The Physical Nature of Spiral Wave Patterns in Sunspots
Juhyung Kang, Jongchul Chae, Valery M. Nakariakov, Kyuhyoun Cho, Hannah Kwak, Kyeore Lee
DDraft version May 23, 2019
Typeset using L A TEX preprint style in AASTeX62
The Physical Nature of Spiral Wave Patterns in Sunspots
Juhyung Kang, Jongchul Chae, Valery M. Nakariakov,
2, 3
Kyuhyoun Cho, Hannah Kwak, and Kyeore Lee Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic ofKorea Centre for Fusion, Space and Astrophysics, Physics Department, University of Warwick, Coventry CV4 7AL, UK School of Space Research, Kyung Hee University, Yongin 17104, Republic of Korea (Received April 2, 2019; Revised April 26, 2019; Accepted May 1, 2019; Published May 20, 2019)
Submitted to ApJLABSTRACTRecently, spiral wave patterns (SWPs) have been detected in 3 minute oscillations ofsunspot umbrae, but the nature of this phenomenon has remained elusive. We presenta theoretical model that interprets the observed SWPs as the superposition of twodifferent azimuthal modes of slow magnetoacoustic waves driven below the surface inan untwisted and non-rotating magnetic cylinder. We apply this model to SWPs of theline-of-sight (LOS) velocity in a pore observed by the Fast Imaging Solar Spectrographinstalled at the 1.6 m Goode Solar Telescope. One- and two-armed SWPs were identifiedin instantaneous amplitudes of LOS Doppler velocity maps of 3 minute oscillations. Theassociated oscillation periods are about 160 s, and the durations are about 5 minutes. Inour theoretical model, the observed spiral structures are explained by the superpositionof non-zero azimuthal modes driven 1600 km below the photosphere in the pore. Theone-armed SWP is produced by the slow-body sausage ( m = 0) and kink ( m = 1)modes, and the two-armed SWP is formed by the slow-body sausage ( m = 0) andfluting ( m = 2) modes of the magnetic flux tube forming the pore. Keywords:
Sun: chromosphere — sunspots — Sun: oscillations — magnetohydrody-namics — waves INTRODUCTIONWave motions are a conspicuous dynamic phenomenon observed in sunspots. The first detectionof sunspot waves in the chromosphere was reported by Beckers & Tallant (1969). Subsequent worksrevealed that the predominant period of the waves is 5 minutes in the umbral photosphere (Bhatnagaret al. 1972), and 3 minutes in the chromosphere (Beckers & Schultz 1972). Sunspot waves were alsoobserved in the transition region and corona with the periods of less than three minutes (e.g. De
Corresponding author: Jongchul [email protected] a r X i v : . [ a s t r o - ph . S R ] M a y Kang et al.
Moortel et al. 2002; Sych et al. 2009; Tian et al. 2014). Furthermore, a radially propagating wavepattern was detected in the sunspot penumbra that is known as running penumbral waves (RPWs;Giovanelli 1972; Zirin & Stein 1972). A comprehensive review of sunspot waves can be found inKhomenko & Collados (2015).The nature of 3 minute chromospheric oscillations has been attributed to upward propagating slowmagnetoacoustic waves (Lites 1984; Centeno et al. 2006). Centeno et al. (2006) clearly showed thepropagating property of the waves by measuring the phase difference between the time series of theline-of-sight (LOS) velocity in the photosphere and that in the chromosphere. In the same context,the RPWs have been interpreted as the slow waves propagating along the inclined magnetic fieldlines (Bloomfield et al. 2007; L¨ohner-B¨ottcher & Bello Gonz´alez 2015).The plausible driving sources of sunspot waves are external p -modes and internal magnetoconvec-tion. The external driving scenario assumes that f - and p -mode waves in a quiet Sun propagateinto a sunspot. A fraction of the energy of the incident f - and p -mode is absorbed by its conversioninto a slow magnetoacoustic mode at the plasma- β equal to one layer (e.g., Cally et al. 1994; Cally& Bogdan 1997; Cally et al. 2003). Zhao & Chou (2013) successfully observed the absorption ofthe f - and p -mode wave energy in a sunspot in the k − ω diagram. In the internal driving model,magnetoconvection occurring inside a sunspot can excite the waves. The radiative magnetohydrody-namics simulations of the magnetoconvection showed that multi-frequency waves can be generatedin a magnetic concentration region such as a sunspot (Jacoutot et al. 2008). Chae et al. (2017) foundthat the wave energy flux was enhanced around the light bridge and umbral dots, and they concludedthat the magnetoconvection may be the driving source of 3 minute oscillations. The internal excita-tion was further supported by Cho et al. (2019)’s identification of several patterns characterized byoscillation centers and radial propagation above individual umbral dots that are under substantialchanges. Recent works suggested that an internal driving source