppler velocities being an order of magnitude larger than those obtained for Ti V and Fe V1. The weakness of the spectral features arising zc from the molecular lines causes the Mol V wavelength measures to be considerably less certain than their atomic transition counterparts, leading to enhanced power at high frequencies.

FIG. 5.ÈMean values and rms Ñuctuations of the strength, zenith angle, and azimuth angle of the magnetic Ðeld vs. slit position across the sunspot. Also shown are scatter plots of the rms Ñuctuations (rms subscript) of each quantity vs. the least-squares-Ðt uncertainties (lsq subscript) of Stokes inversion for that quantity.

VELOCITIES AND MAGNETIC FIELDS IN A SUNSPOT

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TABLE 1 SUNSPOT RMS POWER LEVELS Regiona and Frequency Bandb Umbra : 5 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limb penumbra : 5 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . Center penumbra : 5 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limb umbra-penumbra transition : 5 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . Center umbra-penumbra transition : 5 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 minute . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fe V2c (m s~1)

Fe V1d (m s~1)

Fe V e zc (m s~1)

Mol Vf (m s~1)

Ti V g zc (m s~1)

oBo (gauss)

c (deg)

41 11

42 14

40 13

45 18

48 21

4.0 1.7

0.09 0.03

3.6 1.7

59 20

58 20

41 11

... ...

... ...

6.8 3.0

0.29 0.12

3.0 1.2

48 23

47 22

43 19

... ...

... ...

5.2 2.4

0.27 0.18

2.8 0.9

59 11

60 12

76 18

... ...

... ...

3.0 1.1

0.10 0.04

4.3 1.9

42 7

44 8

48 9

... ...

... ...

4.5 1.6

0.9 0.4

5.1 1.9

103*I /I c c

a Refer to Fig. 8 for positions of the umbra, limb/center umbra-penumbra transition, and limb/center penumbra regions along the ASP camera slit. b The 5 minute band contains oscillation frequencies between 2.0 and 4.5 mHz ; the 3 minute band contains frequencies between 4.5 and 7.0 mHz. c Fe I 630.25 nm : Stokes inversion. d Fe I 630.15 nm : Stokes inversion. e Fe I 630.15 nm : Stokes V zero crossing. f Molecular lines : Stokes V cross-correlation. g Ti I 630.4 : Stokes V zero crossing.

The lower panel of Figure 4 shows the full phase di†erence and coherence spectrum for the umbral measures of Mol V-Ti V . This panel serves to illustrate that determizc nations of relative phases of oscillation in the umbra are valid at frequencies up to about 10 mHz, and strong coherence of oscillations with well-determined phase lags are present in the measures of Doppler shift. Beyond 10 mHz the measurements are dominated by noise all the way up to the Nyquist frequency. Figure 6 displays average umbral power spectra for the time series of all Ðve measures of Doppler velocity in addition to power spectra of continuum intensity Ñuctuations and of the inferred magnetic Ðeld strengths and inclination angles. Because there is apparently no useful information at higher frequencies, these spectra are plotted only out to 10 mHz. The upper, left-hand panel indicates that the inferred Doppler velocities for the two strong Fe I lines are nearly identical and not very di†erent from the Doppler velocity determined only from the zero-crossing of the Fe I 630.15 nm Stokes V proÐle.5 The measured velocity Ñuctuations in the 5 minute p-mode band do not result in power in the umbral continuum intensity at the level at which our measurements would reveal it. From the power spectra plotted in the upper panel of Figure 6, we determined rms velocity Ñuctuations in the 5 minute (2 \ l \ 4.5 mHz) and 3 minute (4.5 \ l \ 7 mHz) bands as follows. First we subtracted the smooth background power associated with the high-frequency noise, which we set equal to the average power in the frequency band 25È50 mHz. We then integrated the residual excess power across each frequency band and from this determined the average rms velocity Ñuctuation in each band. The resulting average rms Ñuctuations are given in Table 1. 5 The lower, right-hand panel of Fig. 2 shows that the Fe I 630.25 nm Stokes V proÐle has a magneto-optical reversal in the core of the line ; hence, no attempt has been made to determine a Doppler velocity based on the zero-crossing of its V proÐle.

Figure 7 shows average phase and coherence spectra between several pairs of quantities measured in the umbra. Note that the phase spectra for pairs of measured velocities (in Fe I, Ti I, and the molecular lines) show nearly zero phase di†erence across the 5 minute band (2È4.5 mHz), consistent with the notion that these oscillations are evanescent waves in the umbral photosphere. Evanescence of the 5 minute oscillations in the quiet photosphere as veriÐed by measured phase delays has long been recognized (Schmieder 1976 ; Schmieder 1978 ; Lites & Chipman 1979), and evanescence is expected to be even more Ðrmly entrenched in the cool umbral photosphere than it is in the quiet Sun. Figure 7 shows that the present data have sufficient signal levels and observing cadence in several photospheric spectral lines to permit phase delay measurements of the weak umbral photospheric oscillations. Above about 5.2 mHz, the phase di†erence increases in a roughly linear fashion, indicating that the waves begin to tunnel through the acoustic cuto†Èfrequency barrier presented by the cool umbral temperature minimum, while su†ering little downward reÑection from the transition region. Indeed, our phase spectra thus provide a measured value of the e†ective acoustic cuto† frequency in the umbra of about 5.2 mHz. This value is in excellent agreement with the theoretical calculations of ° 5 based on the model M atmosphere for the umbral core given by Maltby et al. (1986). The ordering of the line pairs displayed in Figure 7 is chosen so that the phase di†erence is positive for an upward-propagating wave ; i.e., the phase of the velocity in the line formed lower in the atmosphere is subtracted from the phase of the velocity in the line formed higher in the atmosphere. The ordering of the heights of formation of the lines, from lowest to highest in the umbral atmosphere, is thus Fe I 630.25 nm, Fe I 630.15 nm, the molecular lines of TiO and CaH, and Ti I 630.4 nm. We can use the information in the phase spectra in Figure 7 to calculate di†erences in the e†ective height of formation,

