Category: UKSP Nugget

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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114. Hidden Coronal Loop Strands within Hi-C 2.1 Data

Author: Thomas Williams and Robert W. Walsh at the University of Central Lancashire.

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Observational investigations of coronal loop structure have been undertaken since the 1940s [1]; however, due to insufficient spatial resolution of current and previous instrumentation, the definitive widths of these fundamental structures have not been fully resolved. Recent high-resolution data from NASA’s Interface Region Imaging Spectrometer (IRIS; [2]) and the High-resolution Coronal imager (Hi-C; [3]) have led to coronal loop width studies in unprecedented detail.

Hi-C Observations

Recent work with 17.2 nm observations investigates loops from five regions within the field of view of the latest Hi-C (2.1) flight [4,5]. As with [6], coronal strand widths of ~513 km were determined for four of the five regions analysed. In the final region, which exhibits low emission, low density loops, much narrower coronal strands are found of ~388 km width, placing those structures below the width of a single AIA pixel. The fact that these strands are above the resolution limit of Hi-C (220-340 km [5]) suggests that Hi-C may be beginning to resolve a key spatial scale of coronal loops.

Notably, [4] also find example structures that may not be fully resolved within the Hi-C data. These relate to smaller ‘bumps’ or turning points in the cross-section intensity profiles that are larger than the observational error bars but do not constitute a completely isolated strand. Could these be the result of projection effects of overlapping structures along the integrated line of sight for this optically thin plasma, or are they the result of further structures beneath even the resolving abilities of Hi-C?


To answer this, a selection of slices is taken from the Hi-C 2.1 field-of-view (Figure 1) where these non-Gaussian shaped structures are seen. Following the method of [4], each slice is time-averaged for ~60 s and summed across a width of three-pixels to increase the signal-to-noise ratio. Cubic spline interpolation is then employed to generate and subtract the background intensity of each slice. To estimate the number of strands hidden within the cross-sectional profiles, a non-linear least-squares curve fitting method is employed to fit a number of Gaussians to the observed intensity profile. The correct number of Gaussian profiles fitted to each cross-sectional slice is determined by using the Akaike Information Criteria (AIC; [8]) model selection method. AIC evaluates how well a fit is supported by the data by rewarding a fit for the accuracy relative to the original Hi-C data, but punishes each fit as the complexity increases i.e. as the number of Gaussians fitted increases. The use of AIC helps mitigate the potential for over(under)-fitting the Hi-C data as the number of Gaussians fitted is taken as the model best supported by the data.


A total of 183 Gaussian profiles are fitted to twenty-four Hi-C cross sectional slices, the positions of which are shown in Figure 1. A closer view of four sample slices shown in Figure 2 and their cross-sections are displayed in Figure 3. In the cross-section plots, the original intensity (blue) is compared to the best AIC-determined fit (red) and the Gaussian profiles (grey) that generate said fit. Overall, the observed intensity is well reproduced though there are minor discrepancies that can be observed (e.g. Slice 2: positions 5’’ and 13’’). These could be eradicated by fitting more Gaussian profiles, however AIC determines that additional Gaussians are not supported by the Hi-C data.

The full-width at half-maximum (FWHM) of the 183 Gaussian profiles are collated into occurrence frequency plots (Figure 4) with the same spatial binning as [4] along with their 1-σ errors returned from the curve fitting method. As with [4], the most frequent spatial width of the measured strands is ~500 km, whilst the majority lie between 200 – 800 km, yielding a similar median (645 km) to previous 19.3 nm Hi-C width measurements [9].

Furthermore, ~21% of widths exceed 1000 km whilst ~32% of the strands studied are at the SDO/AIA resolving scale of 600-1000 km. From this, ~47% of the strands are beneath the resolving scale of SDO/AIA. The  Hi-C strand widths obtained reveal the presence of numerous strands (~32% of the 183 Gaussian widths) whose FWHMs are beneath the most frequent strand widths seen previously [~513 km; 4] Similarly, ~17% are below an AIA pixel width of 435 km. Comparatively then, only ~6% of the strands are actually at the smallest scale at which Hi-C can resolve structures [220-340 km, 5].


