Magnetic Knots Research Articles

Observations of Twist, Current Helicity, and Writhe in the Magnetic Knots of δ-sunspots Consistent with the Kink Instability of a Highly Twisted Flux Rope

We measure current helicity (H r c ) as well as proxies for twist (α r ) and writhe (W) in the isolated magnetic knots of three delta (δ)-sunspots and report that the observations are consistent with a kink instability acting on a highly twisted flux tube. δ-spots are active regions (ARs) in which positive and negative umbrae share a penumbra. We identify and isolate “magnetic knots,” i.e., opposite polarity umbrae that are in close proximity and forming the δ-configuration, in ARs NOAA 11158, 11267, and 11476 as observed with data from the Solar Dynamic Observatory Helioseismic and Magnetic Imager. We find that H r c , α r , and W have the same sign for each magnetic knot, as predicted in simulations of a kink instability acting on highly twisted flux tubes. The deformed flux tube causing the δ-formation, the magnetic knot, is only a portion of the entire AR and demonstrates the potential for the kink instability to act on a smaller spatial scale within the AR. Each magnetic footpoint contains a single sign of the radial current, J r , which suggests that we are observing the core of the flux rope without return currents. As a counterexample, we analyze one β-spot that shows H r c and α r have the opposite signs of W. While our observations support the formation mechanism of the magnetic knots in δ-spots being the kink instability, a much larger sample is needed to determine confidently the prevalence of the kink instability as the cause of flux tube deformation.

Twisted magnetic knots and links

For magnetic knots and links in plasmas, we introduce an internal twist and study their dynamical behavior in numerical simulations. We use a set of helical and non-helical configurations and add an internal twist that cancels the helicity of the helical configurations or makes a non-helical system helical. These fields are then left to relax in a magnetohydrodynamic environment. In line with previous works, we confirm the importance of magnetic helicity in field relaxation. However, an internal twist, as could be observed in coronal magnetic loops, does not just add or subtract helicity, but also introduce an alignment of the magnetic field with the electric current, which is the source term for helicity. This source term is strong enough to lead to a significant change in magnetic helicity, which for some cases leads to a loss of the stabilizing properties expressed in the realizability condition. Even a relatively weak internal twist in these magnetic fields leads to a strong enough source term for magnetic helicity that for various cases even in a low diffusion environment, we observe an inversion in sign of magnetic helicity within timescales much shorter than the diffusion time. We conclude that in solar and stellar fields, an internal twist does not automatically result in a structurally stable configuration and that the alignment of the magnetic field with the electric current must be taken into account.

High-resolution imaging of solar pores

Context. Light bridges are bright, long, and narrow features that are typically connected to the formation or decay processes of sunspots and pores. Aims. The interaction of magnetic fields and plasma flows is investigated in the trailing part of an active region, where pores and magnetic knots evolve into a complex sunspot. The goal is to identify the photospheric and chromospheric processes, which transform the mainly vertical magnetic fields of pores into a sunspot with multiple umbral cores, light bridges, and rudimentary penumbrae. Methods. Conducting observations with a broad variety of telescopes and instruments provides access to different atmospheric layers and the changing morphology of features connected to strong magnetic fields. While the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory (SDO) provides full-disk continuum images and line-of-sight magnetograms, the fine structure and flows around a pore can be deduced from high-resolution observations in various wavelengths as provided by the Goode Solar Telescope (GST) at the Big Bear Solar Observatory (BBSO). Horizontal proper motions are evaluated applying local correlation tracking (LCT) to the available time series, whereas the connectivity of sunspot features can be established using the background-subtracted activity maps (BaSAMs). Results. Photospheric flow maps indicate radial outflows, where the light bridge connects to the surrounding granulation, whereas inflows are present at the border of the pores. In contrast, the chromospheric flow maps show strong radial outflows at superpenumbral scales, even in the absence of a penumbra in the photosphere. The region in between the two polarities is characterized by expanding granules creating strong divergence centers. Variations in BaSAMs follow locations of significant and persistent changes in and around pores. The resulting maps indicate low variations along the light bridge, as well as thin hairlines connecting the light bridge to the pores and strong variations at the border of pores. Various BaSAMs demonstrate the interaction of pores with the surrounding supergranular cell. The Hα line-of-sight velocity maps provide further insights into the flow structure, with twisted motions along some of the radial filaments around the pore with the light bridge. Furthermore, flows along filaments connecting the two polarities of the active region are pronounced in the line-of-sight velocity maps. Conclusions. The present observations reveal that even small-scale changes of plasma motions in and around pores are conducive to transform pores into sunspots. In addition, chromospheric counterparts of penumbral filaments appear much earlier than the penumbral filaments in the photosphere. Penumbra formation is aided by a stable magnetic feature that anchors the advection of magnetic flux and provides a connection to the surrounding supergranular cell, whereas continuously emerging flux and strong light bridges are counteragents that affect the appearance and complexity of sunspots and their penumbrae.

