transverse oscillations in coronal loops observed with trace

TRANSVERSE OSCILLATIONS IN CORONAL LOOPS OBSERVEDWITH TRACE

II. Measurements of Geometric and Physical Parameters

MARKUS J. ASCHWANDEN, BART DE PONTIEU, CAROLUS J. SCHRIJVER andALAN M. TITLE

Lockheed Martin Advanced Technology Center, Solar and Astrophysics Laboratory, DepartmentL9-41, Bld.252., 3251 Hanover St., Palo Alto, CA 94304, U.S.A. (e-mail: [email protected],

[email protected], [email protected], [email protected])

(Received 24 September 2001; accepted 14 December 2001)

Abstract. We measure geometric and physical parameters of transverse oscillations in 26 coronalloops, out of the 17 events described in Paper I by Schrijver, Aschwanden, and Title (2002). Theseevents, lasting from 7 to 90 min, have been recorded with the Transition Region and Coronal Ex-plorer (TRACE) in the 171 and 195 Å wavelength bands with a characteristic angular resolutionof 1", with time cadences of 15–75 seconds. We estimate the unprojected loop (half) length L

and orientation of the loop plane, based on a best-fit of a circular geometry. Then we measurethe amplitude A(t) of transverse oscillations at the loop position with the largest amplitude. Wedecompose the time series of the transverse loop motion into an oscillating component Aosc(t) and aslowly-varying trend Atrend(t). We find oscillation periods in the range of P = 2–33 min, transverseamplitudes of A = 100–8800 km, loop half lengths of L = 37 000–291 000 km, and decay times oftd = 3.2–21 min. We estimate a lower limit of the loop densities to be in the range of nloop = 0.13–

1.7 × 109 cm−3. The oscillations show (1) strong deviations from periodic pulses, (2) spatiallyasymmetric oscillation amplitudes along the loops, and (3) nonlinear transverse motions of the cen-troid of the oscillation amplitude. From these properties we conclude that most of the oscillatingloops do not fit the simple model of kink eigen-mode oscillations, but rather manifest flare-inducedimpulsively generated MHD waves, which propagate forth and back in the loops and decay quicklyby wave leakage or damping. In contrast to earlier work we find that the observed damping times arecompatible with estimates of wave leakage through the footpoints, for chromospheric density scaleheights of ≈ 400–2400 km. We conclude that transverse oscillations are most likely excited in loopsthat (1) are located near magnetic nullpoints or separator lines, and (2) are hit by a sufficiently fastexciter. These two conditions may explain the relative rarity of detected loop oscillations. We showthat coronal seismology based on measurements of oscillating loop properties is challenging due tothe uncertainties in estimating various loop parameters. We find that a more accurate determinationof loop densities and magnetic fields, as well as advanced numerical modeling of oscillating loops,are necessary conditions for true coronal seismology.

1. Introduction

In Paper I (Schrijver, Aschwanden, and Title, 2002) we describe 17 events withtransverse oscillations of coronal loops observed with the Transition Region andCoronal Explorer (TRACE) during 1998–2001. This unique dataset allows for anumber of dynamical studies. In Paper I we described the triggering and evolu-

Solar Physics 206: 99–132, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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100 M. J. ASCHWANDEN ET AL.

tion of these loop oscillations qualitatively, while we undertake in this Paper IIquantitative measurements to describe the geometric and dynamical parameters, tofirst order. The 3D loop geometry is approximated by segments of inclined circles.We characterize the dynamical motion by an exponentially damped oscillation,and a residual drift of the loop centroid. The oscillation measurements are madeat a single position in each loop, where the oscillation is most prominent. Thisrepresents the first study of quantitative measurements of a comprehensive datasetof transverse loop oscillations observed in the solar corona.

Quantitative measurements of transverse loop oscillations are thought to pro-vide a diagnostic of the kink mode (Aschwanden et al., 1999b; Nakariakov et al.,1999), which gives a relation between the oscillation period P , wavelength λ, andAlfvén velocity vA. The observed parameters (λ, P ) permit us therefore to esti-mate the Alfvén velocity vA(B, ne), which theoretically provides a measurementof the coronal magnetic field strength B, if we can measure the electron density ne.We can estimate lower limits for this electron density ne from the observed EUVbrightness and loop widthw. This is currently the only available method to estimatethe coronal magnetic field strength directly in a well-defined geometric locationin the solar corona. Alternative methods to measure the coronal magnetic fieldare gyro-resonance modeling of radio emission (Lee et al. 1999), but that methodis restricted to strong-field regions (B ≈ 1000–2000 G) above sunspots and isnot applicable at higher coronal altitudes. The most popular methods to quantifythe coronal magnetic field are (potential-field, force-free, constant-α) extrapola-tion methods of photospheric magnetograms (e.g., Gary, 1989), but these methodsare only constrained by lower boundary conditions, while extrapolations into thecorona face large uncertainties due to magnetohydrodynamic (MHD) effects inthe chromosphere and transition region (which results into an unknown degreeof canopy divergence) and unknown electric currents (which causes deviationsfrom potential fields). Geometric tracings of coronal loops and comparisons withthe extrapolated magnetic field clearly show significant deviations (e.g., Gary andAlexander, 1999; Aschwanden et al., 2000) that are not adequately described byany theoretical model, and thus reveal a lack of knowledge of physical conditionsin the solar corona. It is, therefore, desirable to have a better method to determinethe coronal magnetic field, so that unknown parameters (current, nonpotentiality,damping) in theoretical extrapolation models can be constrained.

In this paper we describe first the method of analysis (Section 2), present thenthe observational analysis (Section 3), and discuss the statistical parameter rangesfound and their implications for physical interpretations (Section 4). Conclusionsare summarized in Section 5.

CORONAL LOOP OSCILLATIONS, II 101

2. Data Analysis

The selection and a qualitative description of the data is given in Paper I, containingthe date, start time, end time, cadences, wavelength, active region numbers, andGOES class of associated flares of the 17 observed events of oscillating loops(Table I in Paper I).

2.1. DIFFERENCE IMAGES

From the TRACE movies, which we define by a datacube F(xi, yj , tk) withpixel ranges 1 < xi, yj < n with n = 768 or 1024 and time ranges 1 < tk < m

with m = 18–130, we identify oscillating loops in difference images D(x, y) =F(x, y, tD2) − F(x, y, tD1). The best contrast is obtained when the times of thedifference images are chosen when a transverse oscillation reaches the peak ampli-tude in one direction, say at time tD1, and the second image a half oscillation periodlater, i.e. tD2 = tD1 + P/2. Because loop oscillations generally are exponentiallydamped, the largest amplitude is mostly found during the first detected period, andhence we choose difference images with this combination of tD1 at the first peakor valley, and a time tD2 approximately a half period later. Because we subtract anearlier from a later image, positive (bright) structures in greyscale images mark theposition of the loop in the later image, and negative (dark) structures mark the loopposition in the earlier image. This allows directly to read off the direction of theloop motion from the greyscale images, shown in Figures 3–16.

