Field Line Helicity Research Articles

Using relative field line helicity as an indicator for solar eruptivity

Context. Relative field line helicity (RFLH) is a recently developed quantity that can approximate the density of relative magnetic helicity. Aims. This paper aims to determine whether RFLH can be used as an indicator of solar eruptivity. Methods. Starting from magnetographic observations from the Helioseismic and Magnetic Imager instrument on board the Solar Dynamic Observatory of a sample of seven solar active regions (ARs), that comprises over 2000 individual snapshots, we reconstruct the AR’s coronal magnetic field with a widely used non-linear force-free method. This enables us to compute the RFLH using two independent gauge conditions for the vector potentials. We focus our study on the times of strong flares in the ARs, above the M class, and in regions around the polarity inversion lines (PILs) of the magnetic field, and of RFLH. Results. We find that the temporal profiles of the relative helicity that is contained in the magnetic PIL follow those of the relative helicity that is computed by the accurate volume method for the whole AR. Additionally, the PIL relative helicity can be used to define a parameter similar to the well-known parameter R, whose high values are related with increased flaring probability. This helicity-based R-parameter correlates closely with the original parameter, showing in some cases even higher values. Additionally, it experiences more pronounced decreases during flares. This means that there exists at least one parameter deduced from RFLH that is important as a solar eruptivity indicator.

Resonant alpha particle loss in stellarators

Particle resonances in stellarators can produce islands in the space of passing particle orbits without the presence of an unstable Alfvén mode, provided the period of the resonance matches the period of the equilibrium magnetic field. In this case, the equilibrium itself plays the role of a mode amplitude, and the islands appear on surfaces where the orbital helicity matches the field period. At low energy, these surfaces are given by the field line helicity, but at higher energy, cross field drift causes them to move. The resonances are also felt by trapped particles bouncing back and forth on surfaces with matching helicity. The periodic variation of B along these orbits produces local wells, giving loss due to drift while trapped in a well. Stellarator designs that have equilibrium-induced resonance islands exhibit anomalous alpha particle loss and are unsuitable for reactors.

Poor confinement in stellarators at high energy

High-energy particle resonances can modify particle distributions and even cause significant particle loss. Resonances can be present in any toroidal confinement device and can easily be found numerically. Many stellarators have weak magnetic shear so that large islands and large chaotic regions can be produced by resonant perturbations with small amplitudes. While the choice of the field line helicity profile in the plasma can limit the presence of resonances at low particle energy, the resonance location is energy-dependent, and they can move into the plasma at higher energy. If resonances match the toroidal variation of the equilibrium, they can produce wide islands in the phase space of orbits even in the absence of perturbations due to instabilities. These islands increase in size with particle energy and can seriously affect the confinement of high-energy ions.

Intrinsic winding of braided vector fields in tubular subdomains

Braided vector fields on spatial subdomains which are homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field’s topology—the entanglement of its field lines—is non-trivial, and can play a significant role in the vector field’s evolution. We present a complete topological characterisation of such vector fields (up to isotopy) using a quantity called field line winding. This measures the entanglement of each field line with all other field lines of the vector field, and may be defined for an arbitrary tubular subdomain by prescribing a minimally distorted coordinate system. We propose how to define such coordinates, and prove that the resulting field line winding distribution uniquely classifies the topology of a braided vector field. The field line winding is similar to the field line helicity considered previously for magnetic (solenoidal) fields, but is a more fundamental measure of the field line topology because it does not conflate linking information with field strength.

Evolution of field line helicity in magnetic relaxation

Plasma relaxation in the presence of an initially braided magnetic field can lead to self-organization into relaxed states that retain non-trivial magnetic structure. These relaxed states may be in conflict with the linear force-free fields predicted by the classical Taylor theory, and remain to be fully understood. Here, we study how the individual field line helicities evolve during such a relaxation, and show that they provide new insights into the relaxation process. The line helicities are computed for numerical resistive-magnetohydrodynamic simulations of a relaxing braided magnetic field with line-tied boundary conditions, where the relaxed state is known to be non-Taylor. First, our computations confirm recent analytical predictions that line helicity will be predominantly redistributed within the domain, rather than annihilated. Second, we show that self-organization into a relaxed state with two discrete flux tubes may be predicted from the initial line helicity distribution. Third, for this set of line-tied simulations we observe that the sub-structure within each of the final tubes is a state of uniform line helicity. This uniformization of line helicity is consistent with Taylor theory applied to each tube individually. However, it is striking that the line helicity becomes significantly more uniform than the force-free parameter.