This work outlines a follow-up analysis to [4], in which non-Gaussian shaped loop profiles not fully resolved within the Hi-C data were found. Employing a nonlinear least-squares curve-fitting method, a total of 183 Gaussian profiles are fitted to these partially resolved Hi-C structures. The fact that (i) the FWHM of these Gaussians are  at the same spatial scales as previous high-resolution findings [4,6,9] and (ii) ~94% of strand widths measured are above the Hi-C resolving scale (220-340 km) provides strong evidence that structures with non-Gaussian distributions are likely the result of overlapping structures along the integrated line of sight rather than the result of an amalgamation of strands beneath even the resolving capabilities of instruments like Hi-C.

This work has been published in The Astrophysical Journal and a full-text version can be found here.


  • [1] Bray, R. J., Cram, L. E., Durrant, C., Loughhead, R. E. 1991, Plasma Loops in the Solar Corona (Cambridge: Cambridge University Press)
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  • [8] Akaike, H. 1974, ITAC, 19, 716
  • [9] Brooks, D. H., Warren, H. P., Ugarte-Urra, I.,

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113. Probing small-scale solar magnetic fields

Author: Mykola Gordovskyy and Philippa Browning (University of Manchester), Sergiy Shelyag (Deakin University), Vsevolod Lozitsky (Kyiv Taras Shevchenko University).

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What is known about small scale structure of photospheric magnetic fields?

The magnetic field in the solar photosphere is very inhomogeneous, and has fine structure with spatial scales of about 10km (see [1,2] and references therein for a review). Therefore, magnetograms produced by instruments, such as SDO/HMI or Hinode/SOT, show not the actual magnetic field, but the field averaged over a volume of around 100x100km (spatial resolution of an instrument) x500km (thickness of the photosphere) (Figure 1). To complicate things further, the observed magnetic field is weighted by a function of depth, or the contribution function. Since different spectral lines used for magnetic field measurements have different magnetic field sensitivities and different contribution functions, the maps produced by different magnetographs can differ substantially, particularly in active regions, where the magnetic field is expected to be complex.

To a first approximation, the small-scale magnetic field structure can be described using the so-called two-component model, where photospheric magnetic flux is carried by thin fluxtubes with field strength Breal and the filling factor α (α represents the fraction of volume penetrated by Breal). Hence, the observed magnetic field is

Bobs = α Breal.

Although the two-component approximation sounds like an oversimplification, in fact, it is not very far from the reality. Simulations show that the magneto-convective collapse results in two populations of magnetic field [3]: strong magnetic elements with kG strength (usually concentrated at the photospheric network boundaries) and weak ambient field (Figure 2).

Why is this important?

The difference between the observed and real distributions of magnetic field in the photosphere can affect all sorts of measurements. For instance, in terms of the two-component model, ignoring the filling factor α while estimating the magnetic energy density or the Poynting flux in the photosphere would result in them being underestimated by factor of 1/α. Similarly, the current density would be underestimated by factor of about α-1.5.

Can we do anything about it?

There are a few different ways to evaluate the “real” magnetic field (or the filling factor). Firstly, the classical Magnetic Line Ratio (MLR) method based on the comparison of the Zeeman Effect in spectral lines with different magnetic field sensitivities (see [4] for an in-depth review).

Recently, Gordovskyy et al. [5,6] have developed an alternative method for diagnostics of unresolved field: the Stokes V Width (SVW) method. It links the filling factor with the width of the Stokes V component in some classical magnetometric spectral lines, such as Fe I 5247 Å and 6301 Å. An important advantage of this new method is that, unlike MLR, it requires only one spectral line. The SVW method has been tested using the magnetoconvection models of the photosphere and appears to be, generally, as reliable as the classical MLR method (Figure 3). Comparison of these methods applied to different spectral lines show that MLR is usually more reliable for lower values of Bobs (typically, up to about 500G), while SVW is more reliable for higher Bobs values.

Can we do better?

The methods discussed above cannot properly account for the temperature and velocity variations, which affect spectral line profiles and, hence, the reliability of both the MLR and SVW methods. Stokes inversion yields much better quality [e.g. 7]. The idea of the Stokes inversion approach is to find the line-of-sight distribution of thermodynamic parameters, LOS velocity, magnetic field components, and filling factors (for B and VLOS) providing the best fit to the observed Stokes components of selected spectral lines. However, this approach is computationally expensive, and usually applied only to relatively small patches of the solar surface (order of 100×100 arcsec or so). Therefore, the Stokes inversion can be used for more accurate analysis of smaller areas, while MLR and SVW can be used for fast “on-the-fly” analysis of large areas (a big active region or the whole solar disk) or for calibration of large-area magnetograms.