may be located, below the sunspotphotosphere down to 5 Mm in the sunspot’s flux tube, by analyzing the photospheric fast-movingwave patterns (Zhao et al. 2015; Felipe & Khomenko 2017).Interestingly, recent observational works reported that in the horizontal plane, 3 minute oscilla-tions often appear in sunspot umbrae as one- and two-armed spiral wave patterns (SWPs; Sych &Nakariakov 2014; Su et al. 2016; Felipe et al. 2019). SWPs apparently propagate radially out at thevelocity of around 20 km s − , and also propagate upward (Su et al. 2016). Because these propagatingproperties are similar to RPWs, Su et al. (2016) concluded that observed SWPs could be associatedwith the slow waves propagating along a twisted magnetic field. Sych & Nakariakov (2014), however,pointed out that the magnetic field should be uniformly twisted in low- β plasma of sunspots, andit cannot contribute to the non-uniformity of a SWP. Moreover, the observed SWPs highlight thestructure of the wavefront in a certain horizontal cross section of the magnetic flux tube, which doesnot require the flux tube twisting. Very recently, Felipe et al. (2019) also concluded that althoughthe twist can affect the shape of the observed SWPs, it is not their main cause.In this Letter we present a simple model that SWPs can naturally appear in an untwisted magneticflux tube when non-axisymmetric disturbances from below the surface are taken into account. Weobservationally identify one- and two-armed SWPs in a pore in Doppler velocity maps of the H α line profiles, and develop a theoretical model explaining the appearance of SWPs. In section 2,we describe the observations, and summarize observational results. In section 3 we describe the he Physcial Nature of Spiral Wave Patterns OBSERVATIONWe observed a pore in NOAA 12078 on 2014 June 3 from 16:48:41 to 17:56:32 UT with the 1.6 mGoode Solar Telescope. The target was located at x = 160 (cid:48)(cid:48) , y = − (cid:48)(cid:48) when we started theobservation. In this study, we used the data acquired by the Fast Imaging Solar Spectrograph (FISS)in the H α band, and this is the same data analyzed previously in Chae et al. (2015). The FISSscanned the pore with a spectral sampling of 0.019 ˚Aand spatial sampling of 0 . (cid:48)(cid:48)
16, covering a fieldof view of 20 (cid:48)(cid:48) by 40 (cid:48)(cid:48) . The exposure time was 30 ms, and the time cadence of the data was 20 s.The basic calibration was performed as described by Chae et al. (2013b). We measured the LOSDoppler velocities for all data pixels by using the lambdameter method (Chae et al. 2013a) with thelambdameter chord of 0.4 ˚A. To highlight 3 minute oscillations, we filtered the data in frequency,leaving only the frequencies of 5 . − spiraling . We determined the durationof the SWPs by the visual inspection of the rotating motion. It was found to be about 4 minutesfor the one-armed spiral, and 5 minutes for the two-armed spiral. From the wavelet analysis, weestimated the oscillation period of SWPs at about 120 s at the center of the pore and at about 250s near its boundary. The period averaged over the pore is about 165 s.To identify the spatial fluctuations of the patterns in the azimuthal direction, the discrete Fouriertransform was applied along the dashed line. The Figure 1 shows the time-averaged azimuthal powerspectra of the two SWPs constructed along the two circles marked by the dashed curves. At thesetwo radii, the power of non-zero azimuthal mode m is the largest. In the case of the one-armedSWP, most of the power is concentrated at m = 0 and m = 1 (panel (b)). For the two-armed spiral,the power is concentrated at the m = 0 and m = 2 (panel (d)). These indicate that the SWPs arecomposed of at least two azimuthal modes. We found that during each event, both the azimuthallysymmetric modes ( m = 0) and the non-symmetric mode ( m = 1 or 2) appeared and disappearedtogether. The power of m = 0 mode at the chosen radius fluctuated substantially for the period ofabout 80 s, whereas the power of m = 1 or 2 mode changed slowly with time.We detected such SWPs in other sunspots as well. Roughly speaking, from an one hour observation,two or three SWPs occurred inside each sunspot. The rotation direction of the SWPs did not haveany hemispheric dependence. In some cases, in fact, two SWPs of opposite rotation directions wereobserved in the same sunspot at two different times. Even though such SWPs were detected in anytypes of sunspots, the spiral arms were simply shaped in small axisymmetric sunspots. The detailsof these observational results will be described in a subsequent paper. MODELING
Kang et al. í < D U F V H F 8 7 D 1 R U P D O L ] H G 3 R Z H U E í ; D U F V H F í < D U F V H F 8 7 F P 1 R U P D O L ] H G 3 R Z H U G Figure 1.