2.2. RECONSTRUCTION OF 3D LOOP GEOMETRY

In the difference images we outline the oscillating loop by marking about np ≈7–11 positions along the entire loop, (xi, yi), i = 1, . . . , np, where (x1, y1) and(xnp , ynp ) represent the two footpoints, as best we can determine these (see alsoFigures in Paper I). A finer resolution of the projected loop coordinates is achievedwith a 2-D spline interpolation [Interactive Data Language (IDL) procedureSPLINE_P.PRO). From the measured pixel numbers (xi, yi), the offset of theTRACE image from Sun center (XCEN and YCEN in the FITS header) and theephemerides (heliographic longitude and latitude of Sun disk center, and positionangle) we convert the measured image coordinates (xi, yi) into heliographic coor-dinates (li , bi) (see coordinate transformations in Appendix A of Aschwanden etal. 1999a). At the midpoint between the two footpoints, at coordinates

l0 = (l1 + lnp )/2 , (1)

b0 = (b1 + bnp)/2 , (2)

we define the reference point of a loop plane coordinate system, which has anazimuth angle α of

α = arctan

(bnp − b1

lnp − l1

)(3)


Figure 1. Definition of geometric parameters of circular loop model (thin circle) which approximatesthe observed loop shape (thick curve). The circular model has footpoints on the solar surface atidentical positions as the observed loop, and is defined by a height h0 of the circular loop center anda circular radius rloop. The loop plane is generally inclined with respect to the vertical to the solarsurface.

to the solar east-west direction. The loop baseline, defined as the distance betweenthe footpoints along the solar surface is

Lbase = 2πR�3600

√[(lnp − l1) cos b0]2 + (bnp − b1)

2 . (4)

Approximating the loop geometry with a circular segment in the loop plane, de-fined by a circle with radius r and height h0 of the circular center above the solarsurface (Figure 1), we obtain the loop radius r as function of the height h0 by therelation

r =√(

Lbase

2

)2

+ h20 . (5)

The full quantification of the 3D loop geometry requires 6 parameters, 2 in theloop plane (r, h0), see Figure 1, and 4 to define the relative position and orientationof the loop plane to the heliographic coordinate sytsem (l0, b0, α, ϑ), includingthe azimuth angle α of the loop baseline and the inclination angle ϑ of the loopplane to the vertical (see Figure 2). Thus we have to optimize 2 free parameters(h0, ϑ) to obtain a good match with the observed image coordinates. We achievethis by minimization (using the Powell method) of the differences in the projectedimage coordinates, after transformation from loop coordinates into heliographiccoordinates (li, bi), and then to image coordinates (xi, yi), The results of the six


Figure 2. Definition of the 3D loop coordinate system in the loop plane with respect to heliographiccoordinates (li , bi ) and image coordinates (xi , yi ). The loop plane has an inclination angle ϑ to thevertical plane (dark-gray), which is rotated by the azimuth angle α to the east-west vertical plane(bright-gray).

loop coordinate elements (l0, b0, r, h0, α, ϑ) are listed in Table I, and graphicallyshown in Figures 3–16 (bottom left frame).

2.3. MEASUREMENT OF OSCILLATION AMPLITUDE

In this study we measure the transverse oscillations at a single position of eachloop, chosen at locations where the oscillation is most pronounced in the differenceimages and least confused by the dynamics of adjacent structures (neighboringloops or erupting filaments). After we have chosen a suitable location along a loopin a difference image, say at position (xA, yA), we extract flux data with bilinearinterpolation in a coordinate grid that is aligned with the loop axis at location(xA, yA). This way we obtain 3D data cubes of the flux F(r, s, t), where s is thecoordinate along the loop segment, r in orthogonal direction (radial to loop axis),and t as function of time. Then we obtain the cross-section profiles f (r, t) byaveraging along the loop axis s over some segment %s (typically 10–50 pixels,


which correspond to ≈ 10% of the loop length) to improve the signal-to-noiseratio,

f (ri, tk) = 1

ns

ns∑j=1

F(ri, sj , tk) . (6)

Next we measure, at each time tk, the centroid position of the oscillating loop byfitting a Gaussian to the cross-section,

f (ri, tk) ≈ fback(ri, tk)+ floop(tk) exp

(−[ri − rG(tk)]2

2σ 2G(tk)

), (7)

where the center position rG(tk) of the Gaussian yields the oscillation amplitude,σG(tk) the Gaussian width of the loop, floop(tk) the flux associated with the oscil-lating loop, and fback(tk) the background flux (approximated by a linear function).

2.4. TIME SERIES ANALYSIS

The loop oscillations consist of an oscillatory component, with strong dampingover a few periods. On top of this oscillatory motion there is often also a systematicmotion of the entire loop, which we call a trend. Thus we model the transversemotion of a loop with two components,

rG(t) = Atrend(t)+ Aosc(t) , (8)

where the trend is quantified by a low-order polynomial of first to sixth degree,

Atrend(t) = a0 + a1(t − t0)+ a2(t − t0)2 + a3(t − t0)

3 + · · · (9)

(with t0 the start time of the time series and (t − t0) in units of hours), and theoscillatory component with an exponentially damped sine function,

Aosc(t) = A0 sin

(2π [t − t0]

P− ϕ0

)exp

(− t − t0

td

)(10)

with A0 the oscillation amplitude at time t = t0, P the harmonic oscillation period,and ϕ0 the phase shift. In total, we have 4 + n free parameters (with n the de-gree of the trend polynomial) for this time-dependent model (A0, P, ϕ0, t0, a0, a1,

[a2, . . .]), which is fitted to a time series r0(tk), k = 1, . . . , nt with typically nt =18–130 datapoints. We accomplish these fits with a minimization of the differencesfrom the model to the observed time series rG(tk), using the Powell method (IDLprocedure POWELL). The time series parameters are listed in Table II.

The introduction of the ‘trend function’ is an attempt to separate the (expo-nentially damped) oscillatory loop motion from other superimposed dynamics, forwhich we have no quantitative physical model at this point. We are aware that thedetermination of the so-called trend is not unique, and thus restricts the accuracy


of the determined oscillation parameters. However, because the trend plus the os-cillatory function are fitted simultaneously with a least-square method, the trendis automatically adjusted to yield a best solution for the superimposed oscillatorymotion. A more accurate determination of the oscillation parameters can only beobtained once a physical model for the trend function is available, which has alsoto include projection effects of the 3D dynamics.

2.5. DENSITY MEASUREMENT

From the Gaussian fits to the loop cross-sections we obtained a separation of theloop-associated EUV intensity flux, floop(t), from the background flux of the am-bient plasma, fback(t). From the Gaussian fits we can also extract the full width athalf maximum (FWHM), which we use as an estimate of the diameter of the loops,

w(t) = 2√

2 ln 2 σr(t) . (11)

Because all but one of the movies were taken in only one single wavelength (171 Å)to provide maximum cadence, we cannot determine the temperature of the loops.However, if we assume that the differential emission measure peaks dEM(T )/dTof detectable loops are randomly distributed within the primary FWHM sensitivityrange of the used filter, i.e., T ≈ 0.8–1.2 MK for the 171 Å filter, the responsefunction varies from 100% down to 50% in this range, having an average andstandard deviation of 81% ± 16%. Thus, correcting the peak response functionby this factor 0.81 yields a statistically averaged conversion factor of the flux toemission measure units. [An exception from this rule-of-thumb is extremely brightflare emission at temperatures of T ≈ 20 MK, which can show up in the sidebandof 195 Å, if the hot T ≈ 20 MK plasma has about a � 103 higher emission measurethan at T ≈ 1.5 MK (Warren et al., 1999)]. The peak response of TRACE 171 Å isf171 = 1.10 × 10−26(DN pixel−1 s−1 cm5) %t(s)EM171(cm−5) at Te = 0.96 MK.This is the peak response for coronal elemental (mostly iron) abundances, butis about a factor 3 smaller for photospheric (iron) abundances (see Appendix inAschwanden et al. 2000). With the standard definition of the emission measureEM and the loop diameter w(t) we can then obtain an estimate of a lower limit tothe electron density in the loop, assuming unit filling factor,

nloop(t) �√EMloop(t)

w(t). (12)