Characterizing the Umbral Magnetic Knots of δ-Sunspots

Delta (δ) spots are active regions (ARs) in which positive and negative umbrae share a penumbra. They are known to be the source of strong flares. We introduce a new quantity, the degree of δ (Doδ), to measure the fraction of umbral flux participating in the δ-configuration and to isolate the dynamics of the magnetic knot, i.e., adjacent umbrae in the δ-configuration. Using Helioseismic and Magnetic Imager data, we analyze 19 δ-spots and 11 β-spots in detail, as well as 120 δ-spots in less detail. We find that δ-regions are not in a δ-configuration for the entire time but spend 55% of their observed time as δ-spots with an average, maximum Doδ of 72%. Compared to β-spots, δ-spots have 2.6× the maximum umbral flux, 1.9× the flux emergence rate, 2.6× the rotation, and 72× the flare energy. On average, the magnetic knots rotate 17° day−1, while the β-spots rotate 2° day−1. Approximately 72% of the magnetic knots present anti-Hale or anti-Joy tilts, contrasting starkly with only 9% of the β-spots. A positive correlation exists between ϕ Doδ and the flare energy emitted by that region. The δ-spots obey the hemispheric current helicity rule 64% of the time. A total of 84% of the δ-spots are formed by single flux emergence events, and 58% have a quadrupolar magnetic configuration. The δ-spot characteristics are consistent with the formation mechanism signatures as follows: 42% with the kink instability or Sigma effect, 32% with multisegment buoyancy, 16% with collisions, and two ARs that are unclassified but consistent with a rising O-ring.

Magnetic knot cascade via the stepwise reconnection of helical flux tubes

Abstract

Energy and helicity of magnetic torus knots and braids

By considering steady magnetic fields in the shape of torus knots and unknots in ideal magnetohydrodynamics, we compute some fundamental geometric and physical properties to provide estimates for magnetic energy and helicity. By making use of an appropriate parametrization, we show that knots with dominant toroidal coils that are a good model for solar coronal loops have negligible total torsion contribution to magnetic helicity while writhing number provides a good proxy. Hence, by the algebraic definition of writhe based on crossing numbers, we show that the estimated values of writhe based on image analysis provide reliable information for the exact values of helicity. We also show that magnetic energy is linearly related to helicity, and the effect of the confinement of magnetic field can be expressed in terms of geometric information. These results can find useful application in solar and plasma physics, where braided structures are often present.

Influence of the Magnetic Field on Oscillation Spectra in Solar Faculae

In this work, we studied oscillation parameters in faculae above magnetic knots and in the adjacent to them areas. Using SDO data we analysed oscillations in magnetic strength, Doppler velocity, and intensity signals for the lower photosphere, and in intensity for the higher levels. We found that in the magnetic field strength oscillation spectra in magnetic knots, peaks at a frequency of about 4.8 mHz appear, while there are no such frequencies in the adjacent facular patches of a moderate field strength. On the contrary, Doppler velocity photospheric oscillation spectra are similar for these types of regions: in both cases, the significant peaks are in the 2.5--4.5 mHz range, though the oscillations in magnetic knots are 2--3 times weaker than those at the facular periphery. At the upper photosphere, the dominant frequencies in magnetic knots are 0.5--1 mHz higher than in the medium-field regions. The transition region oscillations above magnetic knots mainly concentrate in the 3--6 mHz range, and those above moderate-field patches concentrate below 3 mHz.

