Transverse loop oscillations via vortex shedding: a self oscillating process
DDraft version February 8, 2021
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Transverse loop oscillations via vortex shedding: a self oscillating process
Konstantinos Karampelas
1, 2 and Tom Van Doorsselaere Department of Mathematics, Physics, and Electrical Engineering, Northumbria University,Newcastle upon Tyne, NE1 8ST, UK Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven,Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium (Received 2020 December 7; Revised 2021 January 13; Accepted 2021 January 14)
Submitted to ApJLABSTRACTIdentifying the underlying mechanisms behind the excitation of transverse oscillations in coronalloops is essential for their role as diagnostic tools in coronal seismology and their potential use as waveheating mechanisms of the solar corona. In this paper, we explore the concept of these transverseoscillations being excited through a self-sustaining process, caused by Alfv´enic vortex shedding fromstrong background flows interacting with coronal loops. We show for the first time in 3D simulationsthat vortex shedding can generate transverse oscillations in coronal loops, in the direction perpendicularto the flow due to periodic “pushing” by the vortices. By plotting the power spectral density we identifythe excited frequencies of these oscillations. We see that these frequencies are dependent both on thespeed of the flow, as well as the characteristics of the oscillating loop. This, in addition to the factthat the background flow is constant and not periodic, makes us treat this as a self-oscillating process.Finally, the amplitudes of the excited oscillations are near constant in amplitude, and are comparablewith the observations of decay-less oscillations. This makes the mechanism under consideration apossible interpretation of these undamped waves in coronal loops.
Keywords:
Magnetohydrodynamical simulations; Solar coronal loops; Solar coronal seismology INTRODUCTIONIn recent years, observations by the Coronal Multi-channel Polarimeter, the Solar Dynamics Observatory,and Hinode spacecraft have already proven the ubiq-uity of transverse perturbations and waves in magneticstructures in the solar corona (e.g. Tomczyk et al. 2007;McIntosh et al. 2011). The importance of these wavesand oscillations is connected to their use in coronal seis-mology (e.g. Nakariakov & Ofman 2001; Nakariakov &Verwichte 2005), as well as their potential role as heat-ing mechanisms for the solar corona (for a review, seeVan Doorsselaere et al. 2020).Kink oscillations of coronal loops have been inten-sively studied ever since they were firstly observed (As-chwanden et al. 1999; Nakariakov et al. 1999). Fol-
Corresponding author: Konstantinos [email protected] lowing the theory of waves in a magnetized cylindricalflux tube (Zajtsev & Stepanov 1975; Edwin & Roberts1983), these observed perturbations have been treatedas standing kink modes (Van Doorsselaere et al. 2008).These first observations were of oscillations with ampli-tudes of a few megameters, which were decaying overtime after an being excited by external energetic phe-nomena (e.g. Nakariakov et al. 1999; Zimovets & Nakari-akov 2015; Nechaeva et al. 2019). The damping of theseoscillations has been attributed to the phenomena of res-onant absorption and phase mixing (Ionson 1978; Hey-vaerts & Priest 1983; Goossens et al. 2011) and havebeen studied both analytically and numerically in 3DMHD setups, where the effects of gravity, radiation, andthe Kelvin-Helmholtz instability (KHi) has also beenconsidered (e.g. Terradas et al. 2008; Antolin et al. 2014;Magyar et al. 2015; Hillier et al. 2019).Alongside those larger-amplitude decaying oscilla-tions, a second category of low-amplitude, transversewaves occurring in coronal loops has also been observed a r X i v : . [ a s t r o - ph . S R ] F e b Karampelas & Van Doorsselaere in recent years. These waves were first detected by Wanget al. (2012) and Tian et al. (2012), and were proven tobe omnipresent in active region coronal loops (Anfino-gentov et al. 2013, 2015), making them possible tools forcoronal seismology (Anfinogentov & Nakariakov 2019;Yang et al. 2020). These decay-less oscillations have anear constant amplitude over the course of many peri-ods, with frequencies equal to that of the fundamentalstanding kink mode (Nistic`o et al. 2013), as well as itssecond harmonic (Duckenfield et al. 2018).While the mechanism exciting these decay-less