The largest uncertainty in this estimate of the density probably comes from thepoorly determined temperature of the loop, the unknown filling factors, and fromelemental abundances. The broad peak of the temperature response curve for theTRACE 171 Å bandpass implies the density could be larger by as much as a factorof ≈ 2. The uncertain filling factor could easily be as low as 0.5, implying wemay underestimate the density by a factor of 2. And the elemental abundances cast


Figure 3. Oscillation event No. 1a on 14 July 1998, 12:45 UT. This loop is identical with the caseanalyzed in Nakariakov et al. (1999). Top left: difference image with rectangular box indicates wheretransverse oscillations are analyzed. Bottom left: 3D geometry of observed loop (thick line) fittedwith a circular model (thin line), specified by the baseline (b), loop radius (r), height of circular loopcenter (h0), azimuth angle of baseline (az = α), inclination angle of loop plane to vertical (th = θ),heliographic longitude (l1) and latitude (b1) of baseline midpoint. The spacing of the heliographicgrid is 5◦ (≈ 60 Mm). Top right: transverse loop position (in pixel units), where the trend is fitted witha polynomial (with coefficients c0, c1, c2, c3, ...). Bottom right: detrended oscillation, fitted with anexponentially decaying oscillatory function (with amplitude A, period P , decay time td , phase ϕ0).t0 is the start time (to which the phase ϕ0 is referenced), and d is the duration of the fitted interval.

an uncertainty factor up to ≈ 1.9 in the electron density. The combination of thesefactors means the lower limit to the density nloop can underestimate the loop densityby an order of magnitude. The real value for the density is most probably severaltimes larger than the lower limit estimated by Equation 12. Clearly, simultaneousspectroscopic measurements of loop density would be beneficial for future studies.


Figure 4. Oscillation event No. 1f on 14 July 1998, 12:45 UT. This loop is located in the same flareregion analyzed in Aschwanden et al. (1999b).

3. Results

We analyzed the oscillations of 26 individual loops from the list of 17 flare events(Table I in Paper I), where we analyzed one oscillating loop in each event generally,except for event 1 (7 loops), event 5 (3 loops) and event 12 (2 loops). The physicalparameters resulting from this analysis are listed in Table I (geometric elements),Table II (time series analysis), Table III (electron density estimates), and statisticalvalues (Table IV). A graphic version of the results from a subset of 14 oscillatingloops are displayed in Figures 3–16. Each Figure contains a difference image thatdiscriminates dynamic from static loops (Figures 3–16, top left), shows their 3Dgeometric orientation (Figures 3–16, bottom left), the time-dependent amplitudewith its trend (Figures 3–16, top right) and its oscillatory part (Figures 3–16,


Figure 5. Oscillation event No. 3a on 23 November 1998, 06:35 UT.

bottom right). Let us now summarize the meaning of the numerical results in thefollowing.

3.1. GEOMETRIC PARAMETERS

The 6 geometric elements that define the best circular approximation of the ana-lyzed loops (based on the least-square fit of the model to the observed coordinatesof the projected loops in image coordinates (x, y)) are given in Table I. We list thedifference of the heliographic coordinates (l0−l�, b0−b�) with respect to the solardisk center, in order to quantify the center–limb angles of the oscillating loops. Theazimuth angles α illustrate the orientation of the loop plane, which is importantwhen the excitation of loop oscillations is studied with respect to a nearby locationof a flare or filament eruption, because it yields the ‘hit angle’ of a shock wave ora flare-initiated signal. The inclination angle ϑ varies generally over quite a range


Figure 6. Oscillation event No. 4a on 4 July 1998, 08:33 UT.

from ϑ ≈ −45◦ to ≈ +45◦ and is essential to correct for the effective scale heightλT = λ

loop

T / cos(ϑ) that determines the density-dependent Alfvénic propagationspeeds vA(h) as function of the loop altitude. The most essential parameter is theloop half length L (last column in Table I), in particular for the kink mode period,which corresponds to the Alfvénic transit time forth and back along the full looplength 2L. In our sample of 26 loops we find loop half lengths of L = 110 ± 53Mm (Table IV).

3.2. TIME SERIES ANALYSIS

The results of the time series analysis are shown in Figures 3–16, where the ab-solute coordinate of the transverse loop motion is shown in the top right panels,approximated with a trend, which is characterized by a polynomial Atrend(t) (Equa-tion 9) with the coefficients listed in the top right panels. After subtraction of the


Figure 7. Oscillation event No. 5c on 25 October 1999, 06:30 UT.

trend, the residual oscillatory components are shown in the bottom right panels ofFigures 3–16, with parameters listed there and also in Table II.

We find transverse oscillation amplitudes with a mean and standard deviation ofA = 2200 ± 2800 km (Tables II and IV). Compared with the loop half lengths ofL = 110 000 ± 53 000 km, the transverse oscillations correspond only to relativeamplitudes of A/L ≈ 1–5%.

Oscillatory periods are found with a mean of P = 5.4 ± 2.3 minutes, coveringa range of P = 2.3 to 10.8 minutes (Tables II and IV), and one extreme case witha period of P = 33.4 min (case 9a). This is a much wider range than previously re-ported (Aschwanden et al. 1999b; Nakariakov et al. 1999). It rules out the necessityof a coupling with the principal helioseismologic period of ≈ 5 min. Moreover, thelower cutoff of the fastest observed period of P = 2.3 min roughly corresponds tothe 4-point resolution limit of a %t ≈ 30 s cadence for this period, and the longest


Figure 8. Oscillation event No. 7a on 23 March 2000, 11:35 UT.

observed periods are close to the maximum length of contiguously observed timeintervals with the TRACE spacecraft (i.e., ≈ 20–40 min of un-interrupted orbitsegments between radiation belt transits and/or South Atlantic Anomaly). Thus theobserved range of periods may be restricted by instrumental limits.

Another interesting property is the damping time or exponential decay time tdof the periodic oscillations. We find a relatively limited time range for the decaytimes, i.e., td = 9.7 ± 6.4 min, which provides quite restrictive constraints on thecoronal damping mechanisms, because we are in principle able to measure decaytimes up to ≈ 90 min, based on the longest (though interrupted) time intervals(case 9a) that exhibited coherent oscillations. On the other hand, not every caseshows an exponentially damped amplitude. Only about a third of the cases exhibitsa clearly damped amplitude. In another two thirds, the damping time could notbe evaluated because there were too few periods or the amplitudes fluctuated too


Figure 9. Oscillation event No. 8a on 12 April 2000, 03:31 UT.

much (e.g., case 7a, Figure 8), or were even increasing with time (e.g., case 11a,Figure 11).

We measured also the maximum transverse speed vmax from the transverse mo-tion A(t) (Equations 8–10), which represents a lower limit of the exciter speed,vexc ≥ vmax. We find a mean of vmax = 42 ± 53 km/s (Tables II and IV), whichcorresponds to a subsonic Mach number of M = vmax/vsound = 0.28 ± 0.35.