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FIG. 6.ÈAverage power spectra in the umbra of several quantities (top to bottom and left to right) : Ñuctuations in velocity measured by the full Stokes inversion of Fe I 630.15 nm (Fe V1) and Fe I 630.25 nm (Fe V2) ; Ñuctuations in velocity measured by the Stokes V zero-crossing in Fe I 630.15 nm (Fe V ) and zc in in Ti I 630.4 nm (Ti V ) ; Ñuctuations in velocity measured in the molecular lines (Mol V) ; Ñuctuations in relative continuum intensity ; Ñuctuations zc magnetic Ðeld strength measured from the full Stokes inversion ( o B o ) and from the separation of the peaks of the Stokes V proÐle ( o B o ) ; and Ñuctuations Vpeak in the magnetic Ðeld inclination t. The lower two panels also display power spectra of the error estimates of the inferred Ðeld strength (d o B o ) and of the lsq inclination angle (dt ) resulting from the least-squares inversion procedure. lsq

or more precisely, the e†ective ““ response height ÏÏ of the four spectral lines to Doppler velocity perturbations in the sunspot umbra. In each of the phase spectra for velocity pairs in Figure 7, we construct a straight line through the

origin having a least-squares Ðt to the phase spectrum in the range 5.5È10 mHz. These Ðtted straight lines are shown in the velocity phase spectra in Figure 7. From the slope of each of these lines and an assumed (constant) value of the

FIG. 7.ÈAverage phase spectra (points) and coherence spectra (curves) in the umbra for various pairs of quantities, as indicated by the legend in each panel. See text for meanings of these quantities.

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sound speed in the umbral photosphere of 6 km s~1, we can determine the di†erence between the e†ective velocity response heights of the two spectral lines. The results of this calculation are as follows :6 z [z \ 13 km , Fe I 630.15 Fe I 630.25 z [z \ 6 km , TiO,CaH Fe I 630.15 z [z \ 70 km , Ti I 630.4 Fe I 630.15 z [z \ 69 km . Ti I 630.4 TiO,CaH A somewhat surprising result of this calculation is that the molecular lines have a velocity response height that is much closer to that of the Fe I lines than that of the Ti I line. This method of determining the relative e†ective velocity response heights of a group of spectral lines can serve as an independent check of a comprehensive radiative transfer calculation of the velocity response functions. Also, using the relative velocity response heights determined by this method, absolute response heights of all the lines in the set can be assigned based on the absolute velocity response height of any one of the lines, as determined by radiative transfer calculations for an accurate model atmosphere. Thus, for example, we could take the umbral value z \ 190 km relative to the location of q \ 1 as Fe I 630.25 by Bruls, Lites, & Murphy (1991), which 500 would calculated then give, for example, z \ 260 km. However, the Ti I 630.4 calculations of Bruls et al. (1991) were for intensity, not Doppler velocity. To be more precise, those calculations should be repeated for the response of all the Stokes proÐles to Doppler velocity perturbations (see Sanchez Almeida, Ruiz Cobo, & del Toro Iniesta 1996). For many years there has been speculation that a subsurface, narrow frequency band oscillation might drive the dramatic chromospheric umbral oscillations nearly universally observed over sunspot umbrae, but here we Ðnd no evidence in our selected photospheric velocity measures for any isolated peaks of power near 5.5 mHz. Previously, Abdelatif, Lites, & Thomas (1986) and others have found evidence for small peaks in umbral power spectra near 5.5 mHz (see Lites 1992 for a review). Lites & Thomas (1985) did not see a substantial peak in the power spectra of their measures of Doppler velocities in the Ti I 630.4 nm line, but they did Ðnd a strong enhancement of coherence between oscillations in that line and in the Fe I 543.4 nm line, which is formed high enough in the umbral atmosphere to show the chromospheric oscillations at those frequencies. More recently, Penn & LaBonte (1993) carried out measurements of Doppler shifts of 18 molecular lines in a sunspot umbra over a period of several days. Their sampling rate is such that their measurements are insensitive to oscillations with frequencies greater than 5.4 mHz, but there is no evidence for any power peaks in their data near their limiting frequency. Because we see neither peaks in power near 5.5 mHz nor signiÐcant peaks in coherence among the measured Doppler shifts from various spectral lines, the present observations suggest that such sharp peaks previously observed in umbral photospheric power spectra may be spurious. Given the greater sensitivity of the present observational mode, it would be very interesting to carry out measurements in the future in which the ASP observations 6 The slight inconsistency of these results is due to di†erences associated with the least-square Ðts.

Vol. 497

are augmented by Doppler measurements in a line sensitive to the umbral chromospheric oscillations, such as Fe I 543.4 nm. 3.2. Umbral Magnetic Field Oscillations The bottom, left-hand panel in Figure 6 shows the power spectra of variations in magnetic Ðeld strength as determined from the full Stokes inversion of the Fe I lines ( o B o , solid curve) and as determined simply by the separation of the peaks of the two lobes of the Stokes V proÐle ( o B o , dotted curve). In the power spectrum for o B o there Vpeak is an apparent power peak spanning the 5 minute band (2È4.5 mHz) and rising signiÐcantly above the apparent noise level. The rms variation in Ðeld strength, determined from the integrated power across the 5 minute band, is 4 G. The power spectrum for o B o has comparable power in the 5 Vpeak minute band, but the signal is noisier, and the power in the 5 minute band rises only marginally above the noise level. The least-squares inversion procedure Ðts analytical proÐles to all four Stokes proÐles of both Fe I lines and hence provides a more accurate measure of o B o than does o B o Vpeak computed from Fe I 630.25 nm alone. Our measured 4 G rms variation in magnetic Ðeld strength associated with the 5 minute umbral oscillations is much lower than that found7 recently by Horn, Staude, & Landgraf (1997). Signals at such small levelsÈa 4 G perturbation of the magnetic Ðeld corresponds to a 0.015 pixel change in the separation of the Stokes V lobesÈmust be regarded with considerable skepticism because the techniques of both instrumentation and analysis can cause ““ cross talk ÏÏ from real Doppler velocity signals into the inferred magnetic Ðeld. For example, small errors in the Ñat-Ðelding of the Stokes polarization images can alter the line proÐle shape, perhaps leading to spurious small Ñuctuations in the inferred vector magnetic Ðeld as the proÐle shifts back and forth across the calibration anomaly. The phase measurements discussed below also suggest that the inferred 4 G rms variation is the result of cross talk and not a real solar oscillation of the magnetic Ðeld strength. (This interpretation is further supported by the theoretical considerations presented in ° 5.) Although we have not identiÐed the source of such cross talk, we were able to rule out errors of the least-squares inversion itself through an examination of the time series of the uncertainties in the Ðeld strength and inclination, d o B o and dt , that are associlsq lsq ated with the inversion. The lower, right-hand panel of Figure 7 presents observed phase and coherence spectra between oscillations in the measured quantities Fe V1 and o B o . We note a slight increase in coherence in the 5 minute band, accompanied by phase di†erences that cluster around 0¡ and phases that tend to cluster around 180¡ at higher frequencies. Phase di†erences of either 0¡ or 180¡ are commonly encountered when measured Ñuctuations result from observational e†ects rather than solar e†ects. For example, a small, magnetized region having a steady Ñow that is caused by seeing to jitter on and o† of the spectrograph slit will produce an apparent time-varying velocity signal that is either in or out of phase with apparent Ñuctuations of the magnetic Ðeld, depending on the sign of the Ñow velocity and the magnetic 7 Although Horn et al. (1997) do not report an rms value for their data, visual inspection of their time series for magnetic Ðeld strength suggests a value of 20È30 G.