Relative field line helicity of a large eruptive solar active region

Context. Magnetic helicity is a physical quantity of great importance in the study of astrophysical and natural plasmas. Although a density for helicity cannot be defined, a good proxy for this quantity is field line helicity. The appropriate quantity for use in solar conditions is relative field line helicity (RFLH). Aims. This work aims to study in detail the behaviour of RFLH, for the first time, in a solar active region (AR). Methods. The target AR is the large, eruptive AR 11158. In order to compute RFLH and all other quantities of interest, we used a non-linear force-free reconstruction of the AR coronal magnetic field of excellent quality. Results. We find that the photospheric morphology of RFLH is very different than that of the magnetic field or electrical current, and this morphology is not sensitive to the chosen gauge in the computation of RFLH. The value of helicity experiences a large decrease, that is ∼25% of its pre-flare value, during an X-class flare of the AR; this change is also depicted in the photospheric morphology of RFLH. Moreover, the area of this change coincides with the area that encompasses the flux rope, which is the magnetic structure that later erupted. Conclusions. The use of RFLH can provide important information about the value and location of the magnetic helicity expelled from the solar atmosphere during eruptive events.

Particle resonances in toroidal fusion devices

Resonances of high energy particles in magnetic confinement devices due to electromagnetic instabilities can strongly modify the distribution, leading to a reduction in fusion power and even discharge termination and particle loss to the device walls through avalanche. The existence of a mode particle resonance depends on properties of the equilibrium, particle trajectories, and perturbation mode harmonic content. Resonance location is a function of particle energy and equilibrium field line helicity. Different methods for finding resonance location and energy dependence are developed. The properties of resonances are discussed using examples from magnetic fusion devices. We show that if mode resonances exist at low particle energy, they very likely also exist at high energy, thus modifying high energy beam particles and fusion products. It is possible for a resonance to appear due to mode induced orbit helicity modification when it is forbidden at low mode amplitude.

Optimal unstirred state of a passive scalar

Given a passive tracer distribution $f(x,y)$ , what is the simplest unstirred pattern that may be reached under incompressible advection? This question is partially motivated by recent studies of three-dimensional (3-D) magnetic reconnection, in which the patterns of a topological invariant called the field line helicity greatly simplify until reaching a relaxed state. We test two approaches: a variational method with minimal constraints, and a magnetic relaxation scheme where the velocity is determined explicitly by the pattern of $f$ . Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of $f$ , and is numerically more robust. We therefore apply the latter method to two complex mixed patterns modelled on the field line helicity of 3-D magnetic braids. The unstirring separates $f$ into a small number of large-scale regions determined by the initial topology, which is well preserved during the computation. Interestingly, the velocity field is found to have the same large-scale topology as $f$ . Similarity to the simplification found empirically in 3-D magnetic reconnection simulations supports the idea that advection is an important principle for field line helicity evolution.

The Minimal Helicity of Solar Coronal Magnetic Fields

Abstract Potential field extrapolations are widely used as minimum-energy models for the Sun’s coronal magnetic field. As the reference to which other magnetic fields are compared, they have—by any reasonable definition—no global (signed) magnetic helicity. Here we investigate the internal topological structure that is not captured by the global helicity integral, by splitting it into individual field line helicities. These are computed using potential field extrapolations from magnetogram observations over Solar Cycle 24, as well as for a simple illustrative model of a single bipolar region in a dipolar background. We find that localized patches of field line helicity arise primarily from linking between strong active regions and their overlying field, so that the total unsigned helicity correlates with the product of photospheric and open fluxes. Within each active region, positive and negative helicity may be unbalanced, but the signed helicity is only around a tenth of the unsigned helicity. Interestingly, in Cycle 24, there is a notable peak in unsigned helicity caused by a single large active region. On average, the total unsigned helicity at the resolution considered is approximately twice the typical signed helicity of a single real active region, according to non-potential models in the literature.