Snapshots of the LOS Doppler velocity maps (left panels), and their time-averaged azimuthalpower spectra in the azimuthal direction along the dashed line (right panels). Blue (red) color representsupflows (downflows), and the saturation amplitude of velocity is 3 km s − . The black contour representsthe boundary of the pore. The cross symbol indicates the center of the dashed line, and this position is setto be the origin. The radius of the dashed line is 2 (cid:48)(cid:48) for the one-armed SWP (a) and 3 (cid:48)(cid:48) for the two-armedSWP (c). To interpret the detected SWPs, we first consider azimuthal wave modes in an untwisted uniformthick magnetic cylinder with the magnetic field along the z direction, following Edwin & Roberts(1983). The observed pore is well compatible with this assumption because it contains a straightfield that is confined to the pore’s boundary. The internally oscillatory solution (body waves) ofthe transverse and longitudinal velocity components in cylindrical coordinates ( r, θ, z ) are given asfollows (Spruit 1982; L´opez Ariste et al. 2016): v r = − ω − k c s ω n A m J (cid:48) m ( nr ) exp i ( kz + mθ − ωt ) , (1) v z = − i kc s ω A m J m ( nr ) exp i ( kz + mθ − ωt ) , (2) he Physcial Nature of Spiral Wave Patterns rd ) D V W : D Y H 6 O R Z : D Y H ~ B β < β = 1 β > ' O D \ H U Figure 2.
Schematic images of the longitudinal velocities v z in the m = 1 mode in the x − z plane. Thedriving source of the wave is located at the center of the bottom. Blue (red) color represents the upflows(downflows). The black solid line indicates the β = 1 layer and the dashed line denotes the detection layer(D layer). Magnetic field lines are shown by the gray arrows. The propagating direction of the fast (slow)wave is shown by the blue (red) arrow. where k is the wavenumber along the field, ω is the frequency, c s is the sound speed, A m is theamplitude of an azimuthal mode m , J m is the Bessel function of the first kind, and J (cid:48) m is its derivative.In this study, we follow the general naming convention for the integer azimuthal modes: sausage modefor m = 0, kink mode for m = 1, and fluting modes for m ≥ n is given by (Edwin & Roberts 1983) n = ( ω − c s k ) ( ω − c A k )( c s + c A ) ( ω − c T k ) , (3)where c A is the Alfv´en speed, and c T is the tube speed, c T = c s c A / ( c s + c A ). For body waves n mustbe positive, and for slow modes the phase speed ω/k lies between the tube speed and sound speed(Roberts 2006).In addition, we assume that the driving source of the wave is located below the photosphere insidethe flux tube. This approach is in line with the suggestion of Zhao et al. (2015) and Felipe &Khomenko (2017) made to interpret the photospheric fast-moving radial wave patterns. In thisscenario a fast mode wave is driven at the high- β region, then it propagates quasi-isotropically to the β = 1 layer (see Figure 2). Thus, the arrival time t A ( r ) at the β = 1 layer is given as a function of Kang et al. í í ; D U F V H F í í < D U F V H F m = 0 m = 1 m = 2 í v z Figure 3.