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FIG. 9.ÈAverage power spectra for regions of the sunspot outside of the umbra : umbra-penumbra transition region (left panels) and penumbra (right panels) shown separately for either side of the sunspot. See Fig. 8 and text for explanation.

Ðeld. The fact that the observed phase di†erences cluster around 0¡ in the 5 minute band and 180¡ at higher frequencies is a hint that the oscillatory power detected in o B o is not solar in origin and that the two frequency regimes su†er from di†erent instrumental e†ects. Theoretical considerations, discussed in ° 5, predict phase di†erences between these two observed extremes.

The bottom, right-hand panel of Figure 6 shows the power spectrum of the magnetic Ðeld inclination angle t (angle the Ðeld makes with respect to the line of sight) as determined by the Stokes inversion of the Fe I lines. There is an apparently signiÐcant power peak spanning the 5 minute band, with an rms variation of 0¡.09 in the range 2È4.5 mHz, corresponding approximately to a 1 G Ñuctuation in the

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FIG. 10.ÈAverage phase spectra (points) and coherence spectra (curves) for regions of the sunspot outside of the umbra : the umbra-penumbra transition region (left panels) and the penumbra (right panels). See Fig. 8 for the location of these regions.

line-of-sight component if the magnetic Ðeld were oscillating only in orientation, not in magnitude. For reasons similar to those given above, we also suspect that this measured 5 minute oscillation in the magnetic Ðeld inclination is due mostly to instrumental and inversion cross talk and not to a real solar phenomenon. 4.

FLUCTUATIONS IN MAGNETIC FIELD AND VELOCITY OUTSIDE OF THE UMBRA

Figure 8 (Plate 6) shows gray-scale plots of the spacetime behavior of Ñuctuations in Doppler velocity Fe V1 (in Fe I

630.15 nm), continuum intensity I , magnetic Ðeld strength c t. The graphs to the o B o , and magnetic Ðeld inclination right of each gray-scale panel are plots of the mean values of each quantity ; these mean values were subtracted from the time series of each quantity to give the Ñuctuations plotted in the spacetime panels. The location of the regions we deÐne as umbra and penumbra, as determined from the mean continuum intensity, are indicated on each graph. The plot of mean velocity Fe V1 shows the typical pattern of the Evershed Ñow : a blueshift in the center-side penumbra and a redshift in the limb-side penumbra.

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The top panel in Figure 8, showing Ñuctuations in the Doppler velocity Fe V1, shows distinct oscillations in the umbra with a period near 5 minutes and with a characteristic herringbone pattern of the phase fronts, with the phase of the oscillation decreasing with radial distance from the center of the sunspot. This characteristic herringbone pattern was noted previously by Thomas, Cram, & Nye (1984). The apparent radial phase speed of these oscillations is of order 100 km s~1, which is much greater than the sound speed or Alfven speed at the heights of formation of the Fe I 630.15 nm line (Bruls et al. 1991). The oscillations at the outer edge of the umbra lag those at the center of the umbra by nearly 2 minutes. An interpretation of the phase behavior of these oscillations in terms of their forcing by p-modes in the surrounding convection zone, and the implications of this behavior for the local helioseismology of sunspots, are discussed in another paper (Bogdan et al. 1998). Although events of larger amplitude are observed in the penumbra in all four measures presented in Figure 8, the perturbations occur mostly at lower frequencies. In Figure 9 we show power spectra computed for the umbra-penumbra transition (left) and for the penumbra (right). Separate curves are plotted for averages over regions on opposite sides of the umbra (see Fig. 8 for the locations of these regions.) Power in the 5 minute band is clearly visible in measures of Doppler velocity, with the side of the sunspot closest to the limb showing the largest power. Since this sunspot was observed so close to disk center, the di†erences in measured quantities on each side of the sunspot may result more from departures of the spot from cylindrical symmetry than from di†erences in the viewing angle with respect to the direction of the magnetic Ðeld. Fluctuations in o B o and t show no remarkable peaks, and what power is present could well be due to instrumental or observational sources. Clearly the penumbra shows much more oscillatory power than the umbra at low frequencies, the source of which was already evident in Figure 8. This will be discussed in ° 6 below. For completeness, we present in Figure 10 phase and coherence spectra in the frequency range 0È10 mHz for the pairs Fe V1È o B o and for Fe V1Èt in the penumbra and in the umbra-penumbra transition region. Generally the measured phase delays for the umbra-penumbra transition, and especially for the penumbra itself, show no distinct coherence. The single exception is a possible narrow peak at 2.5 mHz for the pair Fe V1È o B o in the umbra-penumbra transition. This peak appears on both sides of the sunspot, and there is some evidence for a peak there in the pair Fe V1Èt. This result is tentative, given the weakness of the correlation, but at least on the limb side the phases appear to di†er somewhat from 0¡. Lower frequency perturbations in the magnetic Ðeld and Doppler velocity at small scales are known to be associated with the Evershed e†ect (Shine et al. 1994 ; Rimmele 1994), and it is possible that our peaks in coherence may result from these events. Clearly, more investigation will be necessary to validate the results of Figure 10 and to identify their cause if they prove to be real. 5.

THEORETICAL CONSIDERATIONS

Having apparently detected rms magnetic Ðeld strength variations associated with 5 minute umbral oscillations of order 4 G, which we suspect may be at least partly an artifact of the observing procedure, we now turn to the

