Category: UKSP Nugget

124. The independence of kink oscillation periods on noise

Author: Valery Nakariakov, Dmitrii Kolotkov and Sihui Zhong at the University of Warwick.

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The launching of the TRACE spacecraft ushered in a new era of high-resolution observations, boosting the confident detection of MHD waves and their use for probing unknown physical parameters in the coronal plasma, known as MHD seismology. In particular, this technique is crucial for the estimation of the magnetic field in corona where the Zeeman effect could not be resolved.

Solar coronal loops are known to host several types of waves and oscillations, among which kink oscillations are the most intensively applied in seismology [1]. The kink oscillation period is determined by the time-varying displacement extracted from time-distance maps. Then, the period is used to derive the kink and Aflvén speeds, the magnetic field strength, scale height of loop density, and transverse density structure etc., by assuming that the observed period is interpreted as the period of a standing kink wave.

Kink oscillations appear in two regimes; decaying and decayless. The former is triggered by impulsive energy releases, while the latter appears without a visible driver, and displays ubiquitously in coronal loops. Decayless kink oscillations are a promising tool for routine diagnostics of the quiescent corona. In addition, its undamped nature arouses interest in how the energy is continuously input into the loop system to balance the damping by e.g. resonant absorption. So far, decayless kink oscillations have been interpreted as self-oscillations sustained by a quasi-steady flow moving across the loop footpoints [2], or random motions of loop footpoints [3]. Alternatively, it is described as an apparent periodic brightness because of the Kelvin-Helmholtz instability and resonant absorption.

Typical dynamic processes of the corona are coloured noises manifesting as power-law-like Fourier power spectra of observables. In the context of MHD seismology, it is of interest to understand whether the noise affects the quality of seismological inversions. The self-oscillatory model of decayless kink oscillations could include noises associated with random footpoint and coronal motions which sustain the oscillations. Therefore, it is necessary to know to what extent the kink oscillation period is influenced by the existence of the noise.

In this work, we combine the randomly driven and self-sustained oscillation models of decayless kink oscillations of coronal loops to demonstrate that the oscillation period is almost insensitive to the noisy dynamics around the oscillating loop and its footpoints.


We construct a zero-dimensional model describing the decayless kink oscillations based on a randomly driven Rayleigh oscillator with random coefficients. The model accounts for negative friction responsible for the interaction of the loop with external non-periodic flows, and random motions of footpoints. Both processes could compensate the damping. The governing equation includes multiplicative and additive noises which represent chromospheric and coronal noisy processes. The noises are independent of each other.

In the absence of noise, the governing equation reduces to the standard Rayleigh equation, which has an asymptotically stationary (decayless) oscillatory solution with constant period and amplitude.

In the presence of essentially broadband red noise, the oscillatory patterns are found to be almost monochromatic. For both multiplicative and additive noises of the amplitudes up to 120% of the corresponding model coefficients, the oscillation period is found to depart from the noise-free value by less than two percent (Figure 1).

In the noisy cases, the oscillation amplitude experiences fluctuations around the mean value. Random fluctuations of the coefficients associated with the external flows are found to result in a symmetric modulation of the oscillation pattern, while random footpoint motions lead to antisymmetric modulation. When both multiplicative and additive noises are present, the amplitude modulation is a superposition of the symmetric and antisymmetric modulations, see Figure 2. Such modulations are also qualitatively consistent with the observed behaviour, see Figure 3.

The results obtained justify the seismological diagnostics based on the kink oscillation periods, e.g., the construction of Alfvén speed and magnetic field maps of pre-flaring coronal active regions [4].


In terms of a zero-dimensional model based on a randomly driven Rayleigh oscillator with random coefficients, we investigate the effect of noise on decayless kink oscillations of solar coronal loops. It was found that the period of kink oscillations is practically independent of noises, justifying the coronal seismology techniques based on kink oscillation periods. Moreover, in the presence of noise, kink oscillatory signals experience long-durational amplitude modulations which are consistent with observational findings.

This work has been published in MNRAS. The full version could be found here.


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123. Transverse Oscillations of Spicules and Estimating Their Energy Flux

Author: William Bate at Queen’s University Belfast.

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Spicules are dense and narrow jets of chromospheric plasma which rise from the chromosphere and can reach up to coronal heights. They have relatively short lifetimes, typically less than ten minutes, and are extremely common. Spicules are typically understood to outline near-vertical magnetic field. When the solar limb is viewed in certain wavelengths, such as H-alpha, they appear as a dense forest of straw-like features. This “forest of spicules” is visible in the left panel of Figure 1 above the limb.

One of the major questions of solar physics is the so-called “coronal heating paradox”, where the solar atmosphere is at a higher temperature than would be expected through conventional means. However, it may be more appropriate to call it the “chromospheric heating paradox”, as the energy required to heat the corona is around an order of magnitude less than that required to heat the chromosphere, with an energy flux of 103–104 W m−2 required to balance the radiative losses of the chromosphere [6]. One of the proposed resolutions of this paradox is propagation and dissipation of wave phenomena, also known as the AC heating mechanism (as opposed to the DC heating mechanism believed to be caused by magnetic reconnection).

Transverse Oscillation Property Analysis

These spicules, as dense tubes of plasma, are of particular interest when it comes to the transfer of energy from the photosphere into the higher levels of the solar atmosphere. Due to their topology, with their lengths being much greater than their widths, spicules can generally be described by the magnetic cylinder model [3]. This allows their behaviour to be interpreted and modelled in terms of the magnetohydrodynamic (MHD) modes of a magnetic cylinder.

Perhaps the most straightforward sort of wave behaviour that spicules can exhibit is their transverse oscillations. Spicules can be seen to “sway” across the plane of sky above the solar limb (with motions in along the line of sight likely also present). These can be interpreted as the kink mode of a magnetic cylinder which displaces the axis of the entire flux tube. These transverse oscillations are typically observed having mean periods of 50-300s, displacement amplitudes on the order of 100km, velocity amplitudes on the order of 10 km s-1, often with 50-1000 examples found within an individual dataset. For a more comprehensive review of the properties of these oscillations, see Section 5 of [4].