The average number of periods we see is N = d/P = 4.0 ± 1.8 (Tables II andIV). So we observe on average only two periods per damping time, or four periodsduring the total duration of a detected oscillation. The weakest damping is observedin the first event 1a (Figure 3), with a number of d/P = 9 periods. This is the sameexample analyzed by Nakariakov et al. (1999), who study only the first 5 periods.Just for a comparison, to demonstrate the fidelity of our analysis algorithm, let uscompare the quantitative parameters determined in the two studies. In the study



of Nakariakov, a mean pulse period of P = 256 s, a decay time of td = 870 s,an amplitude of A = 1015 ± 290 km and a loop half length of L = 65 000 kmwere determined (where A is corrected by factor of 2, according to Leon Ofman,private communication). In our study (Figure 3) we find P = 261 s (+2%), a decaytime of td = 1183 s (+35%), an amplitude of A = 760 km (−25%), and a loophalf length of L = 84 000 km (+29%). Clearly, the period P is most accuratelyretrieved, generally with an accuracy of a few percents for cases with multiple pe-riods. Substantial differences in the measured amplitudes can result from differentbackground subtractions: If a stationary loop cannot be separated from an oscillat-ing loop along the same line of sight, the combined amplitude follows the commoncentroid of the flux, which yields then only half the amplitude (if both have equalflux) compared with the oscillating loop alone. Thus, substantial differences in loopamplitudes are not surprising. The contamination with stationary loops, however,does not affect the measurement of the period P . Another side-effect can be caused



by dimming of oscillating loops, which reduces the contrast between the signalof oscillating loops to the background. This side-effect progressively reduces theoscillation amplitude (because of the increasing weight of the underlying stationaryloops) and leads to underestimates of the damping time. This could possibly be thereason for the wide spread of damping times measured among the 7 loops (1a, 1b,. . . , 1g) in case 1.

From our analysis with a decomposition of a trend function and an oscillatorypart we find some new results that have not been recognized before. Eigen-modesin terms of the kink-mode are expected to consist of a (damped) sine-function,without any additional trend. In our analysis we find only a small subset thatis dominated by the oscillatory component, having a negligible trend [cases 1f(Figure 4), 2a, 5b, and 9a]. Most of the cases have either a monotonic trend [cases1a (Figure 3), 1c, 1d, 1e, 4a (Figure 6), 5a (Figure 7), 5c, 6a, 6b, 8a, 12a, 15a


Figure 12. Oscillation event No. 12b on 12 April 2001, 10:16 UT.

(Figure 14), 16a (Figure 15), 17a (Figure 16)], or even a slowly-varying, damped,oscillatory trend [cases 1b, 3a (Figure 5), 7a (Figure 8), 10a (Figure 10), 11a (Fig-ure 11), 12b (Figure 12), 14a (Figure 13)]. This suggests an additionally actingforce during the oscillations or an adjustment response to the impulsive excita-tion. Although the measurements of spatial phase differences are beyond the scopeof this study, the detection of a slowly-varying trend, besides a faster oscillatorycomponent, indicates propagating waves that have a more complex behavior than aclassical eigen-mode with phase-independent transverse oscillations. In Paper I weargued that it is likely that such propagating waves occur because the initial triggerproduces asymmetric displacements along the loop, which leads to a broadbandspectrum of oscillation frequencies, consisting of shorter and longer wavelengthspropagating along the loops (see discussion in Paper I). Apparently, the dampingtime is so short so that the broadband spectrum has not enough time to settle into afundamental eigen-mode.


Figure 13. Oscillation event No. 14a on 15 April 2001, 22:46 UT.

3.3. DENSITY ANALYSIS AND ALFVÉNIC SPEEDS

In Table III we provide the results of the density estimates (according to the methodoutlined in Section 2.5). We find on average a lower limit to the density inside theoscillating loops of nloop � (6.0 ± 3.3)× 108 cm−3. The background density of theambient plasma outside of the loops, necessary for determining the relative ratioof Alfvén speeds inside and outside of the loops, cannot reliably be determined.From the TRACE 171 Å images, it is beyond doubt that the density of plasmaat T ≈ 1 MK is higher inside of the loops than outside of the loops. However,the presence of dense and hot (T > 3 MK) plasma outside of the loops cannotbe excluded. Accurate determination of densities at a wide range of temperatureswould be necessary to provide an estimate of the ratio of Alfvén speeds. The latteris relevant to the propagation of surface waves, as well as to the transition time


Figure 14. Oscillation event No. 15a on 13 May 2001, 03:03 UT.

from oscillatory to chaotic time behavior in impulsively generated MHD waves(Roberts, Edwin, and Benz (1984)).

Let us consider the observed time scales in normalization to the Alfvénic transittime tA (forth and back through a loop with half length L),

tA = 4L

vA. (13)

The Alfvén velocity vA(B, ne) depends on the magnetic field and electron density.We can only infer a lower limit to the densities ne from the observed emissionmeasures EM and widths w of the oscillating loops (Equation 12). In addition, wehave to make an estimate for the magnetic field. We will use an average coronalmagnetic field of B ≈ 30 G, so that an uncertainty factor of 3 covers the entirerange of B ≈ 10–100 G. These Alfvénic transit times are found to have a mean oftA = 150 ± 64 s (Tables III and IV). The mean period is thus by a factor P/tA =


Figure 15. Oscillation event No. 16a on 15 May 2001, 02:57 UT.

2.4 ± 1.2 longer, the decay time is a factor of td/tA = 4.1 ± 2.3 longer, andthe observed duration of the oscillations is a factor of d/tA = 9.8 ± 5.7 longer(Tables III and IV and Figure 17). Since we can only determine a lower limit to thedensity of the loop, the Alfvén crossing time tA could well be larger by a factor ofabout 3. As a result of the uncertainty on the estimates for both the density and themagnetic field, it is difficult to determine the ratio of P and tA, as discussed in thenext section.

4. Discussion

We now discuss the results of our quantitative measurements in terms of various ex-isting theoretical models, i.e. loop oscillations in terms of kink eigen-modes (Sec-


Figure 16. Oscillation event No. 17a on June 2001, 06:35 UT.

Figure 17. Ratios of period P and decay time td to Alfvenic transit time tA = 4L/vA in oscillatingflux tubes. The subset with the four most regular kink-mode oscillations (cases 1f, 2a, 5b, and 9a) isindicated with black diamond symbols.


TABLE I

3-D geometric elements of oscillating loopsa.