475

question of what variations might be expected on theoretical grounds. 5.1. Magnetic Oscillations in the Umbral Photosphere One possible source of variation in the measured magnetic Ðeld might be due to oscillations in the e†ective magnetic response height of the Fe I lines. Imagine an umbral magnetic Ðeld that diverges radially from the sunspot axis with increasing height. Suppose that this magnetic Ðeld remains rigid and Ðxed in position, while oscillatory motion of the gas occurs only along the magnetic Ðeld. Although the magnetic Ðeld does not change, we would measure an oscillation in magnetic Ðeld strength and inclination simply because the e†ective magnetic response heights of the Fe I lines move up and down with the oscillating atmosphere, and the Ðeld strength and inclination vary with height. A simple potential magnetic Ðeld model Ðt to our sunspot suggests that the umbral magnetic Ðeld strength drops o† with height at a rate of about 1 G km~1. From our measured rms velocity of about 40 m s~1 and a period of 300 s, we would expect rms vertical displacements of about 2 km and thus rms magnetic Ðeld strength variations of about 2 G from this mechanism, about half the rms we observe. However, for this mechanism the magnetic Ðeld Ñuctuation should be 180¡ out of phase with the vertical displacement and thus 90¡ out of phase with the vertical velocity. This is not consistent with the phase information shown in Figure 7. An alternative scenario, at the opposite extreme, is one in which a rigid, buoyant sunspot magnetic Ñux tube Ñoats up and down with the p-mode oscillations of the atmosphere and hence involves no relative motion between the magnetic Ðeld and the magnetic response height of the Fe I lines. This picture is probably accurate only for p-modes with wavelengths much larger than the sunspot diameter. In this case, there would be no magnetic Ðeld variations associated with the vertical velocity oscillations. Of course, a sunspotÏs magnetic Ðeld is not perfectly rigid in the presence of umbral velocity oscillations in the photosphere, where the plasma beta is not small. And so intermediate between these two extreme pictures is one in which the Ñuid motions distort the magnetic Ðeld and the resulting Lorentz force reacts back on the Ñuid motions. This selfconsistent interaction between the plasma and the magnetic Ðeld is accounted for through the computation of the magneto-atmospheric wave modes of the sunspot umbral atmosphere (Uchida & Sakurai 1975 ; Antia & Chitre 1979 ; Scheuer & Thomas 1981 ; Thomas & Scheuer 1982 ; Schwartz & Leroy 1982 ; Cally 1983 ; Zhugzhda 1984 ; Gurman & Leibacher 1984 ; Zhugzhda & Dzhalilov 1984 ; Zhukov 1985 ; Zhugzhda, Locans, & Staude 1984 ; Lou 1988 ; Campos 1989 ; Wood 1990 ; Abdelatif 1990 ; Hasan 1991 ; Cally & Bogdan 1993 ; Cally, Bogdan, & Zweibel 1994). Following Scheuer & Thomas (1981), we assume that the umbral magnetic Ðeld is uniform with a nominal strength of B \ 3000 G (see Fig. 8) and aligned with the 0 prevailing constant gravitational acceleration g \ 0.274 km s~2. The plasma is taken to be in hydrostatic balance and is described by the model M umbral core atmosphere of Maltby et al. (1986) between the atmospheric heights of z \ [122 km and ]2126 km. Above z \ 2126 km we append an isothermal 2 ] 106 K corona to the model M umbral core atmosphere, and below z \ [122 km we append a c \ 5/3 adiabatic polytrope that is smoothly Ðtted

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Here, c(z) 4 [cp(z)/o(z)]1@2 and a(z) 4 B /[4no(z)]1@2 denote 0  n speed, respecthe adiabatic sound speed and the Alfve tively, and c is the ratio of speciÐc heats. Although Figure 2 of Cally et al. (1994) is based on an imposed 4000 G magnetic Ðeld, it nevertheless provides a very useful sense of the variations of the sound speed and Alfven speed with height for [2 Mm ¹ z ¹ ]2.5 Mm in our present umbral magneto-atmosphere. Following the method employed by Cally et al. (1994), we solve equations (4) and (5) for the full range of altitudes [O \ z \ O by making use of the asymptotic solutions of these equations for very large Alfven speed [a D exp (z/2H ), c D z0 as z ] O ; H is the constant coronal o o density scale height] and sound speed (c D o z o0.5, a D o z o~0.75 as z ] [O).8 In each of these regimes the equations decouple to yield a vertically propagating slow magneto-atmospheric wave and an exponentially growing/ decaying fast magneto-atmospheric wave (or better, boundary layer). High in the isothermal corona, the slow magneto-atmospheric wave is essentially a vertically propagating acoustic wave, guided by the strong magnetic Ðeld, while at large depths in the polytrope, the slow wave behaves as a transverse, incompressible, Alfven wave. In practice, equations (4) and (5) are solved numerically for a

Ðnite range of z, and at the boundaries of the computational domain, the numerical solution is matched smoothly onto the asymptotic solution for the outward-propagating slow waves and the exponentially declining fast waves. The procedure is thus equivalent to applying ““ radiating,ÏÏ or ““ transmitting,ÏÏ boundary conditions at the base and top of the umbral atmosphere. A nontrivial solution is possible only for a restricted set of the separation parameters u, k, and m. We suppose that the umbral oscillations are excited by the external p-modes, and so we are free to specify the oscillation frequency u and the azimuthal order m and then search for the allowed (complex) horizontal wavenumbers k \ k (u), with n a nonnegative integer, that will permit a n nontrivial solution of the governing equations and the transmitting boundary conditions.9 The result of such a search is displayed in Figure 11. A sequence of modes with increasing horizontal wavelengths, j \ 2n/Re (k), is found for each frequency, l \ u/2n. As the prescribed oscillation frequency is continuously varied, these allowed wavelength eigenvalues trace out an eigencurve as indicated by the solid lines plotted in Figure 11. For frequencies below 4 mHz, these eigencurves are visually indistinguishable from those that would be obtained for our adopted umbral atmosphere but with the magnetic Ðeld absent. For this nonmagnetic case, we would associate with each eigencurve a p-mode of a deÐnite radial order n, the left-most curve corresponding to the fundamental f-mode, the next curve to the right to that of the n \ 1 p-mode, and so forth. By analogy, for the umbral magneto-atmosphere we shall refer to the eigenmodes as n-modes, which also have a deÐnite radial order n that increases from left to right (at low frequencies). For the umbral magneto-atmosphere the n-modes have complex wavenumbers due to the escape of slow magnetoatmospheric waves as o z o ] O. Overplotted at random locations along the eigencurves in Figure 11 are circles whose size indicates the quality factor, Q \ Re (k)/2n Im (k), of the mode. At low frequencies and large horizontal wavelengths, only downward-propagating quasi-Alfvenic slow magneto-atmospheric waves leak energy from the umbral oscillations. For frequencies in excess of B80% of the acoustic cuto† frequency at the umbral model M temperature minimum (B6 mHz for 100 km \ z \ 900 km), upward-propagating slow magneto-atmospheric waves can also escape through the 2 ] 106 K corona. Above 5 mHz it is apparent that the mode Q is no longer a monotonically decreasing function of frequency, but rather, along every eigencurve but one there are several critical frequencies, l \ 5.20 mHz (period P \ 192 s), l \ 6.68 mHz (P \ 150 * s),* and l \ 7.66 mHz (P \ 131 s), where Q attains a local * minimum. This cyclic variation of Q is related to transmission properties of the upward-propagating slow magnetoatmospheric waves through the discontinuity in temperature and density within the transition region (Cally 1983 ; Gurman & Leibacher 1984 ; Cally et al. 1994). Having identiÐed the allowed n-mode oscillations at a given p-mode forcing frequency, we now turn to the question of which n-modes are the favored response of the sunspot umbral atmosphere. This question is best addressed

8 This approach is preferable to the traditional application of reÑecting boundary conditions, since it avoids the creation of a dense distribution of spurious eigenmodes owing to the small vertical wavelength of the slow magneto-atmospheric wave at large depths (e.g., Hasan & ChristensenDalsgaard 1992 ; Banerjee, Hasan, & Christensen-Dalsgaard 1995).