By placing slits at different positions along the length of a spicule, it is possible to create a time-distance diagram showing the side-to-side motion of the spicules as bright streaks against the dark background of space. An example time distance diagram is shown in the right panel of Figure 1, with oscillatory behaviour noticeable in many of these spicules. By tracking the centre of these bright streaks over time and applying tools such as wavelet and/or Fourier analysis it is possible to measure oscillation properties such as displacement amplitude, velocity amplitude and period.

Performing similar analysis for slits placed along the length of the spicule, it becomes possible to analyse how these properties evolve as a function of height. In recent work [1] it has been found that period seems relatively independent of distance along the spicule. However, displacement and velocity amplitude appear to increase with distance (or height as the spicules are near vertical). This is not entirely unexpected, as the density of spicules is thought to decrease exponentially with height. At greater heights, the lower density plasma can be moved further and faster with similar energy to smaller heights. Through further analysis of waves within the same spicule at different heights along the spicule, the phase lag between adjacent heights can be found, allowing for calculation of phase speeds and propagation direction.

Energy Flux Estimates

Using velocity amplitude, phase velocity, and an estimate of plasma density, it is possible to calculate an estimate for the energy flux carried by these waves. It is extremely important that care is taken when choosing what wave mode to interpret the transverse motions of the spicules as. For example, when treating their observed transverse motions as bulk Alfvén waves, [2] estimated the energy flux carried by them to be 4000-7000 W m-2. However, by reanalysing these motions as kink modes and assuming that spicules take up around 5% of the local chromospheric volume (also known as a filling factor), this energy flux estimate was reduced to 200-700 W m-2 [5].

Sticking with the kink mode interpretation and 5% filling factor, another set of transverse waves within spicules have been found to carry significant energy upwards through the solar atmosphere [1]. The energy flux estimations for different heights are presented in Figure 2. These are broken up into the total energy flux in the top panel, the energy flux carried by the upwards propagating waves in the middle panel, and the energy flux carried by the downwards propagating waves in the bottom panel. For all waves examined, it can be seen that there is a decrease in energy flux with height. A black dashed line in the upper panel represents a linear line of best fit, with a gradient of -12,600 W m-2/Mm. This is due to the decrease with height of energy flux carried by the upwardly propagating waves (middle panel), whilst the energy flux carried by the downwardly propagating waves remains approximately constant with height. It can also be seen from that the energy flux of the short-period (<50s, red) waves is greater than that of the long-period (>50s, blue) waves for the full set of propagating waves.

The Future

An important topic of further investigation is what causes this decrease in energy flux with height. This could be due to a number of factors, including but not limited to: physical thermalisation of wave energy into localised heat via dissipation mechanisms; damping of detectable transverse waves through the process of mode conversion; and reflection of the waves downward at varying heights above the solar limb.… continue to the full article

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122. Direct evidence that twisted flux tube emergence creates solar active regions

Author: David MacTaggart at the University of Glasgow, Chris Prior, Breno Raphaldini at Durham University and Paolo Romano, Salvo Guglielmino at INAF Catania.

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Solar active regions are the nurseries of the most important dynamical events in the Sun’s atmosphere, including flares and coronal mass ejections. A fundamental constituent that helps to determine the evolution of an active region is the topology of its magnetic field. By this, we mean how the magnetic field lines are connected and twisted. A long-held assumption of active region formation is that they are created by large tubes of magnetic field with a net twist (rather like a large magnetic slinky). This assumption has been popular for two main reasons. The first is that twisted magnetic fields can survive in the solar convection zone better than untwisted fields, and are thus more likely to reach the photosphere and emerge as active regions. The second is that simulations of the emergence of twisted tubes are able to produce a variety of signatures that are found in solar observations, which match well with proxies of magnetic twist found in observations. However, despite many works suggesting the existence of twisted tubes creating active regions, the question of whether or not this is the case has remained open. This is because there exist plausible competing theories which suggest that the twist in active regions can be created in the atmosphere rather than emerge into it. Further, observational proxies used in support of the emergence of pre-twisted tubes can be reproduced by emerging magnetic fields that are not twisted tubes. What is needed is a direct measure of the magnetic topology.

Magnetic winding

One widely-used measure of magnetic topology is magnetic helicity. The flux of this quantity through the photosphere can be calculated in observations, and many studies of this have been made. No clear signature for twisted tube emergence has been found, however. One reason for this is that magnetic helicity represents a combination of two fundamental quantities, magnetic flux and field line topology, which together can introduce a confound. For example, a highly entangled field with a strong topological measure can have a very weak field strength. Thus, the magnetic helicity of such a field can be small and, therefore, not diagnose the topological complexity of the field clearly. It is not that magnetic helicity is giving an erroneous result, rather it is the interpretation of magnetic helicity as a direct measure of magnetic topology which is not correct in general. In order to find a direct measure, we can renormalize magnetic helicity to produce a quantity called magnetic winding [1]. This quantity is effectively the magnetic helicity with the field strength weighting removed, and thus represents a direct measure of magnetic field line topology.

Signatures in simulations and observations

The flux of magnetic winding can be calculated in both simulations and observations, just like magnetic helicity. The analysis of magnetic winding in simulations of twisted flux tubes emerging through a convective layer and into the solar atmosphere, reveals a consistent “rise and plateau” signature across a range of parameters [2]. This is not the case for magnetic helicity which does not provide a consistent signature, just as in observational studies. An example of the accumulation of magnetic winding in a flux emergence simulation is shown below.

Figure 1(a) shows field lines of an emerging twisted tube below and above the photospheric boundary (shown in red). There is no visible resemblance to a twisted flux tube due to the strong deformations caused by convective motion and the emergence into the atmosphere. That being said, the magnetic winding is robust enough to still detect that the magnetic field has a net twist, and this can be seen in Figure 1(b), which shows the accumulations of the total magnetic winding L, as well as the contributions to magnetic winding due to emergence, Lemerge, and photospheric braiding motions, Lbraid. What this signature is saying is that when the bulk of the twisted tube rises above the photosphere, it remains there. Even though convection drags down part of this field, no more topologically-significant field is passing through the photosphere. Thus, after the rise during emergence, there is a plateau which indicates that the main topological content has passed through the photospheric boundary (where the calculations are performed). Further, Lemerge dominates the contribution to L, which means that the signature is dominated by a pre-twisted magnetic field, rather than one whose twist is generated later by photospheric motions.