Characteristics of oscillations in magnetic knots of solar faculae

The characteristics of oscillations of the intensity, magnetic field, andDoppler velocity in facular magnetic knots are compared with those at the facula periphery, where the magnetic field has a moderate intensity. The analysis is based on images in the FeI 6173 A, 1700 A, and He II 304 A lines obtained with the Solar Dynamics Observatory, which are formed in the lower photosphere, upper photosphere, and transition zone, respectively. The spectra of the oscillations of the longitudinal magnetic field in magnetic knots show peaks at a frequency of about 5 mHz, which are not observed at the faculae peripheries. The spectra of the photospheric oscillations of the intensity and Doppler velocity in magnetic knots and in areas of moderate magnetic field are similar, but the oscillation power is a factor of two to four lower in the knots. The maximum spectral peaks in the HeII 304 A line are mainly distributed in the range 3–6 mHz above magnetic knots and in the range 1.5–3 mHz above periphery regions. It is proposed that this distribution of the oscillations is due to the magnetic-field topologies in faculae: the m

Parallelization of the SIR code for the investigation of small-scale features in the solar photosphere

AbstractMagnetic fields are one of the most important drivers of the highly dynamic processes that occur in the lower solar atmosphere. They span a broad range of sizes, from large- and intermediate-scale structures such as sunspots, pores and magnetic knots, down to the smallest magnetic elements observable with current telescopes. On small scales, magnetic flux tubes are often visible as Magnetic Bright Points (MBPs). Apart from simple V/I magnetograms, the most common method to deduce their magnetic properties is the inversion of spectropolarimetric data. Here we employ the SIR code for that purpose. SIR is a well-established tool that can derive not only the magnetic field vector and other atmospheric parameters (e.g., temperature, line-of-sight velocity), but also their stratifications with height, effectively producing 3-dimensional models of the lower solar atmosphere. In order to enhance the runtime performance and the usability of SIR we parallelized the existing code and standardized the input and output formats. This and other improvements make it feasible to invert extensive high-resolution data sets within a reasonable amount of computing time. An evaluation of the speedup of the parallel SIR code shows a substantial improvement in runtime.

On the groundstate energy spectrum of magnetic knots and links

By using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links, and by using ropelength data from the RIDGERUNNER tightening algorithm (Ashton et al 2011 Exp. Math. 20 57–90) we obtain the groundstate energy spectra of the first 250 prime knots and 130 prime links. The two spectra are found to follow an almost identical logarithmic law. By assuming that the number of knot types grows exponentially with the topological crossing number, we show that this generic behavior can be justified by a general relationship between ropelength and crossing number, which is in good agreement with former analytical estimates (Buck and Simon 1999 Topol. Appl. 91 245–57, Diao 2003 J. Knot Theory Ramifications 12 1–16). Moreover, by considering the ropelength averaged over a given knot family, we establish a new connection between the averaged ropelength and the topological crossing number of magnetic knots.

New energy and helicity bounds for knotted and braided magnetic fields

In this article we present a review of some of the author's most recent results in topological magnetohydrodynamics (MHD), with an eye to possible applications to astrophysical flows and solar coronal structures. First, we briefly review basic work on magnetic helicity and linking numbers, and fundamental relations with magnetic energy and average crossing numbers of magnetic systems in ideal conditions. In the case of magnetic knots, we focus on the relation between their groundstate energy and topology, discussing the energy spectrum of tight knots in terms of ropelength. We compare this spectrum with the one given by considering the bending energy of such idealized knots, showing that curvature information provides a rather good indicator of magnetic energy contents. For loose knots far from equilibrium we show that inflexional states determine the transition to braid form. New lower bounds for tight knots and braids are then established. We conclude with results on energy-complexity relations for systems in presence of dissipation.