Loop l0 − l� b0 − b� α ϑ h0 r L

No. (deg) (deg) (deg) (deg) (Mm) (Mm) (Mm)

1a, Figure 3 −15.6 −27.6 87 7 9 47 84

1b −15.5 −26.0 81 19 −1 24 36

1c −19.0 −23.4 9 48 11 46 85

1d −19.5 −24.5 13 −35 1 55 87

1e −19.7 −24.5 8 −45 7 51 87

1f, Figure 4 −19.6 −24.5 12 −44 11 57 102

1g −19.2 −22.7 16 47 10 45 81

2a 47.4 20.5 42 −24 −59 133 148

3a, Figure 5 82.3 −27.7 152 −12 38 99 195

4a, Figure 6 26.0 −27.3 7 −14 12 74 129

5a −23.1 −21.2 50 −13 0 63 99

5b −23.1 −21.4 51 −11 0 65 103

5c, Figure 7 −22.9 −21.3 47 2 0 53 83

6a 0.4 36.7 130 −27 5 21 38

7a, Figure 8 70.7 16.4 158 22 28 43 99

8a, Figure 9 25.6 −8.6 143 53 −8 30 39

9a 108.2 −12.4 8 −33 72 136 291

10a, Figure 10 72.6 −3.8 157 20 67 77 203

11a 78.9 −6.3 25 −30 43 47 130

12a 40.2 −18.8 1 34 0 66 103

12b, Figure 12 49.2 −20.2 150 49 −71 120 113

13a 76.0 −18.2 18 17 13 43 81

14a, Figure 13 75.2 −21.8 10 −22 42 48 128

15a, Figure 14 −4.7 −18.8 99 117 −10 65 91

16a, Figure 15 22.7 −18.3 177 39 −11 68 96

17a, Figure 16 −48.7 −28.0 1 41 19 33 73

a l0 − l� = heliographic longitude relative to Sun center, b0 − b� = heliographic lati-tude relative to Sun center, α = azimuth angle of loop baseline to east-west direction,ϑ = inclination angle of loop plane to vertical, h0 = height of the circular loop center,r = the curvature radius of the loop, and L = the half length of the loop.

tion 4.1), exciter mechanisms (Section 4.2), impulsively generated waves (Section4.3), and damping mechanisms (Section 4.4).

4.1. OSCILLATIONS IN THE KINK EIGEN-MODE

The classical interpretation of oscillating loops in the solar corona was formulatedin terms of symmetric oscillations (sausage modes) for fast periods, and in terms of


TABLE II

Time series analysis of oscillating loopsa.

Loop t0 d P td %t A vmax NP

No. (s) (s) (s) (s) (km) (km s−1) d/P

1a 980714 1259:57 2262 261 1200 −34 800 20 9

1b 980714 1257:38 1121 265 300 5 2000 30 4

1c 980714 1302:26 906 259 (2000) 18 100 4 3

1d 980714 1257:38 898 316 500 −40 6000 100 3

1e 980714 1305:00 982 242 – −45 3000 60 4

1f 980714 1256:32 1563 277 400 −58 4000 70 6

1g 980714 1302:26 1591 272 849 −62 5000 50 6

2a 980830 1804:43 400 299 (200) 132 4000 70 1

3a 981123 0635:57 3179 522 1200 −48 9000 140 6

4a 990704 0833:17 1576 435 600 −92 700 13 4

5a 991025 0631:04 521 162 – 76 100 5 3

5b 991025 0631:48 424 148 – 13 100 5 3

5c 991025 0628:56 1020 143 200 130 500 20 7

6a 000210 0127:20 979 220 (800) 74 300 10 4

7a 000323 1130:50 1750 615 – 270 1100 13 3

8a 000412 0329:18 555 326 (400) 117 1200 30 2

9a 000825 1446:27 5388 (2004) (1300) −226 9000 20 3

10a 010321 0232:44 2456 423 800 2 700 14 6

11a 010322 0456:06 1890 367 – 352 400 20 5

12a 010412 1018:28 733 342 – 275 800 20 2

12b 010412 1016:03 701 136 – 69 700 100 5

13a 010415 2158:43 1175 649 (700) 423 900 7 2

14a 010415 2246:53 1616 286 – −24 150 6 5

15a 010513 0318:41 664 428 – 132 1500 30 1

16a 010515 0257:00 643 185 200 4 9000 200 3

17a 010615 0632:29 1392 396 400 210 1800 30 3

a t0 is the start UT time of the modeled time series containing loop oscillations (in the formatYYMMDD HHMM:SS), d the total duration of the modeled time interval, P the oscillationperiod, td the e-folding decay (or damping) time, %t = P(ϕ0/2π) the phase time differencewith respect to t0, A the transverse oscillation amplitude, vmax the maximum transverse speedof the combined oscillatory and trend motion, and NP = d/P the number of periods in theanalyzed time interval. Values in parentheses are subject to large uncertainties and are notconsidered in the statistics.


TABLE III

Electron density inside oscillating loops, Alfven speed, Alfven transit time andmagnetic field B a.

Loop x0 y0 nloop8 w vA tA B

No. (′′) (′′) (cm−3) (Mm) (km s−1) (s) (G)

1a −238.4 −508.7 5.7 7.2 2600 130 (13)

1b −224.9 −429.5 5.9 6.7 2600 60 (6)

1c −321.6 −354.7 9.8 7.5 2000 170 (18)

1d −294.2 −475.1 5.9 8.3 2600 140 (11)

1e −291.2 −483.9 4.5 8.5 2900 120 (13)

1f −286.0 −493.9 6.2 7.9 2500 160 16

1g −306.2 −347.1 4.2 7.3 3000 110 (10)

2a 736.9 281.5 8.5 5.4 2100 280 25

3a 944.6 −555.0 3.0 16.8 3600 220 (11)

4a 351.4 −519.1 6.3 7.0 2500 210 (13)b

5a −344.4 −317.9 4.7 7.8 2900 140 (23)

5b −345.5 −324.7 6.8 10.9 2400 170 31

5c −347.4 −352.3 7.2 6.3 2300 140 (27)

6a 19.1 598.1 1.6 8.2 4900 30 (4)

7a 938.8 304.3 17.0 8.8 1500 260 (11)

8a 409.5 −124.8 6.9 6.8 2400 70 (5)

9a −986.5 −445.9 1.3 12.5 5500 210 3

10a 1012.0 −22.5 6.2 9.2 2500 320 (20)

11a 1025.5 −229.4 3.2 6.2 3500 150 (11)

12a 542.9 −315.9 4.7 6.3 2900 140 (11)

12b 686.9 −299.1 4.4 7.0 3000 150 (30)

13a 936.2 −355.7 4.1 8.1 3100 110 (4)

14a 929.1 −456.5 5.1 8.5 2800 190 (17)

15a −112.7 −351.5 4.0 11.4 3100 120 (7)

16a 399.2 −355.9 2.7 6.9 3800 100 (15)

17a −614.2 −505.3 3.2 15.8 3500 80 (6)

a (x0, y0) are the x, y coordinates of the source location where loops oscillations

have been measured, given in units of arcsecs from Sun disk center, nloop8 is a

lower limit to the electron density inside the oscillating loop, w the loop width,vA an estimate for the the Alfven speed, tA an estimate for the Alfven crossingtime and B a lower limit to the magnetic field assuming the oscillations arefree kink eigen-modes (see Equation (16)). All Alfvenic speeds and times arecalculated for a magnetic field of B = 30 G, and using the lower limit to the

density nloop8 . The estimated lower limit to B is given between parentheses for

cases that do not show clear examples of kink eigen-mode type oscillations.Probable values of B are a factor of 3 higher, given uncertainties involved inestimating the loop density.bNakariakov and Ofman (2001) determined a magnetic field of B = 13 ± 9 Gfor this case.


TABLE IV

Average and ranges of physical parameters of 26 oscillating loopsa.