Spatial scales and locality of magnetic helicity

Context. Magnetic helicity is approximately conserved in resistive magnetohydrodynamic models. It quantifies the entanglement of the magnetic field within the plasma. The transport and removal of helicity is crucial in both dynamo development in the solar interior and active region evolution in the solar corona. This transport typically leads to highly inhomogeneous distributions of entanglement. Aims. There exists no consistent systematic means of decomposing helicity over varying spatial scales and in localised regions. Spectral helicity decompositions can be used in periodic domains and is fruitful for the analysis of homogeneous phenomena. This paper aims to develop methods for analysing the evolution of magnetic field topology in non-homogeneous systems. Methods. The method of multi-resolution wavelet decomposition is applied to the magnetic field. It is demonstrated how this decomposition can further be applied to various quantities associated with magnetic helicity, including the field line helicity. We use a geometrical definition of helicity, which allows these quantities to be calculated for fields with arbitrary boundary conditions. Results. It is shown that the multi-resolution decomposition of helicity has the crucial property of local additivity. We demonstrate a general linear energy-topology conservation law, which significantly generalises the two-point correlation decomposition used in the analysis of homogeneous turbulence and periodic fields. The localisation property of the wavelet representation is shown to characterise inhomogeneous distributions, which a Fourier representation cannot. Using an analytic representation of a resistive braided field relaxation, we demonstrate a clear correlation between the variations in energy at various length scales and the variations in helicity at the same spatial scales. Its application to helicity flows in a surface flux transport model show how various contributions to the global helicity input from active region field evolution and polar field development are naturally separated by this representation. Conclusions. The multi-resolution wavelet decomposition can be used to analyse the evolution of helicity in magnetic fields in a manner which is consistently additive. This method has the advantage over more established spectral methods in that it clearly characterises the inhomogeneous nature of helicity flows where spectral methods cannot. Further, its applicability in aperiodic models significantly increases the range of potential applications.

Relative magnetic field line helicity

Context. Magnetic helicity is an important quantity in studies of magnetized plasmas as it provides a measure of the geometrical complexity of the magnetic field in a given volume. A more detailed description of the spatial distribution of magnetic helicity is given by the field line helicity, which expresses the amount of helicity associated to individual field lines rather than in the full analysed volume. Aims. Magnetic helicity is not a gauge-invariant quantity in general, unless it is computed with respect to a reference field, yielding the so-called relative magnetic helicity. The field line helicity corresponding to the relative magnetic helicity has only been examined under specific conditions so far. This work aims to define the field line helicity corresponding to relative magnetic helicity in the most general way. In addition to its general form, we provide the expression for the relative magnetic field line helicity in a few commonly used gauges, and reproduce known results as a limit of our general formulation. Methods. By starting from the definition of relative magnetic helicity, we derived the corresponding field line helicity, and we noted the assumptions on which it is based. Results. We checked that the developed quantity reproduces relative magnetic helicity by using three different numerical simulations. For these cases we also show the morphology of field line helicity in the volume, and on the photospheric plane. As an application to solar situations, we compared the morphology of field line helicity on the photosphere with that of the connectivity-based helicity flux density in two reconstructions of an active region’s magnetic field. We discuss how the derived relative magnetic field line helicity has a wide range of applications, notably in solar physics and magnetic reconnection studies.

Relative field-line helicity in bounded domains

Models for astrophysical plasmas often have magnetic field lines that leave the boundary rather than closing within the computational domain. Thus, the relative magnetic helicity is frequently used in place of the usual magnetic helicity, so as to restore gauge invariance. We show how to decompose the relative helicity into a relative field-line helicity that is an ideal-magnetohydrodynamic invariant for each individual magnetic field line, and vanishes along any field line where the original field matches the reference field. Physically, this relative field-line helicity is a magnetic flux, whose specific definition depends on the gauge of the reference vector potential on the boundary. We propose a particular ‘minimal’ gauge that depends only on the reference field and minimises this boundary contribution, so as to reveal topological information about the original magnetic field. We illustrate the effect of different gauge choices using the Low–Lou and Titov–Démoulin models of solar active regions. Our numerical code to compute appropriate vector potentials and relative field-line helicity in Cartesian domains is open source and freely available.