This magnetic winding signature for twisted tube emergence has now also been found in observations. As an example, Figure 2 shows the winding accumulations for the bipolar region AR11318.

The “rise and plateau” signature can be found for this region (even despite the period of missing data). AR11318 has been studied extensively [3] and many signatures had been found which are highly suggestive of twisted tube emergence though not conclusive. The magnetic winding signature is the last piece in the puzzle to confirm that the region is formed by a twisted magnetic flux tube.


Magnetic winding is a robust measure of magnetic topology that can provide different information compared to magnetic helicity. Since it is a direct measure of field line topology, it has been used to show that pre-twisted tubes can emerge to create active regions. Therefore, the assumptions used in many flux emergence simulations now have an observational basis. For many more details and examples, together with a large set of relevant references, please consult [4]. Further developments of magnetic winding for use in flare prediction can be found in [5].


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121. Quasi-periodic problems; what’s going on with QPPs?

Author: Tishtrya Mehta from The University of Warwick

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What are QPPs?

Solar flares, driven by magnetic reconnection, describe the phenomenon of a rapid and localised energy release from an active region on the Sun. The intensity of the electromagnetic radiation produced increases rapidly, in what we call the impulsive phase of the flare, and then after reaching its maximum value gradually falls back down to its pre-eruptive level in what is known as the decay phase. Flares can also be observed on other stars, where they’re known as stellar flares.

Within the emission associated with these flares we often see another behaviour which we call a quasi-periodic pulsation (QPP), where the flare’s brightness oscillates over its duration. The properties of QPPs vary widely across different flares – they can be long or short lasting, but typically have average periods in the range of several seconds to a few minutes [1]. They may also exhibit amplitude modulation, and the pulsations can vary substantially in shape. QPPs are often observed across many wavelengths as can be seen in the well-reported and discussed ‘Seven Sisters QPP’ shown in Figure 1 [2].

Although it is accepted that QPPs are a common feature of both solar and stellar flares, there is still disagreement as to how prevalent they are, with QPPs being reported in 30 – 90% of solar flares [3] [4]. The lack of consensus on the prevalence of QPPs is partly due to there being no single definition of a QPP. Different analysis techniques will vary in their ability at identifying QPPs of different shapes, periods, and durations, which will substantially change the number of QPPs it finds.

Where are the non-periodic oscillations coming from?

As the name suggests, QPPs are rarely perfectly periodic. In fact many QPPs have been seen to have non-harmonic shapes, from triangular profiles to period or phase shifts over the duration of the flare. Some QPPs exhibit growth in their instantaneous period, the study of which could be key in determining the cause of quasi-periodic behaviour. Many different models have been proposed to explain the origin of QPPs (see [1] and references therein) but so far none have been definitively proven to be able to reproduce all of the variations in QPP behaviours that we observe.

Indeed, several of these models may be responsible but as of yet we don’t know which models are the most likely or accurate. Furthermore, it could be that different models can explain different classes of QPPs. In studying the prevalence and properties of non-stationary QPPs, we move one step closer to a full model of solar flares.

A case study of period growth in QPPs:

Let’s investigate a QPP case study which has shown evidence of period growth.
On the 19th of July 2012, the Sun produced an M7.7 class flare (which has since been reclassified as a X.1 class flare, following the removal of the SWPC scaling factors) and an accompanying coronal mass ejection, which gave rise to some beautiful coronal rain as seen here. This long duration flare lasted almost three hours and was observed by both the Geostationary Operational Environmental Satellites (GOES) X-ray Sensor and Atmospheric Imaging Assembly (AIA) aboard the Solar Dynamics Observatory.

We detrend the flare, measured in soft X-ray (1-8 Angstrom) by GOES-15, using Empirical Mode Decomposition (EMD) and the resulting detrended signal can be seen in the top panel of Figure 2. The detrended flux shows oscillatory behaviour characteristic of the QPP phenomenon. The amplitude of the QPPs is modulated throughout the different stages of the flare; the QPPs’ amplitude grows during the impulsive phase, reaches a maximal value at flare maximum, and then decreases over the decay phase until around 07:10 where the amplitude of the QPP returns to the approximate noise level of the pre-flare signal.

The continuous wavelet spectrum of the normalised QPPs (found by dividing the detrended flux by its Savitzky-Golay Envelope) seen in the bottom panel of Figure 2 shows an increase in instantaneous period of the QPPs. The period grows from about 300 seconds at 06:00 to around 600 seconds at 07:00, with the rate of period growth appearing approximately linear.

So now what?

Now that we’ve found proof of period growth in one QPP it’s not unlikely that we’ll find it in others. So the question remains – what’s causing this phenomenon? Is it due to one, or a combination, of the proposed QPP generation mechanisms already suggested? Or is it a result of something else- such as physical changes in the flaring region? We envisage that more in-depth studies of QPPs and their associated period drifts will hold the key to cracking this conundrum and pave the way for a better understanding of QPPs and flare events.

This UKSP Nugget is based on the work by L. Hayes and T. Mehta and is in final stages of preparation to be submitted for publication


  • [1] Zimovets et al., 2021, SSRv, 217, 5
  • [2] Kane et al., 1983, ApJ, 271, 376
  • [3] Inglis et al., 2016 ApJ, 833 284
  • [4] Dominique et al., 2018, SoPh, 293, 4

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120. NuSTAR observations of weak microflares

Author: Kristopher Cooper, Iain Hannah (University of Glasgow).

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What are microflares?