Decay of helical and nonhelical magnetic knots

We present calculations of the relaxation of magnetic field structures that have the shape of particular knots and links. A set of helical magnetic flux configurations is considered, which we call n-foil knots of which the trefoil knot is the most primitive member. We also consider two nonhelical knots; namely, the Borromean rings as well as a single interlocked flux rope that also serves as the logo of the Inter-University Centre for Astronomy and Astrophysics in Pune, India. The field decay characteristics of both configurations is investigated and compared with previous calculations of helical and nonhelical triple-ring configurations. Unlike earlier nonhelical configurations, the present ones cannot trivially be reduced via flux annihilation to a single ring. For the n-foil knots the decay is described by power laws that range form t(-2/3) to t(-1/3), which can be as slow as the t(-1/3) behavior for helical triple-ring structures that were seen in earlier work. The two nonhelical configurations decay like t(-1), which is somewhat slower than the previously obtained t(-3/2) behavior in the decay of interlocked rings with zero magnetic helicity. We attribute the difference to the creation of local structures that contain magnetic helicity which inhibits the field decay due to the existence of a lower bound imposed by the realizability condition. We show that net magnetic helicity can be produced resistively as a result of a slight imbalance between mutually canceling helical pieces as they are being driven apart. We speculate that higher order topological invariants beyond magnetic helicity may also be responsible for slowing down the decay of the two more complicated nonhelical structures mentioned above.

Horizontal flow fields observed in Hinode G-band images

Context: The interaction of plasma motions and magnetic fields is an important mechanism, which drives solar activity in all its facets. For example, photospheric flows are responsible for the advection of magnetic flux, the redistribution of flux during the decay of sunspots, and the built-up of magnetic shear in flaring active regions. Aims: Systematic studies based on G-band data from the Japanese Hinode mission provide the means to gather statistical properties of horizontal flow fields. This facilitates comparative studies of solar features, e.g., G-band bright points, magnetic knots, pores, and sunspots at various stages of evolution and in distinct magnetic environments, thus, enhancing our understanding of the dynamic Sun. Methods: We adapted Local Correlation Tracking (LCT) to measure horizontal flow fields based on G-band images obtained with the Solar Optical Telescope on board Hinode. In total about 200 time-series with a duration between 1-16 h and a cadence between 15-90 s were analyzed. Selecting both a high-cadence (dt = 15 s) and a long-duration (dT = 16 h) time-series enabled us to optimize and validate the LCT input parameters, hence, ensuring a robust, reliable, uniform, and accurate processing of a huge data volume. Results: The LCT algorithm produces best results for G-band images having a cadence of 60-90 s. If the cadence is lower, the velocity of slowly moving features will not be reliably detected. If the cadence is higher, the scene on the Sun will have evolved too much to bear any resemblance with the earlier situation. Consequently, in both instances horizontal proper motions are underestimated. The most reliable and yet detailed flow maps are produced using a Gaussian kernel with a size of 2560 km x 2560 km and a full-width-at-half-maximum (FWHM) of 1200 km (corresponding to the size of a typical granule) as sampling window.

Decay of trefoil and other magnetic knots

AbstractTwo setups with interlocked magnetic flux tubes are used to study the evolution of magnetic energy and helicity on magnetohydrodynamical (MHD) systems like plasmas. In one setup the initial helicity is zero while in the other it is finite. To see if it is the actual linking or merely the helicity content that influences the dynamics of the system we also consider a setup with unlinked field lines as well as a field configuration in the shape of a trefoil knot. For helical systems the decay of magnetic energy is slowed down by the helicity which decays slowly. It turns out that it is the helicity content, rather than the actual linking, that is significant for the dynamics.

Circular dichroism, magnetic knots, and the spectropolarimetry of the cosmic microwave background

When the last electron-photon scattering takes place in a magnetized environment, the degree of circular polarization of the outgoing radiation depends upon the magnetic field strength. After deriving the scattering matrix of the process, the generalized radiative transfer equations are deduced in the presence of the relativistic fluctuations of the geometry and for all the four brightness perturbations. The new system of equations is solved under the assumption that the incident radiation is not polarized. The induced $V$-mode polarization is analyzed both analytically and numerically. The corresponding angular power spectra are calculated and compared with the measured (or purported) values of the linear polarizations (i.e. $E$ mode and $B$ mode) as they arise in the concordance model and in its neighboring extensions. Possible connections between the $V$-mode polarization of the cosmic microwave background and the topological properties of the magnetic flux lines prior to equality are outlined and briefly explored in analogy with the physics of magnetized sunspots.