New formulae for magnetic relative helicity and field line helicity

We consider a magnetic field occupying the simply connected domain D and having all its field lines tied to the boundary S of D. We assume here that has a simple topology, i.e., the mapping from positive to negative polarity areas of S associating to each other the two footpoints of any magnetic line, is continuous. We first present new formulae for the helicity H of relative to a reference field having the same normal component on S, and for its field line helicity h relative to a reference vector potential of . These formulae make immediately apparent the well known invariance of these quantities under all the ideal MHD deformations that preserve the positions of the footpoints on S. They express indeed h and H either in terms of and , or in terms of the values on S of a pair of Euler potentials of . We next show that, for a specific choice of , the field line helicity h of fully characterizes the magnetic mapping and then the topology of the lines. Finally, we give a formula that describes the rate of change of h in a situation where the plasma moves on the perfectly conducting boundary S without changing and/or non-ideal processes, described by an unspecified term in Ohm’s law, are at work in some parts of D.

Magnetic Flux Rope Identification and Characterization from Observationally Driven Solar Coronal Models

Abstract Formed through magnetic field shearing and reconnection in the solar corona, magnetic flux ropes are structures of twisted magnetic field, threaded along an axis. Their evolution and potential eruption are of great importance for space weather. Here we describe a new methodology for the automated detection of flux ropes in simulated magnetic fields, utilizing field-line helicity. Our Flux Rope Detection and Organization (FRoDO) code, which measures the magnetic flux and helicity content of pre-erupting flux ropes over time, as well as detecting eruptions, is publicly available. As a first demonstration, the code is applied to the output from a time-dependent magnetofrictional model, spanning 1996 June 15–2014 February 10. Over this period, 1561 erupting and 2099 non-erupting magnetic flux ropes are detected, tracked, and characterized. For this particular model data, erupting flux ropes have a mean net helicity magnitude of Mx2, while non-erupting flux ropes have a significantly lower mean of Mx2, although there is overlap between the two distributions. Similarly, the mean unsigned magnetic flux for erupting flux ropes is Mx, significantly higher than the mean value of Mx for non-erupting ropes. These values for erupting flux ropes are within the broad range expected from observational and theoretical estimates, although the eruption rate in this particular model is lower than that of observed coronal mass ejections. In the future, the FRoDO code will prove to be a valuable tool for assessing the performance of different non-potential coronal simulations and comparing them with observations.

The global distribution of magnetic helicity in the solar corona (Corrigendum)

By defining an appropriate field line helicity, we apply the powerful concept of magnetic helicity to the problem of global magnetic field evolution in the Sun's corona. As an ideal-magnetohydrodynamic invariant, the field line helicity is a meaningful measure of how magnetic helicity is distributed within the coronal volume. It may be interpreted, for each magnetic field line, as a magnetic flux linking with that field line. Using magneto-frictional simulations, we investigate how field line helicity evolves in the non-potential corona as a result of shearing by large-scale motions on the solar surface. On open magnetic field lines, the helicity injected by the Sun is largely output to the solar wind, provided that the coronal relaxation is sufficiently fast. But on closed magnetic field lines, helicity is able to build up. We find that the field line helicity is non-uniformly distributed, and is highly concentrated in twisted magnetic flux ropes. Eruption of these flux ropes is shown to lead to sudden bursts of helicity output, in contrast to the steady flux along the open magnetic field lines.

The global distribution of magnetic helicity in the solar corona

By defining an appropriate field line helicity, we apply the powerful concept of magnetic helicity to the problem of global magnetic field evolution in the Sun’s corona. As an ideal-magnetohydrodynamic invariant, the field line helicity is a meaningful measure of how magnetic helicity is distributed within the coronal volume. It may be interpreted, for each magnetic field line, as a magnetic flux linking with that field line. Using magneto-frictional simulations, we investigate how field line helicity evolves in the non-potential corona as a result of shearing by large-scale motions on the solar surface. On open magnetic field lines, the helicity injected by the Sun is largely output to the solar wind, provided that the coronal relaxation is sufficiently fast. But on closed magnetic field lines, helicity is able to build up. We find that the field line helicity is non-uniformly distributed, and is highly concentrated in twisted magnetic flux ropes. Eruption of these flux ropes is shown to lead to sudden bursts of helicity output, in contrast to the steady flux along the open magnetic field lines.