Can we do even better?

The methods discussed above can only evaluate Bobs or α. To find the sizes and shapes of small-scale magnetic elements, we need direct high-resolution observations.

During the last decade the spatial resolution of solar optical observations have improved greatly, currently reaching ~0.1arcsec (70km) [e.g. 8]. The Daniel K. Inouye Solar Telescope (DKIST), which is entering operation this year, and the planned European Solar Telescope (EST) will push this boundary even further: DKIST spatial resolution will be 35km, while EST is expected to resolve scales as small as 20-25km. These two instruments are likely to be game-changers, finally revealing the fine structure of solar magnetic fields.


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112. Particle acceleration and transport in CME eruptions

Author: Qian Xia and Valentina Zharkova at Northumbria University, Newcastle.

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Coronal mass ejections (CMEs) are explosive solar events that involve enormous ejections of plasma and magnetic flux, which drive interplanetary dynamics. CMEs are often associated with a filament channel, in the form of a twisted flux rope or sheared arcade, that stores the required large amount of free magnetic energy. The structures can be destabilised by reconnection or by an ideal process (e.g., torus or kink instability) leading to consequent eruption.

In the magnetic breakout model [1], the energy buildup in the filament channel deforms a coronal nullpoint above the system. It forms the breakout current sheet (CS), as shown in Fig. (1). This CS eventually reconnects, removing the flux overlaying the filament channel, disrupting the force balance, and triggering the eruption onset. A generic vertical flare CS forms beneath the erupting filament and reconnects and drives explosive CME acceleration.

Observations show that a significant fraction of the total released magnetic energy is transferred to high-energy electrons and ions. Energetic electrons in flares can be observed through bremsstrahlung hard X-ray, as illustrated in Fig. (2), and gyrosynchrotron microwave emission from the solar corona and chromosphere. At the same time, a fraction of energetic particles escapes to interplanetary space as solar energetic particles that can be detected by in situ observations. The strong particle energisation during solar flares may be driven by various mechanisms associated with a magnetic reconnection process [3]. By studying particle energisation in different breakout and flare CSs occurring during a CME’s evolution, we can understand the energy release by magnetic field restructuring in these events and the energy transfer to energetic particles, and, thus, determine the properties of the high-energy particles produced.

Numerical approaches

Current computing power is unable to resolve the particle characteristic scale (e.g., proton gyroradius ~ 102 m) in an observable hydrodynamic domain (flare current sheets ~ 106m) [3]. Due to the kinetic effects of particle acceleration, many kinetic simulation codes have been developed, such as hybrid and particle-in-cell approaches. The simulation domains are simplified and restricted to the most interesting area, such as the magnetic reconnection sites or the shock fronts, targeting a single specific process. On the other hand, the test-particle approach implements passively moving particles into magnetohydrodynamic (MHD) simulations (their motion would not change the electromagnetic fields). It allows scientists to access the larger hydrodynamic scale. In this nugget, we outline recent progress with this method, which explores multiple particle acceleration regions simultaneously in a single CME eruption model.

Results and discussion

The ideal MHD code, ARMS, is adaptively refined. The non-uniform grid size becomes smaller near the discontinuities (such as current sheets, shocks) due to the steeper gradients (Fig. 3). This ultra-high-resolution code can produce fine structures, such as magnetic islands in the breakout and flare current sheets.

A large number of test particles are initialised randomly in the green regions of Fig. 4 [5]. The particle acceleration sites identified by the most accelerated particles include the single X-nullpoint, the magnetic islands, and the flare loop-top regions, which are consistent with previous localised kinetic studies. Furthermore, particle re-acceleration in different regions (the blue line in Fig. 4 (3) and (6)) is shown for the first time.

When we look at the particle distributions, we first notice that the protons and electrons are ejected from the X-nullpoint asymmetrically, consistent with the previous kinetic (particle-in-cell) results [6]. After the particles are accelerated, the flare current sheets are more efficient at accelerating particles than the breakout current sheets. What is the reason behind them? Can they contribute to different acceleration mechanisms?