Solar flares release stored magnetic energy into mass flows, heating, and particle acceleration throughout the Sun’s atmosphere and occur in active regions (ARs) [1]. Small flares occur more frequently as energy release decreases with a flare frequency distribution consistent with a power-law. This allows for the possibility that the weaker flares actually contribute more net energy to the solar atmosphere than their brighter, but less frequent, counterparts. The energy release of flares spans decades of energies with microflares having energies between 1026–1028 erg and observed to have <10-6 W m-2 in GOES (1–8 Å) soft X-ray flux, with an A-class microflare being on the order 10-8 W m-2 [2,3]. We use the Nuclear Spectroscopic Telescope ARray (NuSTAR, [4]), an astrophysical focussing X-ray imaging spectrometer capable of observing the Sun, to probe these incredibly weak flares that are often very difficult to identify with other, even solar dedicated, instruments. For an overview of all NuSTAR solar observation campaigns and publications please visit Iain Hannah’s GitHub page.

NuSTAR observation overview: 2018 September 9 and 10

NuSTAR performed six hour-long solar observations on 2018 September 9 and 10, initially targeting a region already observed by the FOXSI-3 sounding rocket [5]. However, we find that AR12721, appearing on September 8, dominates NuSTAR’s field of view (FOV) shown in Figure 1 (panel e) [6]. Creating lightcurves of the extreme ultraviolet (EUV) AR emission (Figure 1, panel a), we find that there is no clear behaviour indicative of a microflare until we produce an FeXVIII proxy (panel b, [7]). Using the NuSTAR X-ray lightcurve, in combination with the FeXVIII proxy, we identify 10 microflares across both days with 7 taking place on September 9 (panel c). None of the NuSTAR observed microflares were easily visible in the corresponding GOES data itself and so were estimated to be GOES sub-A class from NuSTAR spectral analysis. During NuSTAR’s second orbit (10:26–11:26 UTC), AR12721 produces the brightest and weakest NuSTAR microflares of these data and are labelled as microflare 3 and 4, respectively.

Microflare 3: potential non-thermal signatures

The brightest NuSTAR microflare from these data was easily separated into a rise, peak, and decay time for further investigation via spectral analysis (Figure 2, [6]). Microflare 3 occurred in isolation and so a pre-flare time was also obtainable. We find that the FeXVIII proxy matches the lower X-ray range well (2.5–4 keV) while the higher X-ray energy range (4–10 keV) is more impulsive. This could indicate the existence of hot material at the very start of the microflare or the presence of non-thermal emission. During the pre-flare time, we find typical hot AR core temperatures at ~4 MK with the rise phase being the hottest reaching temperatures >7 MK. The microflare then progresses to show a slight, general decrease in temperature while increasing the emission measure by almost an order of magnitude. Microflare 3 is estimated to have an A0.1 GOES equivalent class.

Inspecting the spectral profile of microflare 3’s rise time (~10:29–10:32 UTC), we observe an excess in the residuals >7 keV. This would suggest an additional model is needed to accurately represent the observed spectrum. Fitting an extra thermal model we find that unphysically high temperatures are required and so we then fit a power-law model to represent potential non-thermal emission (Figure 3, [6]). The power-law model removes the residual excess and estimates a non-thermal energy release of the same order of magnitude needed to produce the corresponding thermal model parameters.

Microflare 4: a wee flare

The weakest NuSTAR X-ray microflare to be observed in September 2018 was microflare 4 occurring at ~11:04 UTC [6,8]. Like microflare 3, microflare 4 occurred in isolation and a pre-flare time could be identified. Although its emission is very faint microflare 4 still measures 10–20 arcseconds across in SDO/AIA (Figure 4, [8]). Performing spectral fitting on the microflare time (~11:03–11:05 UTC), and fixing the model that best represents the pre-flare emission, we find that this incredibly faint microflare still reaches temperatures >6.5 MK and has a GOES class equivalent of approximately a thousandth of an A-class (Figure 5, [8]). The estimated instantaneous thermal energy release from microflare 4, at 1.1×1026 erg, makes it the weakest X-ray microflare from an AR currently in literature.

Unlike microflare 3, microflare 4’s spectrum is well represented with two thermal models with no detectable count excess. However, non-thermal emission could still be present, but hidden, within the noise of the data where we have counts and consistent with a null detection where we do not. Investigating the non-thermal models that fit these criteria, we find that there are non-thermal upper limits that could provide the required heating rate to the microflare.


  • Sensitive imaging and spectroscopic X-ray observations are crucial when investigating incredibly weak solar microflares as they are inconspicuous at many other wavelengths.
  • Although the NuSTAR X-ray microflares are extremely faint, sub-A class equivalent flares are not necessarily spatially small and can reach very hot temperatures.
  • Sub-A class microflares can have very strong evidence of, or can be consistent with, non-thermal emission with microflare 3 being one of the weakest non-thermal X-ray microflares in the literature.
  • With NuSTAR, an observatory that is not optimised for solar observation, we are approaching events close to nanoflare energies as opposed to microflare energies.

All microflares NuSTAR observed on 2018 September 9 and 10 are thoroughly discussed in [6], with microflare 4 studied further still in [8].


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119. Solar magnetic vortices

Author: Suzana S. A. Silva , Gary Verth, Viktor Fedun The University of Sheffield, Erico L. Rempel Aeronautics Institute of Technology, Sergiy Shelyag Deakin University, Luiz A. C. A. Schiavo Sao Paulo State University.

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Photospheric flows play a crucial role in the dynamical evolution of the solar atmospheric as they are inherently coupled to magnetic fields, and this may lead to the generation of strong energetic events such as flares, coronal mass ejections, and solar tornados. The wide variety of photospheric plasma dynamics can create barriers in the plasma flow, impacting the magnetic field distribution, thereby influencing important aspects of energy and mass transport in the solar atmosphere. There is particular interest in studying the twisting of magnetic field lines due to photospheric motions, however, there is actually no universally accepted definition for a twisted magnetic flux tube. This inhibits the development of techniques to automatically identify twisted magnetic flux tubes and here we address this fundamental problem.