9 In so doing we favor the view of Thomas, Cram, & Nye (1982), and Moore & Rabin (1985), that external dynamical processes, rather than internal overstable convection, are responsible for the excitation of the umbral oscillations.

to the base of the umbral atmosphere. The zero of the vertical coordinate z coincides with the continuum (500 nm) optical depth unity layer. The linear adiabatic oscillations of this magnetoatmosphere may be written as a superposition over frequencies u, horizontal wavenumbers k, and azimuthal orders m, of the separable velocity eigenfunctions u (r, h, z, t) \ r

G C

Re uü (z) J

D

H

m (kr) [ J (kr) exp [i(mh [ ut)] , m`1 kr m

G

(1)

H

m J (kr) exp [i(mh [ ut)] , u (r, h, z, t) \ Re [iuü (z) h kr m

G

H

u (r, h, z, t) \ Re [iwü (z)J (kr) exp [i(mh [ ut)] , m z

(2) (3)

where J (x) is the Bessel function of order m and where uü (z) and wü (z)m satisfy the following pair of coupled ordinary differential equations (Ferraro & Plumpton 1958 ; Scheuer & Thomas 1981) :

C A a2(z)

B

D C

d2 [ k2 [ c2(z)k2 ] u2 uü dz2 ] ik c2(z)

C

c2(z)

D

d2 d [ cg ] u2 wü dz2 dz

C

] ik c2(z)

D

(4)

D

(5)

d [ g wü \ 0 , dz

d [ (c [ 1)g uü \ 0 . dz

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FIG. 11.ÈMagneto-atmospheric eigenmodes of a realistic umbral atmosphere with an imposed uniform magnetic Ðeld of 3000 G. The solid lines give the loci of eigencurves in the frequency-horizontal wavelength plane, while the size of the randomly overplotted circles indicates the quality factor Q \ Re (k)/ 2n Im (k) \ 1/j Im (k). Each eigencurve has a deÐnite radial order n \ 0, 1, 2, 3, . . . as indicated. The overplotted boxes (dashed lines) show the approximate locations of the oscillations observed with the ASP. Eigenfunctions corresponding to the four Ðlled circles are plotted in Figs. 13 and 14.

using Figure 12 (Plate 7), which shows the spacetime behavior of Ñuctuations in the Fe V1 Doppler velocity (see Fig. 8) Ðltered to pass only oscillations in the 5 minute band (top) or the 3 minute band (bottom). These plots permit us to gain a sense of the range of horizontal wavelengths j that are observed within each frequency band. In the 5 minute band, the oscillations have rather large wavelengths, and the oscillations are coherent well into the outer penumbra. In sharp contrast, the oscillations in the 3 minute band have signiÐcantly smaller spatial scales and show no deÐnite lateral propagation tendencies. Based upon the appearance of the Ðltered oscillations shown in Figure 12, we have drawn two boxes over the n-mode eigencurves of Figure 11 that serve to indicate the approximate locations of the observed umbral oscillations in the theoretical l-j diagram. Within these two boxes we have selected four representative n-modes, denoted by the Ðlled circles, and plotted their eigenfunctions in the four panels of Figures 13 and 14. Since these eigenfunctions are complex, we have partitioned the perturbed velocities and magnetic Ðelds, parallel and perpendicular to the ambient magnetic Ðeld, into their respective amplitudes (Fig. 13) and phases (Fig. 14). The overall amplitude normalization is chosen to be consistent with the observed Fe V1 Doppler power spectrum displayed in the upper, left-hand panel of Figure 6. Accordingly, at the height of formation of the Fe I 630.15 nm Doppler signal

(z B ]100 km), the Ðeld-aligned velocity is set to 42 m s~1 for the 2.97 mHz n mode in the 5 minute band and 13 m s~1 for the n and n2 modes in the 3 minute band. The 4.46 3 lies on 4 the boundary between the 5 and 3 mHz n mode 3 minute bands, and so its velocity has been set to an intermediate value of 26 m s~1 at z \ ]100 km. The zero of phase can be assigned purely as a matter of convenience. As can be seen in Figure 13, for all of these representative umbral oscillations, the expected rms variations of either component of the magnetic Ðeld is at the level of a few tenths of a gauss. In the umbra, where the Ðeld is predominantly vertical, the Ñuctuations in o B o are nearly identical to the Ñuctuations in dB , and so for the representative 5 A minute n-mode the expected rms Ñuctuations in o B o are about 0.5 G. This is an order of magnitude smaller than the observed 4 G Ñuctuations (Fig. 6). In the extreme lower left-hand corner of the 5 minute box shown in Figure 11, a 4 G rms Ñuctuation is possible. But along the n eigencurve 1 the Ñuctuations are only about 1 G, and in the upper, righthand corner of the 5 minute box they are even slightly below a tenth of a gauss. These theoretical results strengthen the idea that our detection of a 4 G rms magnetic Ðeld Ñuctuation is an artifact of the observational procedure. This conclusion becomes more convincing when issues related to phase are considered. The phase diagram for our representative 5

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FIG. 13.ÈEigenfunction amplitudes for the four n-modes indicated by the Ðlled circles in Fig. 11. The thin horizontal line marks the approximate height of formation of the Fe I 630.15 nm line. The oscillatory behavior of dB and dv below the visible photospheric surface (z \ 0) is a signature of the downward-propagating, quasi-Alfvenic, slow magneto-atmospheric wave. Note theMchange of horizontal scales between the left and right sets of panels.