To answer these questions, we adopt the particle drift equations and the fluid description of magnetic field energy changes. These analyses ignore the single-particle motions and instead, focus on the macro scale. For example, the betatron acceleration is related to the change of magnetic field strength, and the first-order Fermi acceleration is related to the shortening of magnetic field lines. The results indicate that different mechanisms dominate different acceleration sites (e.g., 1st-order Fermi acceleration is important in the magnetic islands, the compression of the magnetic field does the trick in the flare loop top, etc.). The amplitudes of the energisation terms explain the different efficiency of acceleration sources. If we look into the change of particle energy distributions, we find that the peak of the distribution starts higher than the loop top and then moves downwards to the flare loop. The transition is consistent with the hard X-ray emissions in the impulsive phase of an X8.2 (a giant explosion) flare event matching the standard CME eruption model. On the other hand, the decrease of particle acceleration efficiency in the decaying phase of the flare is accompanied by the fading of the magnetic guiding field after the impulsive phase.

The numerical studies presented have calculated particle acceleration for a “realistic” CME eruption system rather than a preassumed current sheet. The combination of MHD and test-particle simulation assists the prediction of maximum energy gains of particles in different magnetic configurations, and different phases of the flares. Our model could distinguish the different particle energization mechanisms operating on these macro-scales.


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  • [6] Siversky, T. V., & Zharkova, V. V. 2009, J. Plasma Phys., 75, 619

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111. Increasing occurrence of inverted magnetic fields from 0.3 to 1 au

Author: Allan Macneil, Mathew Owens, Mike Lockwood, Matthew Lang, Sarah Bentley (University of Reading) and Robert Wicks (University of Northumbria)

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Local inversions in the heliospheric magnetic field (HMF) are observed at a range of solar distances and latitudes by in situ solar wind spacecraft. Figure 1 shows a schematic example of inverted and uninverted magnetic field.

Recent observations of numerous rapid, Alfvénic inversions (known as ‘switchbacks’) by the new Parker Solar Probe (PSP) mission at distances down to 0.16 au have led to renewed interest in this phenomenon [2, 3]. Finding the origins of these inversions, and specifically whether they are formed at the Sun or through in-transit processes, is of particular interest. Knowledge of some solar origin for these inversions could help to reveal processes occurring in the corona which contribute to the production of the solar wind, such as jets and interchange reconnection [4, 5]. In this nugget, we describe results which put constraints on the origins of HMF inversions by quantifying the change in inversion occurrence as a function of distance r, as measured by the Helios 1 spacecraft.

Helios Observations of Inverted Field

Helios 1 observed the low-latitude solar wind and HMF over distances of 0.3 to 1 au, over several orbits from 1974 to 1981. For this data set, we classify 40 s cadence magnetic field samples as corresponding to either inverted or uninverted HMF. This is done by combining the magnetic field polarity (defined relative to the nominal Parker spiral direction) with the beam direction of the suprathermal electron strahl (which traces an anti-sunward path along the field). Once all valid data has been classified, we bin the samples into bins of r for analysis.

Evolution of Relative Inverted Field Occurrence

For our binned data, we calculate the fractions of all valid samples which correspond to inverted and uninverted HMF. These results are plotted against r in Figure 2.

The occurrence of inverted HMF increases between 0.3 and 1 au, at the expense of uninverted HMF. The relative number of inverted HMF samples increases by a factor of around 4. This result implies that inverted HMF over this distance range is primarily created through some driving process in the heliosphere. If most inversions formed at the Sun, then we would expect inverted HMF occurrence to instead drop-off with r, as the inversions gradually decay.

Direction of Magnetic Field Deflection

Field and plasma properties associated with inversions can provide evidence as to what mechanisms are driving the creation of inverted HMF. We calculate the azimuthal ‘deflection angle’, ΔϕP, of each sampled magnetic field vector away from the nominal Parker spiral direction. The strahl beam direction is used again to remove magnetic sector dependence, such that |ΔϕP| is 0° when the field is unperturbed, and |ΔϕP| > 90° indicates that the field has been deflected to the point of inversion.

Figure 3 shows that the distribution of ΔϕP gradually broadens with r. This supports the above interpretation that the increase in inverted HMF is driven by the gradual deflection of the field away from the nominal Parker spiral direction, as more samples exceed |ΔϕP| = 90°.

Generation of Inversions

In Figure 4 we show some simple schematics of possible processes, adapted from suggestions in [6], which could generate inverted magnetic fields in the heliosphere.