Due to their geometry, we can define the twisted magnetic flux tube as a vortex in the magnetic field, the M-vortex. This is illustrated in Figure 1, where we see the geometry of a twisted flux tube from a realistic magneto–convection simulation performed using MURaM [1].

We performed our analysis on a time frame series of a simulated solar plage , covering 600 km from the solar surface to the lower chromosphere. The vortex boundaries were determined based on the Integrated Averaged Current Deviation (IACD method [2]. The IACD field was calculated by tracing the magnetic field line for each point starting in a selected height level and computing the current density along the line minus the spatial average of the current. The vortex boundary was defined as the outermost convex contour of the IACD field around the local maxima.

The meaning of the vortex boundary defined by IACD is illustrated in Fig. 2 where we can see the following:

  • The boundary delimits a region where the vortex boundary lines are twisted and preserve their coherence for the length used to compute the IACD field,
  • The magnetic field lines inside the vortex are also twisted, forming a coherent twisted flux tube,
  • The lines traced from points inside the M-vortex (green lines) do not cross the vortex boundary (blue lines).

Therefore, IACD provides a proper physical definition for twisted magnetic flux tubes and automatically detects those structures.


We computed the plasma variables across the M-vortices and averaged them along the azimuthal direction, considering a cylindrical coordinate system where the origin is placed at the vortex centre. We found that inside M-vortices mainly downflows are present and that they locally concentrate the vertical magnetic field and current [3]. Based on the ratio of magnetic energy (Em) and kinetic energy (Ek), we classify the M-vortices as: type I M-vortex (Em/Ek >1); and type II M-vortex (Em/Ek ≤1).

Examples of both types of vortices are given in Figure 3, where we see the magnetic field lines of the M-vortex in blue and the surrounding velocity field lines in red. From Figure 3, we see that the different types of M-vortex also present different geometries. Moreover, we see that the M-vortices appear in regions with shear flows[3].

In order to compare M-vortices and kinetic (plasma flow) vortices, which we denote as K-vortice, we also applied Instantaneous Vorticity Deviation to identify K-vortices for the time interval of the analysis as described in [4]. Figure 4 displays a 3D view of the domain with the detected K- and M- vortices at t= 1424 seconds.

The magnetic field lines of all detected M-vortices are represented by white lines and the red lines show the velocity field streamlines from detected K -vortices. On average, over a time period of 500s, the densities of ~13 and ~28 per Mm2 were detected for M- and K-vortices, respectively. The K-vortices are located in low plasma-β regions and the M -vortices are present in regions of the flow with higher plasma-β. We find that K -vortices and M-vortices have a mean lifetime of 84.7s and 54.1 seconds, respectively, i.e., the plasma supports more K -vortex long-duration structures than M-vortices [3].


  • M-vortices are observed in parts of the intergranular lanes with the plasma-beta >1. K-vortices appear in low plasma-beta intergranular lanes.
  • M-vortices locally concentrate the vertical magnetic field due to the gas pressure gradient between the vortex boundary and its centre, forcing a new magnetic field into the M-vortex.
  • There are two types of M-vortices: they show differences between their shape, magnetic and kinetic energy ratio.
  • M- vortices appear if two conditions are simultaneously present: (i) shear flow, (ii) plasma-beta >1
  • Further numerical studies and high-resolution observations at different spatial and temporal scales are essential to correctly describe the interplay between the magnetic field and K-vortices and the interconnectivity between structures at different height levels.


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118. Tour de France of compressive waves in the Sun’s corona

Author: Dmitrii Y. Kolotkov and Valery M. Nakariakov from the University of Warwick.

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What makes solar coronal waves wavy?

The intrinsically filamentary nature of the coronal plasma plays a crucial role in the dynamics of magnetohydrodynamic (MHD) waves. In particular, the waves could be guided along the plasma inhomogeneities. Due to this waveguiding effect, coronal MHD waves are known to be subject to effective dispersion manifested through the dependence of the wave speed on the wavenumber (frequency), which is most pronounced for fast magnetoacoustic (MA) waves. Similarly to a mass-start in road cycling, where all cyclists begin the race at the same time and position, an impulsively excited ensemble of fast MA harmonics, initially localised in space and time, disperses along the waveguide and forms a quasi-periodic fast-propagating perturbation of the local plasma parameters (see Fig. 1, for example). Such rapidly propagating trains of fast waves, guided by field-aligned plasma non-uniformities, are confidently observed both near the base and at higher heights of the Sun’s corona [1].

We determine characteristic signatures of fully developed fast wave trains in the time domain, obtained from theoretical modelling of fast MA waves in a coronal plasma non-uniformity. We demonstrate the link between the time history of a fast wave train and parameters of the hosting waveguide, which gives important and sometimes unique seismological information about the cross-field structuring of the coronal plasma.

From fast magnetoacoustic tadpoles to boomerangs

We model fast MA waves in a low-β plasma slab stretched along the z-axis in the linear regime (e.g., [2]). In the model, the direction of the equilibrium magnetic field coincides with the slab axis, and the cross-field profile of the plasma density (in the x-direction) is given by a continuous function with varying steepness. Thus, the waveguide is characterised by two free parameters, its depth (i.e. ratio of the plasma densities or Alfven speeds inside and outside) and steepness (i.e. smooth or sharp boundary), which determine the dynamics of fast MA waves in this model.

For smooth transverse density profiles, the group speed of fast MA waves varies with the parallel wavenumber monotonically between the Alfven speeds inside and outside the waveguide, CAi and CAe; and it has a well-pronounced minimum for steeper waveguide boundaries (see Fig. 2). The former means that among the impulsively excited ensemble of fast MA waves in a waveguide with sufficiently diffuse boundaries, all guided parallel harmonics will propagate at their own group speeds (between CAi and CAe), eventually forming a quasi-periodic pattern seen in the Morlet wavelet power spectrum as a tadpole with a narrow-band tail and broader-band head [3]. In steeper waveguides, in contrast, there will be a relatively narrow interval of parallel harmonics propagating at the highest group speeds CAi < Vgr < CAe (a quasi-periodic phase I in Fig. 1); pairs of harmonics with distinctly different wavelengths travelling at the same group speed Vgrmin < Vgr < CAi (a multi-periodic or a peloton phase II in Fig. 1); and a single parallel harmonic trailing behind all other guided harmonics at Vgrmin (Airy phase III in Fig. 1). Such a structure of the wave train is seen as a boomerang shape in the wavelet spectrum with two well-pronounced arms at shorter and longer periods (bottom panel in Fig. 1). The revealed multi-phase structure of fast MA wave trains is consistent with the suggestion of [4], based on the analogy with Pekeris waves in an ocean layer.