Parameter Average Range

Loop half length L 110 ± 53 Mm 37–291 Mm

Loop width w 8.7 ± 2.8 Mm 5.5–16.8 Mm

Oscillation period P 321 ± 140 s 137–694 sb

5.4 ± 2.3 min 2.3–10.8 minb

Decay time td 580 ± 385 s 191–1246 sc

9.7 ± 6.4 min 3.2–20.8 minc

Oscillation duration d 1392 ± 1080 s 400–5388 s

23 ± 18 min 6.7–90 min

Oscillation amplitude A 2200 ± 2800 km 100–8800 km

Number of periods 4.0 ± 1.8 1.3–8.7

Electron density of loop nloop (6.0 ± 3.3)108 cm−3 (1.3–17.1) × 108 cm−3

Maximum transverse speed vmax 42 ± 53 km s−1 3.6–229 km s−1

Loop Alfven speed vA 2900 ± 800 km s−1 1600–5600 km s−1

Mach factor vmax/vsound 0.28 ± 0.35 0.02–1.53

Alfven transit time tA 150 ± 64 s 60–311 s

Duration/Alfvenic transit d/tA 9.8 ± 5.7 1.5–26.0

Decay/Alfvenic transit td/tA 4.1 ± 2.3 1.7–9.6c

Period/Alfvenic transit P/tA 2.4 ± 1.2 0.9–5.4b

a All Alfvenic speeds and times are calculated for a magnetic field of B = 30 G, so scaling toother magnetic field values are vA(B) = vA(B/30 G) and tA(B) = tA(B/30 G)−1.b The most extreme period of P = 2004 s (case 9a) is excluded in the statistics.c Only the 10 most reliable decay times td (with no parentheses in Table II) are included in thestatistics.

asymmetric oscillations (kink modes) for slower periods, e.g., see Roberts, Edwin,and Benz (1984). The first detection of transverse loop oscillations was found to beconsistent with the kink mode, based on the observed transverse displacements, aswell as regarding the statistical relation between the periods and estimated Alfvénvelocities (Aschwanden et al., 1999b). In the following we discuss an interpretationin terms of the kink mode for a standing wave (eigen-mode) in its fundamentalmode with a coronal magnetic field assumed broadly uniform, neglecting higherharmonics and possible admixtures from simultaneously excited sausage modes.The period P of the kink mode is

P = 4L

ck, ck = vA

√2

1 + next/nloop(14)

with L the half length of the loop. For the purpose of these estimates, and sincewe have no knowledge of the exterior density, we take ck ≈ 1.2vA (which is the


average between 1 and√

2 for a background consisting of, respectively, vacuumand loop-like densities). Using the range of measured periods, P = 321 ± 140 s,we could in principle constrain the Alfvén speed vA, and thus the coronal magneticfield strength B (in c.g.s. units),

B = vA√µ nloop

2.18 × 1011= 4L

√µ nloop

1.2 × 2.18 × 1011 P, (15)

where the mean molecular weight for coronal abundances is µ ≈ 1.3 (includinghelium and other elements). Using rounded numbers of our average values for L,Pand the lower limit to nloop, we find the following scaling:

B = 18

(L

100 Mm

)(P

300 s

)−1 √nloop

109 cm−3[G] . (16)

Because each of the parameters has a variation by a factor of about 0.5, the variationof the magnetic field values is about

√3 × 0.5 = 0.87, so we would expect a range

of B ≈ 3–30 G for the required magnetic field values. This is confirmed by thevalues in the rightmost column of Table III. Since we used a lower limit to thedensity for nloop, and we have estimated in Section 2.5 that the density could wellbe an order of magnitude larger, we find that B ≈ 3–90 G. Nakariakov et al. (1999)inferred a more restricted range of magnetic field values, B ≈ 4–30 G. In all thesemethods, a constant magnetic field strength value B is assumed along the loops,ignoring variations of the magnetic field B(s) along the loop length coordinate s.In reality, the magnetic field varies considerably along the loop, with much largervalues (� 100 G) in the lower regions of the loops. In addition, the uncertaintyand variation along the loop of the loop density is typically not included. Theseuncertainties render it difficult to check the validity of the kink mode using onlythe relationship between the observed period and loop length and the estimates fordensity and magnetic field.

Dulk and Lean (1978) compiled a larger amount of coronal magnetic fieldmeasurements (using interplanetary in-situ measurements, Zeeman effect measure-ments in active region prominences, extrapolations from photospheric magneto-grams, microwave, decimetric, and metric radio bursts) and found an average heightdependence above active regions of

B(R) = 0.5

(R

R�− 1

)−1.5

(G) (1.02 � R/R� � 10) . (17)

Using the height range of our measurements, h ≈ L/(π/2) = 70 ± 35 Mm,which corresponds to R/R� ≈ 1.06 ± 0.03, Dulk’s formula (Equation (17)) yieldsa range of B ≈ 20–90 G for the magnetic field strength at the heights of theoscillation sources. This magnetic field range overlaps significantly and thus isnot conflict with the range of values inferred from kink oscillations (see also ratioP/tA in left frame in Figure 17). Under the assumptions of filling factor unity,


high coronal elemental abundances and loops at temperatures between 0.8 and1.2 MK, the lower range (3–30 G) of magnetic field values applies. Under theseassumptions, loops oscillating in kink mode would have relatively low magneticfield strengths, especially since these low average values need to account for thehigh values expected in the lower parts of the loops.

An inspection of Figure 17 shows puzzling results. The most striking featureis that no obvious dispersion relation is present as expected for pure kink-modeoscillations. We marked the subset (of 4 cases) with regular, symmetric kink-modeoscillations with diamonds in Figure 17, but even those do not show the expectedkink-mode dispersion relation. The lack of the expected dispersion relation canpartly be blamed on our inability of determining the magnetic field and ambientloop densities, but it might also reflect an intrinsic property of impulsively gener-ated MHD waves, which do not have sufficient time to settle into an eigen-mode,in particular given the fact that the oscillations are strongly damped within 2–4periods (Figure 17, right frame).

We reanalyzed loop oscillations at similar positions here as in Aschwanden etal. (1999b), cases 1b–1f, and in Nakariakov et al. (1999), case 1a, and indeedfind characteristics that are consistent with eigen-modes: (1) the middle part ofthe loops (near the apex) shows the largest transverse oscillation amplitudes, and(2) the centroid position of the oscillating loops shows either no trend (Figure 4),or only a linear drift with time (Figure 3). However, the same behavior is foundonly in a small subset of the analyzed events (marked with black diamonds inFigure 17), while most of the other cases reveal a significant ‘driving’ velocity(vmax) and a varying drift (or trend) during the course of the oscillations, whichcannot be understood in the simplest concept of ideal kink mode oscillations. Thus,some other oscillation mechanisms should be considered as well.

4.2. EXCITER CRITERION

Only about 6% of flares show oscillating loops (Paper I), so the occurrence ofloop oscillations is quite rare. What is the reason for this selective criterion? Letus first consider the speed of the exciter. If a loop is displaced by a velocity thatis much smaller than its transverse speed when oscillating in its eigen-mode, theloop gets slowly displaced up to the maximum stretch and is adjusting to thenew force equilibrium. In contrast, when the exciter speed is much faster thanthe eigen-mode speed (Equation (18)) of the loop, the loop overshoots the newforce equilibrium position and swings back in an oscillatory motion. Thus, we mayformulate a criterion for the excitement of oscillations by the requirement that theexciter velocity has to be faster than the velocity of the eigen-mode oscillation(using Equation (14)),

vexc > vmax = 2πA

P= vA

π

2

A

L

√2

1 + next/nloop, (18)


which yields a criterion for the minimum magnetic field,

B ≤ 10

(vexc

1000 km s−1

) √ne

109 cm−3

√1 + next/nloop

2

L

A[G] . (19)

In the statistical average of our analyzed loops we find a maximum transverseoscillation speed of vmax ≈ 42 km s−1. We do not know what the actual exciterspeed vexc is. In fact, in case 1, a propagation speed of vexc ≈ 700 km s−1 wasinferred from a radially spreading disturbance (Aschwanden et al., 1999b), so itcan be much higher than the transverse speed of the oscillating loops. Neverthe-less, Equation (19) shows that loops with weaker magnetic fields B have a higherlikelihood to oscillate than loops with strong magnetic fields, which have a suf-ficiently fast Alfvén speed to adjust instantaneously to transverse displacements,while loops with lower magnetic fields are excited faster than they can restorethe equilibrium, and thus overshoot and swing back, the well-known pattern of anoscillating system.