oscil-lations is still not identified, different explanations havebeen studied over the years. In Antolin et al. (2016),they were treated as line-of-sight effects created by theKHi vortices from impulsively oscillating coronal loops.Decay-less oscillations have also been modeled numer-ically as driven standing waves from footpoint drivers,both from monoperiodic (e.g. Karampelas et al. 2017,2019; Afanasyev et al. 2019; Guo et al. 2019; Shi et al.2021) and broadband drivers (Afanasyev et al. 2020).Another interpretation was considered in Nakariakovet al. (2016), in the form of a self-sustained oscillation.Unlike periodically driven oscillations, where the inputof energy is done periodically and the frequency is im-posed by the driver, self-oscillations are excited from(near) constant drivers, and the oscillation frequency isset by the system itself, and not the external driver. Inshort, self-oscillations are processes that can turn a non-periodic driving mechanism into a periodic signal. Sucha process was modeled in Karampelas & Van Doorsse-laere (2020), where a slow constant flow around a loop’sfootpoint eventually led to the excitation of a weak os-cillation. Despite the numerical limitations, that studyhas provided a first proof-of-concept for this process ina 3D simulation.In the current work we will continue to explore theconcept of self-oscillations of coronal loops, by consid-ering the excitation of a transversely polarized wavethrough the mechanism of Alfv´enic vortex shedding. Ev-idence of vortex shedding in the solar corona was re-ported in Samanta et al. (2019), in the vicinity of ashrinking loop in a post-flare region. Vortex sheddingdue to solar wind has also been proposed by Nistic`o et al.(2018), to explain the observed oscillations of cometaryplasma tails. In a 0D model first proposed by Nakari-akov et al. (2009), it was described how vortices gener-ated by an upflow passing by a loop can excite an oscilla-tion through a quasi-periodic horizontal force. The phe-nomenon of Alfv´enic vortex shedding has already beenexplored numerically for a bluff body in 2D in Gruszeckiet al. (2010). However, a coronal loop will behave dif-ferently than a fixed and rigid bluff body, thus making a 3D study essential. In this work we will study forthe first time the excitation of kink-mode oscillations byvortex shedding for a full 3D setup of a coronal loop.We will focus on the decay-less regime, although thismethod can also be applied to decaying oscillations. Wewill show that the mechanism under consideration canbe a possible interpretation of these undamped loop os-cillations. Finally, we will explore how this mechanismis affected by the characteristics of both the flow and theoscillator, making it essentially a self-oscillation that canbe initiated in solar coronal loops. NUMERICAL SETUPWe use a model of a straight flux tube of (minor)radius R = 1 Mm and length L = 200 Mm. This modelcorresponds to a semicircular coronal loop with a majorradius or ∼
64 Mm. The radial density profile for ourmodel is given by the relation ρ ( x, y ) = ρ e + ( ρ i − ρ e ) ζ ( x, y ) , (1) ζ ( x, y ) = 0 . − tanh(( (cid:112) x + y /R − b )) , (2)where b sets the width of the boundary layer. We con-sider b = 20, which gives us an inhomogeneous layer ofwidth (cid:96) ≈ . R . The index i ( e ) corresponds to the in-ternal (external) values with respect to our flux tube.The coronal background (or external) density is equalto ρ e = 0 . × − kg m − , three times lower thanthe loop (internal) density ( ρ i ). Similarly to Karam-pelas & Van Doorsselaere (2020), the temperature variesacross the tube axis (in the xy -plane), ranging from0 . .
35 MK outside (see Figure1), effectively modeling a loop during a cooling phase,as observed for loops in thermal nonequilibrium (Fro-ment et al. 2017). The scale height for our model( H ∼
55 Mm) is comparable to the major radius or thecorresponding coronal loop, which allows us to approxi-mate the temperature as constant with height, along theflux tube. For our primary setup, we consider a straightmagnetic field B z , parallel to the loop axis ( z -axis) withinternal and external field values of B zi = 22 . B ze = 22 . x ∈ [ − ,
17] Mm, y ∈ [ − ,
20] Mm and z ∈ [ − , δx, δy, δz ) = (80 , , z = −
100 and z =100 Mm, while z = 0 is the location of the loop apex.This resolution allows us to study the motion of the loop ortex-shedding driver − − − x (Mm) . . . . D e n s it y ( − kg m − ) . . . . . . T e m p e r a t u r e ( K ) Figure 1.