Snapshots of the simulated parallel velocity component for the azimuthal wave modes m = 0,+1 and +2 at t = 0 in x − y plane. Speeds are normalized by the amplitude of each mode. The animationfollows the azimuthal wave modes from t = 0 to 160 s. (An animation of this figure is available.) the transverse distance r from the center of the source, t A ( r ) = √ r + d v fast , (4)where d is the depth of the source and v fast is the averaged propagation speed of the fast wave inthe high- β region. For simplicity, here we have assumed the constancy of the propagation speedand neglected the effect of refraction and reflection. After arriving at the β = 1 layer, a portion ofthe fast wave is converted to the slow wave (Cally 2001) which then propagates along the field. Forthat reason, we can observe the radially propagating wave patterns when the slow mode reaches thedetection layer. With the use of this effect, we can re-write the Equation (2) as follows: v z = − i kc s ω A m J m ( nr ) exp i ( kz + mθ − ω ( t − t A ( r ))) . (5)As the wave frequency is constrained by the observation, we can derive the wave numbers k foreach azimuthal mode m from the dispersion relation of (Edwin & Roberts 1983) ρn e (cid:0) ω − k c A (cid:1) K (cid:48) m ( n e R ) K m ( n e R ) = ρ e n (cid:0) ω − k c A,e (cid:1) J (cid:48) m ( nR ) J m ( nR ) , (6)where the subscript e represents the exterior of the flux tube, K m is the modified Bessel function ofthe second kind, K (cid:48) m is its derivative and R is the radius of the tube, which is 5 (cid:48)(cid:48) in our case. Wetake ω = 2 π/
160 s − from the observation, c s = 9 km s − from Maltby et al. (1986), c A = 300 km s − from Khomenko & Collados (2006), c s,e = 1 . c s and c A,e = 0 . c s from Edwin & Roberts (1983), thenthe k is approximately 4.36 × − rad m − for all azimuthal modes.Substituting these parameters into Equations (1) and (2), the ratio between the amplitudes of v z and v r is estimated as v z /v r ∼ × for all azimuthal modes. It means that every azimuthalslow-body mode is predominantly longitudinal in the chromosphere. Figure 3 shows snapshots of v z for m = 0, 1, and 2 modes in the x − y plane with d = 1600 km and v fast = 20 km s − . Forthe case of m = 0, the ring-like pattern is generated, and this ring apparently propagates radially he Physcial Nature of Spiral Wave Patterns W V W V W V W V W V í ; D U F V H F í < D U F V H F W V W V W V W V W V v L O S k m s − í v z Figure 4.
Time evolution of observed (top) and simulated (bottom) one-armed SWP from 17:17:20 UT to17:20:00 UT. The observed Doppler maps are filtered in frequency bands from 5.5 to 9 mHz. The speeds insimulation are normalized by the maximum value. The boundary of the pore is shown by the solid line inboth cases. The animation orients the observed data to the left and the simulation to the right; the observeddata is presented in 20 s increments while the simulation runs smoothly from t = 0 to 160 s. (An animationof this figure is available.) outward. On the other hand, m = +1 and +2 modes produce apparently rotating patterns in thecounterclockwise direction with one- and two-armed structures, respectively. As the ring-like patternof m = 0 mode propagates radially, the power of this changes with time and radius, while the powerof non-zero modes depends only on the radius because the patterns of these modes do not move out ( see the online animated version of Figure 3).To reproduce the observed one-armed spiraling pattern, we summed up perturbations with m = 0and m = 1, which are the most powerful modes according to the Fourier analysis, with the amplituderatio of A /A = 0 .
54, the source depth of d = 1600 km and averaged propagation speed of v fast =20 km s − . In addition, we introduce the reference time t and reference angle θ terms to set theorigin of the simulation, then the t is replaced by t − t , and θ is substituted by θ − θ in Equation 5.Figure 4 indicates that the temporal evolution of the one-armed SWP from the observation (top)can be fairly well modeled by the simulation (bottom) with t = −
20 s and θ = 170 ◦ . Like theobservation, the simulation can make the one-armed SWP. The red or blue arms abruptly changethe trajectory to inward around x = 2 (cid:48)(cid:48) , y = 1 (cid:48)(cid:48) in both the observation and the simulation.We can successfully model the observed two-armed SWP as well. Because the wave power isconcentrated at m = 0 and 2, we reproduce this pattern by summing up v z of m = 0 and m = 2with the amplitude ratio of A /A = 0 .