Panels a to c show that the action of convecting plasma in the solar wind can drive inversions into the field. The angle between the background Parker spiral field and the radial propagation of these elements means that inversions can only be generated through a deflection in the positive ΔϕP direction. Meanwhile, inversions created by waves and turbulence can result from deflection of the field in either direction. Comparison of the wings of the distributions in Figure 3 at the positive and negative extremes reveals that there is no strong bias towards inversion through either clockwise or anti-clockwise deflection. Thus, of the presented processes, only waves and turbulence are consistent with our observations. We note that the schematics here do not account for more complex possible effects, such as the interaction between stream shears and turbulence [7] or the expansion of inverted structures [8].


We have shown that the occurrence of inverted HMF gradually increases over the distance range 0.3 to 1 au. This indicates that most of these inversions are being actively driven into the HMF, instead of being a remnant of some process at the Sun. Analysis of the azimuthal deflection angle of inverted HMF suggests that waves and turbulence may be the dominant process in creating these inversions. While these results demonstrate that in situ driving of inversions takes place, they do not rule out that inversions may also be generated by processes at the Sun. This is particularly true for the frequent near-Sun switchbacks observed by PSP. These results raise an interesting question: as the switchbacks which dominate the PSP encounters become common on approaching the Sun, at what distance does the occurrence of inverted HMF start to (presumably) increase?


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110. Flare/CME Cartoons

Author: Hugh Hudson, Nicolina Chrysaphi, and Norman Gray at the University of Glasgow.

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A “Grand Archive of Solar Flare Cartoons” has long existed on the Web [1], but without updates within the past decade because of the unfortunate loss of a password, and because the original quite primitive HTML made it hard to access new items. Now, announced here for the first time, a brand-new Archive [2] at last replaces it! The new version contains almost 400 entries (see Figure 1 for some examples), each with a new or good-as-new refurbished descriptive blurb, usually containing links that let the user hop around seeking that exactly perfect concept (which almost never exists, alas). The blurb links to the original source.

Note that some mission creep has occurred: originally inspired only by the venerable and too-often-cited CSHKP model, the Archive has gone far beyond that in an effort to capture lateral thinking and more physically relevant items. Note please that we capitalize Archive in an effort to clothe our pretty lightweight subject with some gravitas. We would not describe these toons as funny ha-ha, at least not by intent.

Often the inspiration for an important bit of science appears first as a sketch on a bit of crumpled paper, perhaps in a bar somewhere like the Eagle Pub in Cambridge (UK). We think that a good cartoon represents a sort of intuitive interpolation formula, in that it captures some crucial new aspect of the science and allows extrapolations of that idea into some useful further direction or other. Note that many of the cartoons in the Archive do not actually do that very well. These often suggest the possibility (probability?) of obsolescence as a matter of course. In fact, one could argue that perpetuating even the most brilliant cartoon may actually serve to stifle innovation and lead to a stale cartoon-chasing style of research.

Of what use is such an Archive?

Does the Archive really serve any useful purpose, or does it merely ossify outdated concepts of little generality? Both, we think. We offer this Archive mainly as an educational matter for the benefit of the Archivist really, but many eager users of the old Archive [1] have (if faintly) praised it. The typical comment notes last-minute deadline pressure for writing a presentation or a proposal. The Archivist has in fact sometimes sat glumly through seminar presentations that seemed to consist mainly of cartoons, and has no statistical basis for judging the success rate for any of the proposal efforts.

Access to the Archive

A recent example (shown in Figure 2) shows how a cartoon can neatly suggest a specific physical mechanism within a global structure. This one also pops up in the segment of the thumbnails view of the Archive shown in Figure 1, clickable in its direct form though not here. In addition to this thumbnails view, the Archive also offers various list options; the chronological list view starts in 1905. The Archive embraces half a dozen varieties of cartoon, but tries to avoid snapshots of numerical simulations wherever possible. Though, of course, a good simulation really just fleshes out somebody’s idea of the important physics.


The Archive continues to grow gradually as further brilliant ideas appear (or sometimes, just as the graphics get better). A successful new entry must satisfy at least one basic requirement: it needs to have appeared in a regular journal with a better-than-average impact factor. If you have a really new and interesting cartoon in such a state that does not currently appear on the Archive, please email the Archive Accessions Department directly. Note again that the Archive does not presently include intentionally funny items.


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109. Kink oscillations of sigmoid coronal loops

Author: Norbert Magyar and Valery M. Nakariakov at the University of Warwick.