Density (in black) and temperature profile (inblue) of the flux tube cross-section along the x -axis at time t = 0. The grid points are depicted as dots on the two curves. and the development of larger scale flow instabilities, likevortex-shedding, in the xy -plane.The side boundaries in the y direction, as well as at x = 17 Mm are set to have Neumann-type, zero-gradientconditions for all quantities. On the “left” side bound-ary (at x = − v y and v z components ofthe velocity field. For the x velocity component at x = − v x = 5 × m s − , for a total duration of ∆ t = 1012 s. This leadsto the development of a horizontal flow along the x di-rection, which is also free to evolve along the y directionas well (see Figure 2). After a time t = 1012 s we switchthe boundary condition for v x at x = − z = − z = 100 Mm), we apply zero-gradient conditionsfor the pressure, density, and the three components ofthe magnetic field. The v z velocity component (alongthe axis of the loop) is set as antisymmetric, to pre-vent any outflows from the top and bottom boundaries,where the bases of the loop are located. Inside the loop( (cid:112) x + y ≤ R ), the v x and v y are set as antisymmetric,to fix the loop endpoints. Outside the loop at z = − z = 100 Mm we set the v x and v y components aszero-gradient (outflow conditions) in order to let the flowevolve freely along the xy -plane, in a way that allows forthe development of vortices.All calculations were performed in ideal MHD in thepresence of numerical dissipation, using the PLUTOcode (Mignone et al. 2012). We use the second-order characteristic tracing method to calculate thetimestep, and the finite volume piecewise parabolicmethod (PPM) with a second-order spatial global accu- Figure 2.
Schematic representation of the flow along the x direction, originating from the “left” side boundary at x = − racy and the Roe solver. Finally, to keep the solenoidalconstraint on the magnetic field, we employ Powell’s 8-wave scheme. RESULTSFollowing the basic idea from Nakariakov et al. (2009),we try to model an upflow around a coronal loop, orig-inating from the propagation of a CME. To that end,we initiate a flow from one of the side boundaries (at x = − t = 1012 s or t = 4 · P ,where P = 253 s is the period of the fundamental kinkmode (Edwin & Roberts 1983) for a loop with the char-acteristics of our primary setup, as described in Section2. A schematic representation of that flow, is shownin Figure 2. Driving with a constant background flow,equal for all heights, is a rather unlikely physical sce-nario, which can render a straight flux tube unstable.This is due to the strong excitation of higher harmonicsdue to the spatial profile of the flow over the loop height.To prevent this, we “switch off” the side boundary driverfor t > x = − P ) of the vortex shedding, the size (here diameter, d )of the blunt body and the flow velocity ( V ): St = dP V . (3) Karampelas & Van Doorsselaere
Figure 3.
Contour plots of density ( × − kg m − ) for our loop, at the apex. The velocity field is also overplotted. Fromleft to right, starting from the top panels, we show the contours at the first six snapshots of the simulation, between t = 0 and t = 189 . .
62 s. An animation of the density evolution is included in the online version of this manuscript.
Figure 4.
Top panels: contour plots of density ( × − kg m − ) for our loop at the apex, with the overplotted velocity fieldin white. Bottom panels: contour plots of the plasma z -vorticity ( × . t = 1012, 1265 and 1518 s. An animation of the z -vorticity ( ω z ) evolution is included in the onlineversion of this manuscript. In Gruszecki et al. (2010) it was found that the Strouhalnumber in MHD for coronal parameters has values be-tween 0 .
15 and 0 .
25. Considering a loop with diameter d = 2 Mm (minor radius of 1 Mm) and a period forthe fundamental kink-mode equal to 253 s, flow speedsof V ∼ −
50 km s − would be required for the initia-tion of vortex shedding with the same periodicity. Thiswould be essential in order for the loop to resonate withand be driven by the vortices. Assuming an averagevalue of St ∼ . v x = V = 50 km s − , in order to excite an oscillationwith a frequency close to that of the fundamental kinkmode.In Gruszecki et al. (2010), a bluff body was used,which would interact with the flow, but would not be af- fected by it. For our 3D setup, where our obstacle is nota bluff body but a loop allowed to oscillate, we expectedthat the background flow will deform the initial circularloop cross-section. Indeed, this can be seen in the panelsof Figure 3, where the first six snapshots of the simula-tion are shown, between t = 0 and t = 189 . .