minute n-mode is found in the lower, left-hand panel of Figure 14. The phase of the vertical velocity shows little variation with height, consistent with the evanescent character of the 5 minute oscillations in the solar atmosphere.10 Because the magnetic Ðeld perturbation is related to the horizontal Ñuid velocity through the magnetic induction equation, and because u and k are largely real numbers, it follows that dv and dB must have identical phases and be M with AdB . However, the relative phase 90¡ out of phase M between these three quantities and the vertical velocity is not predictable by such an elementary argument. Rather, it is a powerful diagnostic of the mode coupling that occurs below the visible umbral surface where a(z) B c(z). In the 5 minute box, the phase angle between the vertical velocity and the vertical magnetic Ðeld Ñuctuation is a function of the mode frequency, with a somewhat weaker dependence upon wavelength. At 2 mHz, dB leads dv by as much as A ]59¡, and, consistent with the Aupper, left-hand panel of Figure 14, this phase angle is as small as ]17¡ by 4.5 mHz. The lower right-hand panel of Figure 7 indicates weak coherence between the observed Fe V1 Doppler velocity and magnetic Ðeld Ñuctuations in the 5 minute band, and consequently the phase di†erence is rather poorly deter10 The increase of phase experienced by dB and dv with increasing A depth is a consequence of the generation of the downward-propagating, quasi-Alfvenic, slow magneto-atmospheric wave just below the visible umbral photosphere.

mined. There is at best a very slight tendency for the Doppler velocity to lead the magnetic Ðeld by about 20¡, while theory requires that the Doppler velocity lag the magnetic Ðeld by B40¡È50¡. Taken together, these various pieces of evidence lead us to conclude that the enhanced magnetic Ñuctuations observed in the 5 minute band do not represent an actual detection of Ñuctuating magnetic Ðelds in the sunspot umbral photosphere. No signiÐcant peaks are present in the power spectrum of umbral magnetic Ðeld Ñuctuations in the 3 minute band (see Fig. 6). Figure 13 indicates that, on purely theoretical grounds, any real magnetic Ðeld Ñuctuations in the 3 minute band would probably lie below the 0.1 G level and thus would be even harder to detect than their counterparts in the 5 minute band. 5.2. Chromospheric Umbral Oscillations Although this paper is based upon observations made exclusively in the umbral photosphere, it seems worthwhile to consider the implications Figures 13 and 14 have for oscillatory phenomena in the umbral chromosphere. From Figure 13 it is clear that a drastic increase in the chromospheric vertical velocity occurs between 4.5 and 5.2 mHz. This is because the upward-propagating, slow magnetoatmospheric waves are able to tunnel through the 6 mHz acoustic cuto†Èfrequency barrier of the cool temperature minimum layers. Based on Figure 13, one may predict the Doppler velocities expected for observations made with the

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FIG. 14.ÈEigenfunction phases for the four n-modes indicated by the Ðlled circles in Fig. 11. The thin horizontal line marks the approximate height of formation of the Fe I 630.15 nm line. The steady phase increase of dB and dv below the visible photospheric surface (z \ 0) is a signature of the downward-propagating, quasi-Alfvenic, slow magneto-atmospheric wave. Note theMchange of horizontal scales between the left and right sets of panels and the change of vertical scale relative to Fig. 13.

Fe I 543.4 nm line that forms at a nominal height of z B 600 km (Lites & Thomas 1985). In the 5 minute band, Doppler Ñuctuations of order 60È80 m s~1 are anticipated in this line if 40 m s~1 Ñuctuations are observed concurrently in the photospheric Fe I 630.15 or Ti I 630.4 nm lines. In the 3 minute band, 80È100 m s~1 rms Doppler velocities are expected for the Fe I 543.4 nm when 13 m s~1 is recorded with Ti I 630.4 nm. Given the uncertainties in the contribution functions for these various lines, these predictions are in outstanding overall agreement with the data reported by Lites & Thomas (1985). This agreement suggests that the diverse character of the oscillatory power spectra obtained at photospheric and chromospheric levels (Thomas 1985 ; Lites 1992) is in fact consistent with the presence of a single realization of excited n-modes. The shift in the peak of the chromospheric power spectra into the 3 minute band is a natural consequence of the drastic change in the character of the n-mode eigenfunctions between 4.5 and 5.0 mHz, as illustrated by the upper two panels of Figure 13. Similar abrupt changes are also echoed in the phase diagrams of Figure 14 and in others that we do not explicitly present in this paper. Based upon this more complete sequence of phase diagrams, we observe the following. At 5 mHz, dv begins to show a slight increase (B6¡) of phase at the base Aof the corona relative to its value at the base of the photosphere. By 5.3 mHz, the phase di†erence between

these locations is nearly 170¡, and that value continues to increase very slowly, reaching 225¡ at 6.5 mHz. As indicated by the lower, right-hand panels of Figures 13 and 14, most of the phase increase takes place in a very thin layer of the atmosphere where the amplitude of dv is severely reduced. A upper, right-hand This ““ node ÏÏ is just starting to form in the panels of Figures 13 and 14, and from Figure 11 it is clear that this formation is associated with the Ðrst minimum in the quality factor Q. The partial downward reÑection of the upward-propagating, slow magneto-atmospheric wave from the base of the corona is responsible for this quasistanding wave pattern in the photosphere and chromosphere. With increasing frequency this node moves to higher altitudes. At the next critical frequency of 6.68 mHz, a second node forms near the base of the photosphere and an additional 180¡ of phase separates the vertical oscillations at the photosphere and coronal base. The process repeats again at 7.66 mHz and sequentially at successively higher frequencies. The formation of nodes in dv near the base of the photoA is paramount in shaping sphere with increasing frequency the various resonator theories of sunspot umbral oscillations (e.g., Zhugzhda et al. 1987). Suppose the Doppler power spectral density at the photospheric base is maintained at a constant value between the frequencies of 5.0 and 5.4 mHz through some unspeciÐed process. Then the