An additional criterion that contributes to the excitation of transverse loop oscil-lations is the location of the loop near magnetic nullpoints or separators. Becausemagnetic field lines diverge to different magnetic polarities along separators, alittle disturbance at the separator or nullpoint has a high amplification factor in thediverging magnetic field, and thus will translate into larger transverse amplitudesof oscillating loops (Schrijver and Brown 2000). This point is discussed in thecontext of individual oscillation events in Paper I. So, we can summarize that therareness of oscillating loops is possibly due to three reasons: (1) the requirement ofa sufficiently fast exciter, (2) the anchoring of loops near magnetic separators, and(3) the presence of relatively weak magnetic fields. The latter cannot be excludedfrom our estimates of the kink period.

4.3. IMPULSIVELY GENERATED WAVES

Examining the 17 TRACE movies described in Paper I, one sees in a number ofcases irregular dynamics on top of transverse kink eigen-mode oscillations, withdiscernable phase differences along the loops (e.g., cases 3, 7, 9, 10, 11, 13, 15).Because any deviation from a symmetric kink eigen-mode can be decomposed intopropagating wave functions, the observed irregular oscillations can be interpretedin terms of propagating waves. Such propagating waves can easily be produced byasymmetric excitation (as simulated in Figure 6 of paper I), which is likely whenthe triggers, i.e., flare-initiated coronal mass ejections or filament destabilizations,are located close to one footpoint of loops susceptible to oscillations (see also dis-cussion at end of Section 3.2). Our analysis in this paper quantifies the oscillatorymotion only at one single location of the loop, usually where the amplitude islargest or where the confusion from adjacent loops is minimal. Measuring phasedifferences along loops turned out to be rather difficult because of the manifold


confusion from adjacent loops, and thus, the evidence for propagating waves ismostly based on visual inspection of movies at this point.

Propagating MHD waves, rather than standing waves, will result whenever dis-turbances are generated impulsively, because the motions have insufficient time toreflect from the far end of the loop, or in open field regions (Roberts, Edwin, andBenz (1984). Such waves are known as Pekeris waves in oceanography (Pekeris,1948), and their evolution was applied to solar fluxtubes with minor modificationsby Roberts,

Edwin, and Benz (1984). Modeling in terms of impulsively generated MHDwaves seem to be a promising avenue to understand our observations, given thecomplexity of the time profiles with significant deviations from strictly periodicdamped oscillations. Because of the difficulty of analytical solutions for this prob-lem, numerical MHD simulations have been performed in a series of studies byMurawski and Roberts (1993a–d, 1994). These simulations show non-periodic andstrongly damped pulses due to energy leakage, which is more severe for sausagemodes than for kink modes. The numerically simulated time profiles that resemblethe observations the most, are those from Figure 2(a) and Figure 5(a) in Murawskiand Roberts (1994), which show a similar evolution as the cases shown in Figure 3a(case 3a), Figure 10 (case 10a), or Figure 15 (case 16a). These cases all show aninitial large displacement (seen in the trend function in our analysis) with a periodlength of several Alfvén crossing times, followed by a few strongly damped pulsesthat have a period closer to one Alfvénic crossing time.

In summary, impulsively generated MHD waves seem to reproduce the irreg-ularities of the observed time profiles much better than the pure kink eigen-modeoscillations. This opens up a new diagnostic of the trigger mechanism, in termsof impulsively generated MHD waves excited by nearby erupting filaments, flares,and coronal mass ejections.

4.4. DAMPING OF OSCILLATIONS

As a next step we discuss the decay phase. We found a restricted range of dampingtimes, i.e., td ≈ 3–21 min, compared with the maximum duration of detectedoscillations (dmax ≈ 90 min). Decay of coronal oscillations was recently discussedin the review of Roberts (2000). Viscous and ohmic damping, optically thin ra-diation and thermal conduction, loop curvature, are all found to be insufficient toexplain the observed rapid decay within a few periods. However, the presence ofdensity inhomogeneities on small scales is expected to strongly enhance dissipationthrough resonant absorption. Roberts (2000) estimates the damping time due todensity inhomogeneities, using the mechanism of phase mixing, to

td =[

6(2L)2l2

νπ2v2A

]1/3

, (20)


with L the loop half length, l ≈ (2L)/10 a scale of variation in the Alfvén speed,and ν = 4 × 109 m2 s−1 the coefficient of kinematic viscosity. Inserting our meanvalues for loop half lengths, L = 108 m, and Alfvén speeds, vA = 2.9×106 m s−1,we obtain damping times of td = 660 s or 11 min, which agrees well with theobserved range of td = 580 ± 385 s, or 9.7 ± 6.4 min, respectively. The Reynoldsnumber for this value of viscosity is R = LvA/ν = 7 × 104.

Instead of using the kinematic (compressional) viscosity, a more appropriate ap-proach is to use the shear viscosity coefficient. Nakariakov et al. (1999) interpretedthe observed damping time of trapped resonant Alfvén waves in terms of viscousdissipation of velocity gradients or resistive dissipation by currents generated bymagnetic field gradients. They infer a coronal dissipation coefficient that is a factor∼ 109 higher than the classical value, corresponding to a Reynolds number that isa factor of ∼ 10−9 lower. This interpretation, however, remains still a controversialissue, because a corona with such high dissipation coefficients would be subject tooverefficient heating (Hudson and Kosugi 1999).

Alternatively, the strong damping seems to be well-reproduced by the numericalsimulations of impulsively generated MHD waves, where the rapid damping is aconsequence of the nonlinear evolution of leaking and trapped kink mode waves.

Recent papers on the oscillating loops observed with TRACE have all underes-timated the leakage of wave energy from the coronal volume into the footpoints.Nakariakov et al. (1999) and Roberts (2000) both estimated that the wave leakageat the footpoints is 2 orders of magnitude too small to account for the observeddamping in case 1. These estimates are based on a paper by Berghmans and DeBruyne (1995) which, just like many other papers on this topic, however assumesthat the Alfvén speed vA varies as a stepfunction at the ends of the loops, with a highand constant coronal value vAc and a much lower value of vAph in the photosphere.With this simplification, it is easy to calculate the amplitude reflection coefficientof downward propagating coronal waves impinging on the photospheric footpointsby

R =∣∣∣∣vAc − vAph

vAc + vAph

∣∣∣∣ . (21)

The time scale τd corresponding to decay of the loop oscillation due to waveleakage through the ends of the loops is (Hollweg, 1984)

td = 4L

(2 − R1 − R2)vAc= 2τA

2 − R1 − R2, (22)

in which L is the half length of the loop, τA = 2L/vAc = tA/2 is the Alfvéncrossing time from one footpoint to the other and R1,2 are the energy reflectioncoefficients at both ends of the loops. Note that the factor of 2 in the numeratoris missing in Equation (15) of De Pontieu, Martens, and Hudson (2001). For typ-ical densities and magnetic field strengths in photosphere and corona, R ≈ 0.99


Figure 18. The ratio td/P of the decay time td and the wave period P is shown as a function of halfloop length L (Mm). Triangles are data for various oscillating loops. Solid lines are the expectedfootpoint losses from Equation (25) for various values of the chromospheric scale height h in Mm.