Observational example

The high time resolution traditionally available in the radio band seems to be most suitable for the detection of these boomerang-shaped fast MA wave trains in observations.

For example, [5] observed signatures of a fast wave train in the decimetric radio emission at 973-1025 MHz, with a frequency drift of Δft = 8.7 MHz/s towards lower frequencies which could be interpreted as an upward propagation of the wave train at the speed about 870 km/s (see Fig. 3). The shape of the wavelet spectrum of this wave train is seen to change from a tadpole at higher frequencies (lower heights) with a narrow-band tail around 81-s oscillation period and a broader-band head to a boomerang at lower frequencies (higher heights) with two well-pronounced arms around 81 s and 30-40 s. These observational properties are consistent with the theoretical scenario described in our work for the development of a fast MA wave train in a waveguide with a steep transverse density profile.

Prospects for coronal seismology

A synergy of observations of such boomerang-shaped wave trains with theory offers a unique possibility for probing simultaneously the plasma waveguide depth (Alfven speed ratio CAe/CAi) and steepness. According to the model, one can estimate CAe/CAi from the observed duration of phase I (see Fig. 1). Likewise, the duration of peloton phase II gives the ratio CAi/Vgrmin which is sensitive to the waveguide steepness. Thus, the time history of fast-propagating quasi-periodic wave trains carries important information about the transverse structuring of the coronal plasma which opens up interesting perspectives for coronal seismology, using high-resolution and high-sensitivity observations from existing (e.g. AIA/SDO, LOFAR) and upcoming (e.g. SKA, METIS/SO, ASPIICS/Proba-3) instruments.

The work has been published in MNRAS, 2021, DOI.


  • [1] Li et al. 2020, SSRv, 216, 136
  • [2] Hornsey et al., 2014, A&A, 567A, 24
  • [3] Nakariakov et al. 2004, MNRAS, 349, 705
  • [4] Roberts et al. 1983, Nature, 305, 688
  • [5] Meszarosova et al., 2011, SoPh, 273, 393

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117. A numerical tool for obtaining wave eigenvalues in non-uniform solar waveguides

Author: Samuel Skirvin, Viktor Fedun & Gary Verth from the University of Sheffield.

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The modern ground- and space-based instruments (DST, SST, DKIST, SDO, Hinode, Solar Orbiter) provide solar physicists with ample observations of solar plasma processes, i.e. magnetic bright points, spicules, plasma flows, structure of magnetic fields etc. at different temporal and spatial scales. However, direct measurements of important plasma properties such as e.g. magnetic field strength in the corona, using traditional observational techniques is incredibly difficult. Fortunately, magnetohydrodynamic (MHD) waves which permeate almost all structures observed in the solar atmosphere can be used as a proxy to determine the properties of the plasma, through a tool known as solar magnetoseismology. Therefore, advanced theoretical modelling becomes essential to explain the ever increasing quality of observational results and provide more accurate information about MHD wave propagation and solar atmospheric plasma properties.

MHD waves in a spatially non-uniform plasma

When it comes to an analytical description of wave properties in a solar plasma, the traditional technique of solving the linearised MHD equations for small perturbations is usually adopted. This method ultimately obtains a dispersion relation which relates the frequency of the wave and its wavenumber along with known characteristic properties of the background plasma. Pioneering work [1] first applied this in a solar context for a uniform magnetic slab model and provided an analytical description of magnetoacoustic kink and sausage modes split into two physical categories namely surface and body waves. In a non uniform magnetic waveguide however (see Figure 1) the governing differential equations develop coefficients which are spatially varying, along the coordinate of inhomogeneity. As a result, the governing equations now have no known closed form analytical solution and consequently no dispersion relation can be obtained, therefore, a numerical approach must be adopted.

Numerical approach

We present a numerical approach based on the shooting method and bisection method to obtain the eigenvalues for a magnetic slab with an arbitrary non-uniform background plasma and/or plasma flow [2]. Real frequencies are obtained such that information about trapped modes of the system can be analysed, complex frequencies such as those in the leaky or continuum regimes are left for future work. The initial wave phase space is used as the domain to find eigenvalues that provide exact solutions to satisfy the relevant boundary conditions of the waveguide, namely the continuity of perturbation of radial displacement and total pressure. Additional information from the definitions of the sausage mode and the kink mode are utilised to obtain the relevant (anti-)symmetric eigenfunctions. The governing equations describing these properties are derived and solved numerically as no analytical solution exists without making simplifying assumptions about the model. The values of wave frequency and wavenumber that satisfy both boundary conditions simultaneously will be classified as a solution and used in further analyses of the wave modes.

This is an extremely powerful numerical tool as, provided the initial equilibrium is stable, a wave analysis of any non-uniform or non-linear plasma can be investigated without the need for a dispersion relation. It should also be noted that this numerical approach is not limited to a purely planar geometry, a cylindrical or spherical geometry would only modify the mathematical vector operators used in the initial analytical description – the physics of the numerical tool still remains the same.

Non-uniform plasma density in a coronal slab

We investigate the properties of magnetoacoustic waves in a coronal slab with a non-uniform background plasma flow modelled with the profiles in Figure 2. A sinc(x) profile models the spatial distribution seen in intensity images of magnetic bright points [3]. The width of the Gaussian profiles is determined by a parameter W, where a smaller W indicates a more inhomogeneous profile. The numerical algorithm obtains the eigenvalues plotted on the dispersion diagram (Figure 3) for each case which allows the resulting eigenfunctions for total pressure and horizontal perturbation of velocity to be calculated.