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Doppler power spectral density at chromospheric levels must necessarily have a very large peak in the vicinity of the critical frequency, 5.2 mHz, where the eigenfunction develops its Ðrst node in the photosphere. On the other hand, no such chromospheric power enhancement would be detected if instead there were a compensating deÐcit of power at the photosphere precisely at 5.2 mHz. In fact, this second scenario is much more likely than the Ðrst. For if the umbral oscillations are forced by the horizontal Ñuid motions of the neighboring convection or the impinging p-modes, then, since the eigenfunctions plotted in Figure 13 show no corresponding photospheric nodal structure in dv , there is no reason to expect anything but a rather M uniform distribution of power between 5.0 and 5.4 mHz in the horizontal Ñuid motions at the photospheric base. Hence it follows that no enhanced chromospheric response is present at 5.2 mHz, but instead there is a lack of vertical oscillatory motion at 5.2 mHz in the photosphere simply because the eigenfunction chooses to have a node there. This idea that the chromospheric ““ resonance ÏÏ periods of 192, 150, and 131 s are better regarded as photospheric oscillation gaps or deÐcits merits further observational study. The short duration of the ASP data used here rules out the search for these deÐcits in photospheric Doppler power spectra. The 50 hr data set employed by Penn & LaBonte (1993) would be better suited to the task (*l B 4.5 kHz), but it is severely compromised by having a low Nyquist frequency of only 5.4 mHz. The observed phase di†erences between the various Doppler velocities, as shown in Figure 7, also share this same property of exhibiting a sudden change in character at around l \ 5 mHz. Below 5 mHz the lack of phase di†erences is consistent with the evanescent character of the 5 minute oscillation (see lower left-hand panel of Fig. 14). For frequencies above about 5 mHz the observations show the steady increase of phase with frequency that we used to infer the relative di†erences in the velocity response heights of the various spectral lines (see ° 3.1). This behavior is manifestly inconsistent with the theoretical phase plots for our two representative 3 minute modes. The observations do not show the reÑection of the vertically propagating acoustic waves from the base of the corona that is present in the theoretical model. Interestingly, this same absence of downward reÑection is also found for p-modes with frequencies in excess of about 5.3 mHz that are observed in the quiet Sun (Kumar et al. 1994). The lack of downward reÑection observed in the 3 minute band suggests a deÐciency of our theoretical model. Two remedies immediately come to mind. First, we might incorporate nonadiabatic e†ects above z B [100 km, and second, we could take account of nonlinearities that develop due to the rapid increase of dv with altitude. Both A addressed by the of these physical processes are admirably recent numerical simulations of Carlsson & Stein (1995, 1997). These papers treat the nonlinear steepening of upward-propagating acoustic disturbances, taking careful account of non-LTE radiative transfer and Ñuctuations in atomic level populations. Although these authors envisaged applying their Ðndings to the internetwork quiet SunÈ presumably below the magnetic canopy (Anzer & Galloway 1983 ; Fiedler & Cally 1990)Èthe 3 minute eigenmodes plotted in Figure 13 indicate that their results are particularly germane to the umbral atmosphere where the strong vertical magnetic Ðeld imposes the essential features of one-

Vol. 497

dimensional gas dynamics upon the excited oscillations. If the work of Carlsson & Stein is applied literally to the problem at hand, then one may begin to understand the physical basis for the steady increase of phase with height and the nonlinear sawtooth shape of velocity oscillations observed in the umbral chromosphere (Lites 1992). 6.

PERSISTENT MOTIONS AND EVOLUTION IN THE SUNSPOT

Returning to the observations, we now consider the more persistent, low-frequency behavior of the velocity and magnetic Ðeld Ñuctuations. The presence of some interesting low-frequency components of the Ñuctuations is already evident in Figure 8, where in all four panels we see a pattern of features migrating radially inward in the umbra and radially outward in the penumbra and surrounding quiet photosphere. In order to examine the low-frequency behavior of these Ñuctuations, we subjected each time series to a low-pass Ðlter of functional form f (l) \ [1 ] cos w(l [ l )]/2 in the range l \ l \ 2l , where l \ 1.195 0 w \ n/l , with f \0 1 (l \ l )0and f \ 0 0(l [ 2l ). mHz and 0 power at frequencies 0 at and above 2.39 0 This Ðlter rejects all mHz. The resulting Ðltered time series are shown in the gray-scale plots of Figure 15 (Plate 8). Note the strong tendency, in all four quantities, for an inward radial drift in the inner penumbra and umbra and an outward radial drift in the outer penumbra and moat. In all four quantities, but especially in Fe V1 and in the Ðeld inclination t, we see features migrating continuously inward from the penumbra well into the umbra. In continuum intensity, the bright features moving inward from the penumbra into the umbra (second panel from the top in Fig. 15) no doubt correspond to the ““ peripheral ÏÏ umbral dots observed by Muller and others (Muller 1973, 1992 ; Kitai 1986). The speed of this inward drift is consistent with that measured by Kitai (1986), i.e., about 0.4 km s~1. In the panels depicting Doppler velocity, magnetic Ðeld strength, and Ðeld inclination in Figure 15, we see features moving radially inward from the penumbra all the way to the center of the umbra. This pattern of convergence toward the center of the umbra is most noticeable in velocity and Ðeld inclination. One might expect a strong correlation among the inward-migrating features in the four quantities in Figure 15. However, when we tested this by blinking pairs of these spacetime images, we found no obvious correlation or anticorrelation between the streaks, except for a weak correlation in the case of velocity Fe V1 and Ðeld inclination t. Similarly, scatter plots of pairs of the four quantities also fail to show any strong relationships among them. The inward- and outward-migrating features in Figure 15 may well be signatures of convective processes in the sunspot umbra and penumbra. Numerical simulations of magnetoconvection with an inclined magnetic Ðeld generally show horizontal propagation of the convective pattern (Hurlburt, Matthews, & Proctor 1996 ; Weiss 1997). The inward-migrating features in the inner penumbra and umbra are most likely due to the convective rise of individual, thin magnetic Ñux tubes. A magnetic Ñux tube lying along the outer boundary (the magnetopause) of the sunspotÏs overall magnetic Ðeld conÐguration is heated by radiative exchange with the surroundings and then, as a result of increased buoyancy, Ñoats upward while tethered

No. 1, 1998

VELOCITIES AND MAGNETIC FIELDS IN A SUNSPOT

somewhere deeper below the surface in the umbra (Jahn & Schmidt 1994). The footpoint where such a Ñux tube intersects the base of the photosphere thus moves radially inward. Recent numerical simulations of this process by Schlichenmaier, Jahn, & Schmidt (1997) indeed produce bright footpoints migrating inward at speeds of 1 km s~1 or less, much like what is seen in intensity Figure 15. The lack of any strong correlations among the observed features in intensity, velocity, magnetic Ðeld strength, and Ðeld inclination could simply be due to the fact that these rising Ñux tubes are very thin, with diameters below our limit of spatial resolution (as is the case in the numerical simulations just mentioned). We also note in Figures 8 and 15 that o B o decreases systematically in some portions of the sunspot during the duration of this time series, notably in the upper half (center-side) of the umbra and in the lower (limb-side) penumbra. The umbral decrease amounts to about 150 G. We have no reason to believe that these decreases are not solar in origin, although the possibility that they arise from some aspect of the data interpretation must remain a possibility. In the umbra, as we have noted above, the observed secular changes might be associated with convective interchange, and this process or other processes governing the slow evolution of the spot could alter the Ðeld strength in the region of the sunspot deÐned by our narrow slit. Penumbral structure also evolves on these timescales, and because the slit was oriented through the center of the sunspot, it is entirely possible that the Ñuted penumbral Ðeld structure is evolving so as to cause the changes in o B o we observe. 7.