54, the reference time of t = 30 s, and the reference angleof θ = 30 ◦ . In this simulation, the source is located at 1600 km below the β = 1 layer andthe averaged phase velocity is about 20 km s − . Figure 5 and associated animation represent thetemporal evolution of the two-armed SWP. The observation and simulation show quite similar two-armed spiraling features. The two blue and red arms abruptly move inward around x = − (cid:48)(cid:48) , y = 2 . (cid:48)(cid:48) x = 1 (cid:48)(cid:48) , y = − . (cid:48)(cid:48) DISCUSSION
Kang et al. W V W V W V W V W V í ; D U F V H F í < D U F V H F W V W V W V W V W V v L O S k m s − í v z Figure 5.
Similar to Figure 4, but for the case of two-armed SWP from 17:44:07 UT to 17:46:47 UT. (Ananimation of this figure is available.)
In this Letter, for the first time, we have presented a model that can explain the observed SWPsas slow magnetohydrodynamic (MHD) waves in an untwisted magnetic field. In our model, theapparently rotating pattern is associated with the superposition of non-zero- m azimuthal slow modes.A non-zero- m mode has a right-handed (left-handed) helical shaped wavefront for the case of positive(negative) m . As this wave propagates upwardly along the straight field in a vertical magnetic fluxtube, the wave pattern observed at some height shows an apparent rotation in the counterclockwise(clockwise) direction. This kind of a rotating wave pattern was observed for the case of m = 1 kinkmode (Jess et al. 2017), and the related vortex dislocations were detected in a time-distance mapalong the slit placed in the center of the axis (L´opez Ariste et al. 2016).The spiral structures and outward propagating wave patterns are formed by the internal drivingsources, i.e. situated inside the magnetic flux tube forming the umbra, which are placed below thephotosphere. Beacause the wave propagates quasi-isotropically in the high- β region, the longer thehorizontal distance from the wave source to the observation point, the later the wave arrives. Thedifference in the arrival times in the photosphere results in an apparent radially moving ring patternin the case of m = 0 (sausage) mode. In non-zero- m modes, the trailing spiral arm structures areformed because of the wave patterns rotate earlier as it is closer to the axis of the waveguiding fluxtube. The number of arms depends on the absolute value of m . Thus, the observed apparent rotatingspiral arms are not caused by the wave propagation in the azimuthal direction, but by the oblique,spiral-shaped wavefront of vertically propagating perturbations.Because of the abrupt spiraling motion of the one-armed spiral, Su et al. (2016) proposed that thispattern may be caused by the reflection at a light bridge. In our case, however, there was not lightbridge at all and, nevertheless, such SWPs were detected. Our simulation clearly shows that thespiraling patterns are formed by the superposition of the wavefronts of an m = 0 and a higher- m modes. The one-armed SWP is generated by an m = 0 sausage mode and an m = 1 kink mode, andthe two-armed SWP is formed by an m = 0 sausage mode and an m = 2 fluting mode.We surmise that the driving source of a SWP may be associated with the downflows caused bythe local magnetoconvection inside the sunspot. According to the 3D radiative MHD simulation ofKitiashvili et al. (2019), acoustic waves can be generated by the converging downflows at 1.5 Mm he Physcial Nature of Spiral Wave Patterns β plasma, the kink wave in a sunspot is mainly characterizedby parallel, field-aligned plasma flows. The radial flows, v r , in this wave are quite small, because the ω − k c s factor in Equation (1) tends to zero as the phase speed is about the sound speed. Anotherdifference is connected with the wave polarization. Kink oscillations of coronal loops are usuallylinearly polarized, while the spiral wave structure in a sunspot requires the kink oscillation to becircularly polarized; i.e. the azimuthal wavenumber is m = +1 or m = −
1. The sign is determinedby the sense of rotation of the wavefront.Because the mechanism does not require additional assumptions such as the flux tube twisting orrotation, we expect that such SWPs may be generally detected in any sunspots. As we accumulate theobservation of those patterns, we can infer more physical parameters in sunspots such as propagatingspeed of fast wave and depth of the wave driving source. Furthermore, those wave patterns can beconsidered as the evidence of the internal excitation of 3 minute oscillations in sunspots. Further studyof the SWPs may provide us with the clues to how magnetoconvection inside a sunspot generatessuch waves.We greatly appreciate the referee’s helpful and constructive suggestions and comments. This workwas supported by the National Research Foundation of Korea (NRF-2017R1A2B4004466). V.M.N.acknowledges support from the STFC consolidated grant No. ST/P000320/1, and the BK21 plusprogram through the NRF funded by the Ministry of Education of Korea.REFERENCES