62 s. Eventually, vortex shedding is initiated, as wecan see for the density and z -vorticity in Figure 4, forsnapshots at t = 1012, 1265 and 1518 s. Although vortexshedding is indicated from the evolution of the velocityfield and the vorticity, the loop cross-section only showsvague signs of displacement in the y direction, perpen-dicular to the flow.To test whether vortex shedding can initiate an oscil-lation, we tracked the center of mass of the loop cross-section at the xy -plane at every height along the z axis. ortex-shedding driver Figure 5.
Left panels: displacement of the loop center of mass at each height, for the duration of the simulation. The top panelis for the full time-series, and the bottom panel is for the detrended time series. Right panels: the corresponding normalizedpower spectral density plots for the two signals.
The results are plotted in the top left panel of Figure 5,where the temporal evolution of the loop displacementalong the y -direction is shown along the loop length.We observe a clear oscillatory pattern, which providesus with the first proof-of-concept for the validity of themechanism proposed in Nakariakov et al. (2009). Thisoscillatory pattern occurs on top of a mean displacementof the loop center of mass at the later stages of the sim-ulation. By plotting the normalized power spectral den-sity along the loop in the top right panel of Figure 5, wecan identify the spatial and temporal harmonic struc-ture of our oscillator. This height-frequency ( z − f ) di-agram shows a local maximum ∼ .
004 Hz, which is theeigenfrequency of the fundamental kink mode for ourloop f = 1 /P ∼ . P = 253 s. Due to thelimits imposed by the relatively short time series, andthe overall broadband nature of our driving mechanism,many additional frequencies are shown to be excited.We can see the inclusion of some very low frequenciespresumably due to the measured mean displacement ofthe loop center of mass. Identifying some of these addi- tional frequencies in a future study could be very usefulfor coronal seismology.In order to remove the effects of the loop mean dis-placement along the y direction from our power spec-tra density plots, we detrend our time-series using the scipy.signal.detrend command for Python. This per-forms a linear least-squares fit to the data, and thensubtracts that result (i.e. the mean displacement) fromthe initial data (i.e. the overall displacement). As wecan see from Figure 6 for the displacement of the cen-ter of mass across the background flow at the apex, thesignal is detrended once the oscillation starts, althoughwe end up with falsely pronounced values at the verybeginning of the simulation.Applying that detrending to the entire time series weget the detrended displacement of the entire loop overtime, at the bottom left panel of Figure 5. As we seefrom that panel, the oscillation amplitudes on the de-trended signal are of the order of 0 . Karampelas & Van Doorsselaere
Figure 6.
Solid black line: displacement of the center ofmass at the apex, in the y-direction. Dashed blue line, theoscillating signal after the mean displacement is subtracted. model of Nakariakov et al. (2009) to the level of 3Dsimulations. In addition, we do not see an obvious andconsistent decay of this amplitude over time, due to thecontinuous presence of the background flow. Despite thesimplicity of this model, this agreement indicates thatvortex shedding can potentially be a mechanism sus-taining decay-less oscillations in loops. From the powerspectra on the bottom right panel of the same figure,we can clearly see that the frequency of the fundamen-tal kink mode is the one excited the most. Finally, it issafe to assume that once the background flow weakenssubstantially, vortex shedding will not be able to sustainthe oscillation and the amplitude will decay.In the same figure, we see that there is an additionalband of frequencies near f = 2 mHz in the normal-ized power spectral density plot. From the equationof the Strouhal number, and assuming that its valuesare St ∼ .
2, this frequency can be obtained for veloci-ties of the order of 20 × m s − . As we can see fromthe velocity profile at the apex (Figure 7; also hintedfrom the velocity field in Figure 4), the velocity field infront of the loop drops to values around 20 −
25 km s − .This could potentially explain the peak near 2 mHz, asthe frequency imposed by the flow. Additionally, thefact that the loop cross-section changes over time willinevitably affect the frequency imposed by vortex shed-ding, explaining the width of the frequency band around2 mHz. Also, it is possible that the value of the Strouhalnumber in 3D environments in the presence of mag-netic field needs is different from the one calculated inGruszecki et al. (2010) for the 2D case. A full 3D pa-rameter study was outside the scope of this work, andtherefore not addressed here. However, the aforemen-tioned frequency band near 2 mHz seems to be dictated Figure 7.