CONCLUSIONS

From our present observations and theoretical model, along with comparisons with previous work, we draw the following principal conclusions : 1. Oscillations in umbral magnetic Ðeld strength and inclination associated with the 5 minute oscillations in the umbra are very weak. Our measured rms amplitudes of Ñuctuations in magnetic Ðeld strength and inclination over the 5 minute band are about 4 G and 0¡.09, respectively, and for several reasons we suspect that the actual solar oscillations are smaller than this. The observed phase relation between Doppler velocity and magnetic Ðeld strength suggests that our measured magnetic Ðeld Ñuctuations are at least partly due to cross talk in the observing procedure and not to a real solar oscillation. Moreover, the magnitude of oscillations in magnetic Ðeld strength predicted by the

481

eigenmodes of a theoretical model of these umbral oscillations, after matching the eigenmodes to the observed velocity Ñuctuations, is at most only about 0.5 G, and the theoretically predicted phase relation between velocity and magnetic Ðeld does not show up in the observations, where instead the phase relationship is more consistent with cross talk between the two signals. These considerations all suggest that the actual rms amplitude of 5 minute oscillations in the umbral magnetic Ðeld strength is at most about 0.5 G. 2. Our theoretical model of umbral oscillations also predicts amplitudes of chromospheric umbral oscillations in excellent agreement with earlier observations (e.g., Lites & Thomas 1985). The well-known shift of power peaks from the 5 minute band into the 3 minute band in going from the umbral photosphere to the umbral chromosphere is a natural consequence of the drastic change in character of the eignemodes of oscillation (n-modes) between the frequencies 4.5 and 5.0 mHz, due to increased tunneling through the acoustic cuto†Èfrequency barrier. 3. With our high-quality measurements of umbral velocity oscillations above the acoustic cuto† frequency, we were able to determine the relative velocity response heights of photospheric spectral lines from the phase di†erences between velocities in these lines, assuming the oscillations propagate vertically at the local sound speed. To our knowledge, this is the Ðrst time this has been done for oscillations in a sunspot umbra. 4. Small, distinct features in the Ñuctuations of continuum intensity, Doppler velocity, magnetic Ðeld strength, and magnetic Ðeld inclination are seen to migrate radially inward from the inner penumbra all the way to the center of the umbra, at speeds of a few tenths of a kilometer per second. We suspect that these moving features are signatures of the convective interchange of magnetic Ñux tubes in the sunspot, although we failed to Ðnd any strong correlation among the features in the di†erent quantities, which may indicate that these features are smaller than our limit of spatial resolution. We are grateful to the observers at the NSO/Sacramento Peak Vacuum Tower Telescope, Steve Hegwer, Richard Mann, and Brian Armstrong, for their assistance with these observations. We also thank R. Casini, K. D. Leka, and O. Steiner for helpful and informative discussions and comments. J. H. T. was supported by NASA grant NAGW-2123.

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FIG. 1.ÈContinuum perspective image of the NOAA 7912 sunspot observed on 1995 October 15 (bottom), showing the location of the ASP spectrograph slit (width \ 0A. 75). The portion of map displayed is 104A ] 74A. Superposed yellow contour lines mark the transitions between the umbra and penumbra and between the penumbra and quiet Sun. Overlying the continuum perspective is a color plot of the zenith angle of the magnetic Ðeld as derived by the inversion procedure. The ““ hair ÏÏ on the zenith angle plane indicates the orientation (but neither the strength nor the polarity) of the vector Ðeld. The location of the ASP slit and the boundaries between the umbra, penumbra, and quiet Sun shown in the lower continuum perspective are also overplotted in the rendering of the zenith angle. Spacetime scans of the magnetic Ðeld strength and the magnetic azimuth and zenith angles at the ASP slit location are shown in the three overlapping panels at the top of the Ðgure. LITES et al. (see 497, 465)

PLATE 4

FIG. 2.ÈSample Stokes I, Q, U, and V proÐles of the spectrum near 630 nm along the ASP slit at one instant of time. In order of increasing wavelength across the upper, left-hand panel (Stokes I), the six visible absorption features are Fe I 630.15, telluric O 630.20, Fe I 630.25, telluric O 630.30, Fe I 630.35, 2 are crosshairs placed across 2the ASP slit to aid in and Ti I 630.40 nm. The two narrow, dark horizontal lines that run across the top and bottom of this panel tracking and registration of consecutive images. The bottom two panels are the same images of Stokes I and V spectra as the top two panels but scaled so as to bring forward the weak molecular lines of TiO and CaH that occur in the sunspot umbra between the stronger lines. LITES et al. (see 497, 465)

PLATE 5

FIG. 8.ÈGray-scale plots showing the spacetime behavior of Ñuctuations in ( from top to bottom) Doppler velocity Fe V1 (in Fe I 6302.5 nm), continuum intensity I , magnetic Ðeld strength o B o, and magnetic Ðeld inclination t. In each case, bright represents a positive Ñuctuation and dark, a negative Ñuctuation c in the quantity plotted. Vertical streaks are due to seeing variations. The graphs to the right of each panel are plots of the mean values of each quantity. Thicker segments of the lines in the graphs deÐne the umbral and penumbral regions, with quiet Sun outside these regions and umbra-penumbra transitions in between the thicker segments. The direction toward the center of the disk is indicated in the top, right-hand panel. LITES et al. (see 497, 474)

PLATE 6

PLATE 7

LITES et al. (see 497, 477)

FIG. 12.ÈGray-scale plots showing the spacetime behavior of Ñuctuations in Doppler velocity Fe V1 (in Fe I 6301.5 nm), similar to the upper panel of Fig. 8, except that the time series has been Ðltered to pass only the 5 minute frequency band (2 mHz \ l \ 4.5 mHz, top) or the 3 minute frequency band (4.5 mHz \ l \ 7 mHz, bottom). Bottom, right-hand panel displays time-averaged continuum intensity I . Note the di†erence in scaling of the amplitudes of the two frequency ranges. c

FIG. 15.ÈGray-scale plots similar to those in Fig. 8, except that in each case the Ñuctuations have been passed through a low-pass Ðlter that rejects all power at frequencies above about 2.4 mHz. These plots thus show the low-frequency, more persistent components of each Ñuctuating quantity. Note the strong tendency for a radial inward drift of features in the inner penumbra and umbra, persisting all the way to the center of the umbra, and a radial outward drift in the outer penumbra and surrounding quiet photosphere. The graphs to the right of each panel are plots of the mean values of each quantity. LITES et al. (see 497, 480)

PLATE 8