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In solar physics, coronal loops have long been in the spotlight. As building blocks of the closed solar corona, understanding their structure and evolution is akin to understanding the coronal heating problem. Moreover, oscillations detected in loops can serve as natural diagnostic probes of their physical properties through coronal seismology, a field in which the properties of coronal plasmas are inferred from observed wave properties and wave theory. Since the first observation of kink oscillations of coronal loops and the first application of coronal seismology [2], models have continuously improved to account for effects such as loop curvature, density stratification, loop cross-sectional variations, cooling, elliptic cross-sections, and so on. A frequently observed property of loops is non-planarity, i.e. exhibiting a helical or sigmoid shape [3]. The effects of coronal loop helicity on their standing kink oscillations were previously investigated analytically, but only through their effect on the local stratified equilibrium density [4]. Here, for the first time, we simulate kink oscillations of sigmoid coronal loops and investigate the ability of coronal seismology to assess the sigmoidity.

Numerical model

Our 3D numerical model consists of a background coronal plasma in a hydrostatic equilibrium in which we embed a coronal loop of higher density. The magnetic field is a force-free dipole with constant ⍺ parameter, adapted from [5]. The ⍺ parameter controls the helicity of the field lines (according to ∇ X B = ⍺B ). We add a higher density loop by tracing a single magnetic field line, and then using it as a central axis to construct a tube. The origin of this single field line, which varies depending on ⍺, is chosen in order to maximise the sigmoidity of the resulting loop while keeping it in the simulation domain. See Figure 1 for an example of a sigmoid flux tube.

The pulse is a horizontally polarised velocity perturbation varying sinusoidally along the loop, which aims to excite a fundamental standing kink mode. However, note that this initial perturbation probably does not coincide with the eigenfunction of the fundamental kink, which is not known. Therefore, while preferentially exciting the fundamental kink mode, other modes are also excited to a small degree, including leaky waves. The resulting kink oscillation of the loop is shown in Figure 2.

The ideal MHD equations are solved in a 3D rectangular domain using MPI-AMRVAC 2.0 [6], with a finite-volume approach. We applied a splitting strategy for the magnetic field, with the time-independent force-free magnetic field considered as a background field. Thus we only solve for the (nonlinear) perturbed magnetic field components. For this, we used the newly-implemented HLLD solver adapted for magnetic field decomposition described in [7].


Analysis of the oscillation properties is based on synthetic 171 Å intensity and Doppler shift images with the lines of sight corresponding to the coordinate axes. After measuring the oscillation properties, we proceed to infer magnetic field estimates seismologically by calculating the theoretical kink period. We do this using the WKB approximation, as the kink speed varies along the loop. From this the magnetic field intensity is determined using the measured oscillation period, loop length, and estimations of the density.

The results are shown in Figure 3. We have considered a range of an order of magnitude for the precision to which the average internal density can be determined, while the density ratio (internal to external density) is taken to range from 1.5 to 10. In the simulation, the average density ratio is close to 2. Here we assume that the measurements of the length of the loop and of the oscillation period are exact.

For the simulation with no sigmoidity, despite the measured and theoretically calculated periods being close to each other, the seismologically predicted magnetic field value is lower than the average value. This can be understood in the following way: as the displacement amplitude of the fundamental mode has a maximum near the apex, the oscillation period is more sensitive to the weaker magnetic field near the apex rather than near footpoints. With increasing sigmoidity however, the predicted magnetic field shows an increasing trend with respect to the average value. This observation might allow for the seismological determination of the sigmoidity of a coronal loop, if some other method to determine the average magnetic field is available, such as force-free extrapolations. In this sense, the free magnetic energy in a coronal loop could be estimated seismologically.


We propose that the dependence of the magnetic field estimate on the loop sigmoidity could be exploited seismologically in order to measure the non-potentiality, i.e. the free magnetic energy in coronal loops. However, for this method to work, the determination of the average magnetic field along the loop is needed, as well as an accurate measurement of the density along the loop. The external/internal density ratio only weakly impacts the results. On the other hand, we demonstrated the robustness of the seismological method, even when applied to non-planar or sigmoid coronal loops. For all values of sigmoidity considered, the estimation of the magnetic field is within the extremal magnetic field values measured in the loop, despite considering an order of magnitude accuracy for the average density determination.


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