In Figure 4 the eigenfunctions for the different cases of non-uniform equilibria are shown for the slow body sausage and kink modes. It can be seen by comparing the uniform and extreme non-uniform cases that additional nodes and points of inflexion are present for the non-uniform equilibrium case which may cause difficulty when interpreting the wave modes of observed/simulated perturbed velocity fields in non-uniform waveguides.


  • A numerical approach has been developed to obtain the eigenvalues for magnetoacoustic waves in an arbitrary symmetrically non-uniform magnetic slab.
  • The algorithm is heavily tested against numerous well known analytical results and successfully obtains the correct eigenvalues for both the sausage and kink modes in the long and short wavelength limits.
  • Investigations of non-uniform plasma and background flow modelled as a series of Gaussian profiles reveal that slow body modes are more affected by the non-uniform equilibria. For investigation, analysis and discussion into the non-uniform background flow case, refer to [2].
  • Additional nodes and points of inflexion appear in the resulting eigenfunctions which may be of interest to observers when interpreting observational results of MHD waves in the highly structured non-uniform solar atmosphere.
  • Future work investigating similar cases to those considered in this work but in the case of a cylindrical model can be found in [4].


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116. Pushing GREGOR to the limit: observing weak, small-scale magnetism in the photosphere with the GRIS-IFU

Author: Ryan J. Campbell, Mihalis Mathioudakis, Peter H. Keys, Chris J. Nelson, Aaron Reid (Queen’s University Belfast), Manuel Collados, Andrés Asensio Ramos (Instituto de Astrofísica de Canarias) and David Kuridze (Aberystwyth University).

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Small-scale magnetism in the photophere

In the quiet solar photosphere, we observe granulation as the dominant pattern. Granulation is generated by convective cells rising from the convection zone and characterized by expansive granules and condensed intergranular lanes (IGLs). Magnetism in this layer of the atmosphere is continuously replenished, with flux balanced by the processes of emergence, fragmentation, coalescence and cancellation [1]. In the network, the magnetic field strength, B, is typically large (B > 1000 G), and the vector highly vertical, with respect to the solar normal, but in the internetwork (IN) it has recently been revealed that the field is much weaker (B < 1000 G) and more inclined [2,3]. Observing the temporal evolution of these dynamic structures has previously been a major challenge, with most studies relying on slit-spectropolarimeters that scan in an X-Y plane to build up an image and which cannot, except in an 1D sit-and-stare configuration, build up a high-cadence time-series.

The diagnostic potential of spectropolarimetry

To observe these weak IN fields, we take advantage of the Zeeman effect, describing how spectral lines become split into several components under the action of a magnetic field. Magnetic fields also cause the white, unpolarized light that would otherwise be emitted from the Sun to become polarized. Measurement of Stokes I gives us the intensity, while Stokes Q and U are two independent states of linear polarization and Stokes V concerns circularly polarized light. At disk-centre, the action of inclined fields will result in linearly polarized light (and thus Stokes Q or U), while vertical fields generate circularly polarized light (and thus Stokes V).

The weak IN magnetic field is hidden to the Zeeman effect at low spatial resolutions, due to cancelling of opposite polarity Stokes signals, and therefore requires very high-resolution observations. As the polarization signals produced by weak fields have such low amplitudes, the Zeeman sensitivity of the employed spectral lines and signal-to-noise (S/N) of the observations is also critical. Information about the strength of the magnetic vector is contained both in the amplitude of the polarization profiles and in the splitting of each of their red and blue lobes, while to constrain the inclination of the vector, γ, we need to measure both linear and circular polarization. To constrain the azimuthal angle, Φ, we need to measure signals in both Stokes Q and U. The large effective Landé g-factor (higher value = more magnetically sensitive) and near infrared wavelength of the Fe I line at 1564.9 nm makes it an effective Zeeman diagnostic. In particular, studies (e.g. [4]) have shown the unique ability of this line to record linear polarization signals generated by horizontal fields, which are typically weaker than the circular polarization signals.

Observing in the near infrared

The new GRIS-IFU (GREGOR Infrared Spectrograph Integral Field Unit) image-slicer mounted at the 1.5 m GREGOR telescope provides the ideal instrument for this purpose. The GRIS-IFU has a very small (3’’ by 6’’) field of view (FOV) but can build up a larger image by ‘mosaicing’. When designing an observing sequence, one must optimize a number of competing factors. If the exposure time is too high, the target can evolve while collecting photons, resulting in further Zeeman cancelling and thus lower measured amplitude of polarization signals. The choice of FOV in the IN is critical, as it is possible with a small FOV to observe relatively low levels of polarization; previous studies with very large FOVs have shown there are enormous regions of the IN apparently devoid of magnetic flux without sufficient S/N [2].

We present in Figure 1 observations representing the highest spatial resolution near infrared time-series available to date. We chose a 3 by 3 mosaic resulting in a FOV of 9’’ by 18’’. The datasets additionally have a high spectral dispersion (40 mÅ/pixel) and a 64 second cadence. Figure 1 shows the observables from 40 frames from the two datasets recorded. Readers are directed to [5] (equations 2, 3) for definitions of the wavelength-integrated linear (LP) and circular (CP) polarizations.

An inverse problem

By employing the Stokes Inversions based on Response functions code (SIR, see [6]), we are able to infer the local thermodynamic, kinematic and magnetic properties of the atmosphere. We consider two inversion setups: scheme 1 (S1), where a magnetic atmosphere (model 2) is embedded in a field free medium (model 1), and scheme 2 (S2), with two magnetic models and a fixed 30% stray light component. Two-component models must be employed as we are not typically resolving the observed small-scale magnetic structures. We therefore quantify the fraction of the pixel element occupied by a given model using its filling factor, α. Also shown in Figure 1 are the line-of-sight velocities, magnetic flux densities and inclination angles returned by S1 inversions.