Profile of the v x ( × m s − ) velocity at theapex, for part off our datacube depicting the loop at t =1012 s. by the Strouhal number of the flow, as shown in thefollowing paragraph.As the final step, we want to test whether driving anoscillation via vortex shedding is a self-oscillating pro-cess. For that, we have considered a secondary setup,identical to the one described in section 2, but with in-ternal and external magnetic field values of B zi = 45 . B ze = 45 .
64 G, respectively, doubling the Alfv´enspeed and halving the period of the fundamental kinkoscillation. In this new setup, we chose to drive thebackground flow for the whole duration of the simula-tion ∆ t = 2536 s, because the stronger magnetic fieldmakes the loop “stiffer,” delaying the start of the os-cillation. This new loop has a fundamental kink modefrequency of ∼ . SUMMARY AND CONCLUSION ortex-shedding driver Figure 8.
Plot of the normalized power spectral density forthe oscillation of the loop with internal and external mag-netic field values of B zi = 45 . B ze = 45 .
64 G (sec-ondary setup).
Although the idea that vortex shedding can excitestanding waves in coronal loops has been proposed inNakariakov et al. (2009), this mechanism had not beenproperly explored in a full 3D MHD model. The phe-nomenon of Alfv´enic vortex shedding has already beenstudied for a bluff body in 2D by Gruszecki et al. (2010).However, a coronal loop will interact differently with abackground flow than a fixed and rigid bluff body, andthus a study in 3D was essential. In the current work wesee that a nonrigid loop will be deformed when interact-ing with a background flow. In addition, we see for thefirst time in a 3D simulation that the vortex sheddingwill eventually force the loop into an oscillation in thedirection perpendicular to the flow, as was first proposedin Nakariakov et al. (2009).The long duration of the vortex-shedding driving andthe oscillation amplitudes comparable to the those foundin Anfinogentov et al. (2015) make this mechanism agood candidate for generating decay-less oscillations.However, once the background flow is no longer present,the oscillations would start decaying. Thus, also decay-ing oscillations could potentially be generated via vortexshedding for short-lived background flows.The mechanism of vortex shedding induced oscilla-tions seems to fall into the self-oscillation processes.These, as was described in Jenkins (2013), are pro-cesses that can turn a nonperiodic driving (like a back-ground steady flow) into a periodic signal (like a looposcillation). Although vortex shedding has a preferred periodicity, as described by the flow Strouhal number(Gruszecki et al. 2010), the spectral densities of two dif-ferent loops have clearly shown strong peaks around thecorresponding frequencies of their fundamental stand-ing kink-modes and not near the peak dictated by theStrouhal number. This shows that the oscillating loopimposes its own frequency, which is a characteristic ofself-oscillations. Future studies could also help identifysome of the additional frequencies and harmonics ob-served, which could be important for coronal seismology.Despite its successes though, our very simple modelstill only provides a proof-of-concept for sustainingdecay-less oscillations through the vortex shedding, andadditional studies are necessary. Although the results ofGruszecki et al. (2010) for the Strouhal number seem tomatch the hydrodynamical case, a full parameter studyin 3D MHD should be performed, as this was outside thescope of the current study. In order to compare with ob-servations, gravitationally stratified loops and realisticflow profiles should be considered. Changing the flowspeed would change the location of frequency peak dic-tated by the Strouhal number, but it could also havecatastrophic effects on the loop cross-section, should theflow be too strong or the loop not “stiff” enough (i.e.having a weaker magnetic field). In addition, the loopcharacteristics are expected to affect the resulting oscil-lation amplitudes, and thus the strength of the differentfrequency peaks.ACKNOWLEDGMENTSThe authors would like to thank the referee for theirhelpful comments. K.K. has received support for thisstudy through a postdoctoral mandate from KU LeuvenInternal Funds (PDM/2019), by a UK Science and Tech-nology Facilities Council (STFC) grant ST/T000384/1,and by a FWO (Fonds voor Wetenschappelijk Onder-zoek – Vlaanderen) postdoctoral fellowship (1273221N).T.V.D. is supported by the European Research Council(ERC) under the European Union’s Horizon 2020 re-search and innovation program (grant agreement No.724326) and the C1 grant TRACESpace of InternalFunds KU Leuven. The computational resources andservices used in this work were provided by the VSC(Flemish Supercomputer Center), funded by the Re-search Foundation Flanders (FWO) and the FlemishGovernment – department EWI.REFERENCES
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