On the groundstate energy of tight knots

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots — up to 10 crossings — is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.

Cross-fertilization of Ferreira's Hopfions and electromagnetic knots

The interrelation between Ferreira's Hopf solitons of a conformal nonlinear σ model and the electromagnetic knots found by Ranada et al. is investigated. It is shown that the electromagnetic knots yield exact solutions of the conformal nonlinear σ model different from those obtained by Ferreira. Conversely, It is discussed that Ferreira's solutions realize magnetic knots. The energy associated with these two kinds of knots are compared. The structure of the electric charge distribution and the electric current density associated with the magnetic knots is investigated.

Long-period oscillations of the line-of-sight velocities in and near sunspots at various levels in the photosphere

New observational data on long-period oscillations of the line-of-sight velocities detected via the Doppler shifts of spectral lines observed at various heights in and near sunspots are presented. The sunspots and nearby magnetic elements oscillate with periods ranging from 40 to 80 min. The oscillations in the line-of-sight velocities persist over the entire observation session (up to four hours). These results support theoretical models in which this phenomenon represents natural long-period oscillations (vertical-radial displacements) of entire magnetic elements (sunspots, pores, and magnetic knots) about some stable equilibrium positions.

Evolution of Magnetic Nonpotentiality in NOAA AR 10486

The active region NOAA 10486 was one of the most flare-productive regions during solar cycle 23. In this paper, we focus our analysis on the daily evolution of magnetic nonpotentiality for this region from 2003 October 26 to October 30. Daily averaged values of three nonpotential parameters: magnetic shear angle, line-of-sight current, and current helicity of the selected regions along the main neutral lines are calculated using the vector magnetograms obtained at Huairou Solar Observing Station. The magnetic flux evolution and proper motion of magnetic features are also studied in detail for the initial brightening regions of the two large X-class flares (X17 on October 28 and X10 on October 29). The main results are as follows. (1) Three parameters of the magnetic nonpotentiality in the photosphere obviously increased at the impulsively brightening flare sites from at least 1 day before the two large X-class flares, and most of the three parameters decreased after the flares around the main flare regions. (2) The increase of magnetic flux and complex proper motions of the magnetic knots were found to be simultaneously occurring at the non—potentiality developing sites along the main neutral lines. Such increases and motions of the magnetic flux can be explained by the model of emerging twisted flux ropes, and we conclude that the emergence of twisted magnetic flux ropes are a main reason for the development of nonpotentiality along the neutral line and for the production of strong X-class flares.

The Contrast of Magnetic Flux Concentrations at Near‐Infrared and Visible Wavelengths

To reconcile discrepancies in the observed contrast of magnetic flux concentrations, we compute synthetic filtergrams from solar magnetoconvection models in four different passbands: two continuum bands, at 1626 and 575 nm, the G band, and the line wing of the Ca II K line. Magnetic flux concentrations are selected by their G-band brightness. In the near-infrared the selected flux concentrations appear inconspicuous with intensities that are on average below that of the synthetic average quiet Sun, while their contrast in the visible passband is mostly positive. When we account for limited telescope resolution and imperfect seeing, the contrasts of bright points in the visible become increasingly negative as a result of mixing with the dark intergranular lanes in which they are embedded. The correlation in bright-point intensity between different passbands becomes increasingly poor from the G band to the visible and the near-infrared. This is caused in part by the varying amount of spectral lines in each of the bands. Further, we find that the largest magnetic field concentrations, representative of micropores or magnetic knots, are dark in all four passbands. The brightenings in the Ca II K line wing do not coincide with magnetic flux concentrations on the spatial scale of the simulation. Finally, we find that the rms intensity contrast in all four passbands depends on the amount of magnetic field present in the simulation, with higher contrast for lower average magnetic field strength.