Evolution of field line helicity during magnetic reconnection

We investigate the evolution of field line helicity for magnetic fields that connect two boundaries without null points, with emphasis on localized finite-B magnetic reconnection. Total (relative) magnetic helicity is already recognized as an important topological constraint on magnetohydrodynamic processes. Field line helicity offers further advantages because it preserves all topological information and can distinguish between different magnetic fields with the same total helicity. Magnetic reconnection changes field connectivity and field line helicity reflects these changes; the goal of this paper is to characterize that evolution. We start by deriving the evolution equation for field line helicity and examining its terms, also obtaining a simplified form for cases where dynamics are localized within the domain. The main result, which we support using kinematic examples, is that during localized reconnection in a complex magnetic field, the evolution of field line helicity is dominated by a work-like term that is evaluated at the field line endpoints, namely, the scalar product of the generalized field line velocity and the vector potential. Furthermore, the flux integral of this term over certain areas is very small compared to the integral of the unsigned quantity, which indicates that changes of field line helicity happen in a well-organized pairwise manner. It follows that reconnection is very efficient at redistributing helicity in complex magnetic fields despite having little effect on the total helicity.

Lower and upper bounds on the energy of braided magnetic fields

Let ℬ be a class of topologically equivalent regular magnetic fields occupying a cylindrical domain of axis Oz and produced by ideal MHD deformations of the uniform field , and let ℬff ⊂ ℬ be constituted of the fields of ℬ which are force-free. We derive topological lower and upper bounds on the energy of the fields in ℬ and in ℬff, respectively. We also establish new formulae for the field line helicity and the relative magnetic helicity of the fields in ℬ, and give a sufficient criterion of ideal MHD linear stability for the fields in ℬff.

Off-axis neutral beam current drive for advanced scenario development in DIII-D

Modification of the two existing DIII-D neutral beamlines is planned to allow vertical steering to provide off-axis neutral beam current drive (NBCD) peaked as far off-axis as half the plasma minor radius. New calculations for a downward-steered beam indicate strong current drive with good localization off-axis so long as the toroidal magnetic field, BT, and the plasma current, Ip, point in the same direction. This is due to good alignment of neutral beam injection (NBI) with the local pitch of the magnetic field lines. This model has been tested experimentally on DIII-D by injecting equatorially mounted NBs into reduced size plasmas that are vertically displaced with respect to the vessel midplane. The existence of off-axis NBCD is evident in the changes seen in sawtooth behaviour in the internal inductance. By shifting the plasma upwards or downwards, or by changing the sign of the toroidal field, off-axis NBCD profiles measured with motional Stark effect data and internal loop voltage show a difference in amplitude (40–45%) consistent with differences predicted by the changed NBI alignment with respect to the helicity of the magnetic field lines. The effects of NBI direction relative to field line helicity can be large even in ITER: off-axis NBCD can be increased by more than 30% if the BT direction is reversed. Modification of the DIII-D NB system will strongly support scenario development for ITER and future tokamaks as well as provide flexible scientific tools for understanding transport, energetic particles and heating and current drive.

On the asymmetries of ELM divertor power deposition in JET and ASDEX Upgrade

An analytical expression was derived for describing the divertor target power during ELMs based on the model discussed in [W. Fundamenski, R.A. Pitts, Plasma Phys. Control. Fus. 48 (2006) 109] where the power load arises from a Maxwellian distribution of particles released into the SOL region. The paper discusses a comparable simple extension of the model by introducing a non-zero characteristic velocity of the Maxwellian distributed particles. This way the experimentally observed temporal evolution as well as the in/out energy imbalance can be described. The extended model named free-streaming-particle (FSP) approach predicts a dependence of the ELM in/out energy balance of the pedestal Mach number as well as an inversion of the in/out balance by a change of the field line helicity as observed experimentally. From the FSP approach the value for E τ IR (see text) is predicted to be 18–25% in good agreement to the experimental data. A comparison to the various discussed mechanism for the ELM in/out asymmetry is presented.