(from Equation (21)). This high value of the reflection coefficient implies very lowfootpoint leakage of, e.g., standing Alfvén waves in a coronal loop with td ≈ 50τA.

In a recent paper, De Pontieu, Martens, and Hudson (2001) point out that a morerealistic treatment of the footpoint regions leads to considerably higher footpointlosses. In reality, the footpoints of a loop consist of a transition region, chro-mosphere and photosphere, so that the Alfvén speed drops much more gradually(than a mere stepfunction) from the coronal to the photospheric level. Followingthe earlier work of Hollweg (1984), we assume a constant Alfvén speed vA inthe corona, and an Alfvén speed which varies exponentially with height in thefootpoints of the loop (i.e., vA ∼ ez/2h, with h assumed constant). The followingexpression is then obtained for the amplitude reflection coefficient R of Alfvénwaves, launched in the corona, impinging on the chromosphere:

R2 = J 20 + Y 2

0 + J 21 + Y 2

1 − 4/(πα)

J 20 + Y 2

0 + J 21 + Y 2

1 + 4/(πα), (23)

where the argument of the Bessel functions is α, with α = 2hω/vAc in whichω = 2π/P . For solar applications α is quite small, with h of the order a fewhundred km, vAc ≈ 2000 km s−1 and ω < 1 for the wave frequencies that aretypically considered. In the case of small α (� 1), Equation (23) becomes

R ≈ 1 − πα . (24)


Equations (22) and (23) imply more penetration of the coronal waves into thechromosphere, and thus higher footpoint leakage of coronal Alfvén waves thanEquation (21). In the presence of chromospheric ion-neutral damping these equa-tions become more complicated, and the reflection coefficient can be lowered byup to 20% for certain wave periods P (De Pontieu, Martens, and Hudson, 2001).

In the absence of such damping, we can use Equations (22) and (24) to obtain:

td ≈ τA

πα= LP

4π2h.(25)

The dependency on vA drops out and we find td/P ≈ L/(40h). Figure 18 showstd/P for various oscillating loops for which the decay time could be measured. Forcomparison with the footpoint losses expected from Equation (25) we have plottedthese for different values of the scale height h. The observed damping can thus beexplained by footpoint losses, but only if the chromospheric/transition region scaleheight is quite variable from case to case, and between the values 0.4 and 2.4 Mm.Such a range of values is greater than would be expected from hydrostatic chro-mospheric models, which typically predict values between 0.15 and 0.3 Mm (DePontieu, Martens, Hudson, 2001). One possible explanation for the larger valuesof h suggested by the td/P ratio of the longer loops (L ∼ 100 Mm) might be thatthe chromosphere is most probably not in hydrostatic equilibrium. Observations ofchromospheric spicules indicate that the chromospheric scale height might be aslarge as several Mm (Johannesson and Zirin, 1996). Within the framework of theassumptions, the footpoint losses thus show an approximate correspondence withthe observed damping of the oscillating loops (but see also Ofman, 2002, for acontrary argument).

5. Conclusions

We carried out quantitative measurements of geometric and physical parametersof 26 oscillating loops observed with TRACE. We analyzed the time series oftransverse oscillations and quantified the periods, decay times, and amplitudes ofthe oscillatory component, and quantified the superimposed nonlinear trend withpolynomial functions. Based on this analysis we arrive at the following conclu-sions:

(1) The loop oscillations are consistent with (symmetric) eigen-mode oscilla-tions in the kink mode in only a few cases. Most of the loops do not show spatiallysymmetric amplitude oscillations, often the oscillation is restricted to one half sideof the loop, while the other half remains immobile.

(2) Most of the observed loop oscillations conform better to the evolution ofimpulsively generated MHD waves, which propagate forth and back through theloops, leading to irregular periods and rapid decay of the oscillation amplitude dueto partial trapping and leakage of kink mode waves. Numerical MHD simulations


reproduce some of the observed features, such as the large initial pulse with aduration of several Alfvénic transit times (visible in our trend function), the non-periodicity of the subsequent pulses, the strong damping, the spatially asymmetricevolution, etc. The trigger mechanism of impulsively generated MHD waves seemsto be caused by eruptive filaments, flares, and coronal mass ejections, which allwere witnessed in the 17 observed events of oscillating loops.

(3) The measurements of the properties of the oscillating loops are unfortunatelytoo uncertain to serve as selection criterion between the two models (impulsively-driven free oscillations versus separator current-driven oscillations) suggested inPaper I. Although it seems clear that in many cases the kink eigen-mode is notdominant, the presence of impulsively generated transverse waves cannot be ex-cluded. The lack of numerical predictions of the second model of Paper I, makesit impossible to exclude strong sensitivity to changes in the field sources for fieldlines near separatrices as a cause of the oscillating loops.

(4) The rapid decay of the oscillations previously posed a problem for the trans-verse waves, and led to suggestions of extreme ad hoc assumptions, such as 9orders of magnitude higher viscosities. In this paper, we found that the decay timeof the oscillations can be explained by wave leakage at the footpoints through achromosphere with a scale height of h ≈ 400–2400 km. The footpoint losses hadbeen underestimated in earlier studies based on stepfunctions of the Alfvén speedbetween the corona and chromosphere.

(5) The promise of performing some type of coronal seismology based on mea-surements of oscillating loops (Roberts, Edwin, and Benz, 1984; Roberts, 1986)proves to be hard to fulfill with current observations and methods. Our estimatesof loop density and magnetic field are too crude to allow distinction between thetwo proposed models, and as a result, coronal loop oscillations cannot reliablybe used to estimate these coronal parameters. Simultaneous spectroscopic densitymeasurements will benefit future studies.

Although we made a first attempt to quantify the most important geometric andphysical parameters in all oscillating coronal loops observed so far, our analysisis still very preliminary, restricted to a time series at a single spatial point alongthe loops. Because it has become clearer that these oscillations involve spatiallypropagating waves, future studies should concentrate on spatio-temporal analysisalong the entire loops. We anticipate that a better theoretical understanding ofthese dynamic phenomena requires detailed numerical modeling with nonlinearMHD codes (e.g., as started by Murawski and Roberts 1993a–d, 1994). Morerealistic models than uniform slabs should also include the hydrostatic denstityscale heights, the magnetic field variation along the loops, asymmetric loops, andperhaps the curvature of the loops. The diagnostic potential of loop oscillationscould further be enhanced by fitting numerical results to observed time profilesor by predicting the temporal evolution based on measured physical parameters,which includes also criteria which loops should oscillate and which not.


Acknowledgements

We thank the referee, Bernie Roberts, and David Alexander for helpful commentsand suggestions. The TRACE team includes scientists from Lockheed Martin Ad-vanced Technology Center, Stanford University, NASA Goddard Space Flight Cen-ter, the University of Chicago, Montana State University, and the Harvard-Smithsonian Center for Astrophysics. Work was supported by NASA contractNAS5-38099 (TRACE).

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transverse oscillations in coronal loops observed with trace

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