We find patches of linear polarization with peak magnetic flux densities of the order of 130−150 G and find that linear polarization appears preferentially at granule-IGL boundaries. It is clear that the evolution and fate of these features are highly dependent on granular motions. The weak magnetic field appears to be organized in terms of complex ‘loop-like’ structures, with transverse fields (γ ~ 90 deg, i.e. in the plane of the solar surface) often flanked by opposite polarity longitudinal fields (γ ~ 0,180 deg, i.e. pointing towards or away from the observer, respectively). How many of these loops can you spot in the video?

We reconstructed our observed profiles by, first, the application of principle component analysis (PCA) to remove noise, in the same manner as implemented by [7], and second, the application of a relevance vector machine (RVM) to remove fringes and other defects (see details in [5]). Another reason to reconstruct our profiles in this way is to reduce the influence of noise on our inversion results.… continue to the full article

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115. Revealing the dynamic & magnetic nature of chromospheric vortices

Author: Juie Shetye* and Erwin Verwichte at the University of Warwick
(*now at New Mexico State University).

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Rotational motion is prevalent in nature, from maelstroms in rivers, airplane turbulence, to weather tornadoes and cyclones. In the universe, we find rotation in the vortices of Jupiter’s atmosphere, in accretion disks of stars and in spiral galaxies. The constant motions of the Sun’s surface creates giant tornadoes in the solar chromosphere. The tornadoes are a few thousand kilometers in diameter, and like their namesakes on Earth they carry mass and energy high up into the atmosphere. They are therefore keenly studied as energy channels to explain the extraordinary heating of the solar corona. The main building block of solar tornadoes are tangled magnetic fields. However, it is notoriously difficult to measure the magnetic field in the Sun’s chromosphere. We present our recent efforts in observing the dynamics and magnetic field of chromosphere vortices, also known as swirls [1].

The solar chromosphere is a dynamic and inhomogeneous layer where the plasma-β varies from larger to much smaller than unity within only a few megameters. Such inhomogeneity results in a combination of physical processes threaded by magnetic fields that give rise to a plethora of structures in the chromosphere. Rotating, braiding and twisting of the magnetic field anchored in the photosphere’s granular velocity field gives rise to swirling structures. The first of these structures were reported as early as 1908 [2]. Since their discovery, their potential for efficiently transferring mass and energy across the solar atmosphere has been recognized [3,4]. They may provide continuous coronal heating, especially if they appear in sufficiently large numbers. Numerical simulations show that, individually, they have an estimated net positive Poynting flux of 440 Wm-2, more than adequate to heat the quiet Sun corona. However, this is observationally unverified [5].

We present our recent efforts to observe chromospheric swirls. We use ground-based observations from the CRISP instrument on the Swedish Solar Telescope [6], and from the IBIS instrument on the Dunn Solar Telescope [7]. We focus on observation channels centred on the chromospheric spectral lines Ca II and Hα. In those channels, swirls appear as temporary bright circular or spiral structures (see Fig. 1).

Multi-channel observations of swirls

The Ca II and Hα spectral lines are formed at different temperatures and densities in the solar atmosphere. Thus comparing the appearance, morphology, dynamics and associated plasma parameters between both channels sheds light on the swirl’s vertical structure (see 5 out of 13 examples shown in Fig 1). Traditionally, such swirls have been observed in Ca II only. We have chosen to classify chromospheric swirls according to whether they appear above a single or multiple magnetic concentrations (MCs) located in the intergranular lanes in the photosphere. MCs are visible in Fe I Stokes-V at least five minutes before the appearance of a swirl and undergo morphological and dynamic changes in precursor and during the appearance of a swirl. In some cases the morphology of MCs transforms from stretched and fractured to circular and compact. Several minutes after this transition, a chromospheric swirl appears with a central brightening in Ca II and Hα, followed by a bright circular ring at these wavelengths. The typical time delay and the formation height of Ca II of 1500 km or less, suggests a travel speed of around 10 km s-1. This is comparable with the typical value of the Alfvén speed in the chromosphere (Fig 2 shows a simplified cartoon for the scenario).

Signatures of acoustic oscillations in swirls

We wish to elucidate the role swirls play in generating and modifying chromospheric acoustic oscillations [5]. Superimposed on the swirl pattern are periodic variations in intensity with a typical period of 180 s, consistent with three-minute chromospheric acoustic oscillations. We have discovered that during the occurrence of the swirl the acoustic oscillation is temporarily altered, i.e. reduced in period to 150 s and with an increased or decreased local intensity. A change of period may be due to a change in the acoustic cavity dimensions, additional magnetic effects, or due to an increase in temperature. If we assume that this change is solely due to temperature, based on the period change, we estimate an upper limit of the temperature increase of 44%.

Phase analysis between the signals in the wings and in the core of the Ca II line show that the blue wing precedes the line core, which in turn precedes the red wing. This pattern is always present and is not altered by the appearance of the swirl. The absence of half-period oscillations in the line-core compared with the wings suggests that the velocity time series is not sinusoidal, but rather asymmetric. This seems consistent with the physical picture of three-minute acoustic waves forming shocks below the formation height of Ca II around 1500 km. The phase relations are more complex in Hα, which is perturbed during the appearance of some of the swirls. Forward-modelling of line-formation in simulations of swirls in the presence of acoustic waves would be needed to establish the exact physical picture.

Chromospheric magnetic signature of vortices

In collaboration with colleagues from Italy, we have revealed for the first time the magnetic nature of a chromospheric swirl using spectropolarimetry in the Ca II line using observation with IBIS at the DST. Those observations show two long-lived vortices (swirls) that each rotate clockwise inside a 10 arcsec2 quiet-Sun region. For both vortices we have extracted the circular and linear polarisation signals. The circular polarization signals are 5-10 times above the noise level. These signals provide information about the nature of the magnetic field in vortices in the chromosphere. This marks the first time such a measurement has been achieved for vortices. Importantly, the vortices have oppositely signed circular polarisation, which indicates that they have opposite magnetic flux. A combination of maps of the Doppler velocity, plane-of-sky optical flow field, line core intensity point and MHD modelling all point to the physical picture where the two vortices form a magnetic dipole.… continue to the full article

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