Why to study magnetic helicity?

What is magnetic helicity?

Magnetic helicity is a quantity which, while possessing key properties to become a potentially strong flare predicator, remains poorly measured and understood in the solar context. The HeliSol project is a project aiming at improving our understanding of this physical quantity, improving the theory of its measurements on the Sun and initiating the application of these methods in the solar eruptions forecast context. The researches performed through HeliSol can provide results which are expected to be routinely used by flare forecast methods in a 10-20 years range.

Magnetic helicity is a quantity that measures the entanglement of the magnetic field. Magnetic helicity is related to the “linking number” of the 19th century German mathematician and physical scientist Carl Gauss, which measures the number of times closed curves wind around each others. Magnetic helicity estimates the level of twisting, writhing and shearing of magnetic field lines. The HeliSol project aims at improving our understanding of this quantity that allows a characterization of the solar magnetic field. The magnetic field is indeed the core element that governs most aspects of solar activity: the magnetic energy is the main energy source of the most dramatic events taking place in the solar atmosphere, as well as the quantity which structures and drives solar plasma in interplanetary space.

Why is magnetic helicity important in solar physics?

Magnetic helicity is an important quantity to understand the global dynamics of the solar magnetic field as it can restrict the amount of energy that can be stored and eventually released during a solar eruption. Magnetic helicity has also a particularly interesting property in the framework of Solar Physics as it is one of the few invariants in Magneto-Hydro- Dynamics (MHD). More specifically, magnetic helicity is strictly conserved in ideal MHD when the plasma is a perfect conductor (Elsasser 56). Helicity is also believed to be quasi-conserved when non-ideal processes, leading to the violent solar flares, are acting (cf. Section 1.3.3). Ideal MHD is the most common paradigm used to describe the evolution of magnetic structures in the Sun. Magnetic helicity conservation property thus should permit the tracking of this quantity throughout the evolution of the magnetic field in the Sun (e.g. review of Démoulin 07, Pevstov et al 14).


Figure1: Evolution, storage & transport of magnetic helicity from the solar interior towards the Heliosphere illustrating the articulation of the different work packages of the HeliSol project.

Solar magnetic field is known to be created within the solar interior, transported through the upper layers of the solar interior via discrete magnetic structures – magnetic flux tubes (see Figure 1, lower-central part). Last 20 years of research on magnetic flux tube transport in the solar interior have demonstrated that the rising flux tubes must be formed of twisted / intertwined magnetic field lines (e.g. review of Cheung & Isobe 14). These twisted flux tube are hence helicity carrying structures. Without helicity these flux tubes would not be able to survive the rise through the solar convective zone and would be dispersed.

These twisted flux tubes eventually emerge in the solar atmosphere, forming solar active regions/sunspots where magnetic flux, energy and helicity slowly accumulate (see Figure 1, lower-central part). Eventually, magnetic energy is very impulsively liberated, inducing a burst of electromagnetic radiations: a solar flare. Both the existence of a certain amount of magnetic energy and magnetic helicity are necessary conditions for the trigger of solar flares as suggested by observational results (Nindos, 04, Tziotziou et al. 12).

Solar flares are frequently (and quasi-always for the most violent) accompanied by   the explosive ejection of matters within the solar system: solar eruption or coronal mass ejections (CMEs). This ejected material travels through the interplanetary space and possibly interacts with the Earth’s own magnetic field and may generate geomagnetic storms. Local magnetic field measurements by satellites orbiting Earth have demonstrated that these CMEs are formed of large twisted magnetic structure travelling through space (see Figure 1, right part). CMEs are thus carrying away the magnetic helicity that had been stored in active regions (e.g. Chandra et al. 10).

A conjecture is thus that solar eruptions are the necessary consequence of the conservation of magnetic helicity (Rust 94, Low 96): CMEs, the most important drivers of geomagnetic activity on Earth, occur because, overall, the Sun must get rid of the magnetic helicity it has been generating in its interior. Magnetic helicity thus appears as an appealing quantity to be tracked and surveyed in the solar corona in the framework of space weather (cf Section 5 of Pevtsov et al 14). This justifies the first subquestion of the HeliSol project: How magnetic helicity is injected and accumulates in the solar atmosphere?

To address this question, the HeliSol will perform case studies of observed active regions in parallel with state of the art numerical simulations of magnetic flux emergence in the solar atmosphere (Leake et al. 13, 14). For each type of data (observational and numerical), using our state-of-the art methods developed in the last years, we will simultaneously monitor the evolution of helicity through the solar surface and the evolution of its content in the atmosphere above active regions. These case by case studies will enable us to better understand the properties of the magnetic helicity in active centers. However, no systematic study of helicity in active region is yet possible because of the inherent difficulties to measure magnetic helicity in the Sun.

Scientific barrier: how to estimate magnetic helicity?

Despites its potential importance, study of this quantity has been overall relatively limited (cf; reviews of Démoulin 07, Démoulin & Pariat 09, Pevtsov et al. 14). This is related to the fundamental difficulties to estimate this quantity, even in controlled environments (laboratory or numerical experiments). Indeed, unlike classical quantities magnetic helicity is a non-local integral quantity. Because of the freedom of the choice of the gauge on the magnetic vector potential magnetic helicity density is not a uniquely defined quantity. Magnetic helicity was first defined and measured only in magnetically bounded system, i.e. systems which are bounded by a surface to which the magnetic field is everywhere tangent. Such condition precludes the direct utilization of magnetic helicity in most application to natural plasma, where no such domain naturally exists.

Hopefully, Berger & Fields 1984 have demonstrated that a derived quantity, called relative magnetic helicity, defined as the difference between the magnetic helicity of the studied domain with the helicity of a reference field, could be generally used to estimate magnetic helicity in any volume. This seminal work opened the way for the study of magnetic helicity in natural plasmas.

Even with this new theoretically useful quantity, magnetic helicity (for simplicity this term will be kept hereafter while usually referring to relative magnetic helicity) is not easily measured in the solar environment. Regular measurements of magnetic helicity in the solar atmosphere have only started in the last 15 years.

Magnetic helicity normally require the precise knowledge of the three-dimensional (3D) structure of the magnetic field. However, in the solar context magnetic field is only measured on the 2D visible-light emitting “surface” (photosphere) of the Sun. Two-dimensional maps of the magnetic field distribution at the Sun photosphere are now produced daily by space satellites (e.g. HMI instrument onboard the SDO mission) and by ground based observatory. No direct estimation of magnetic helicity is therefore possible and complex indirect methods are required. Therefore, the very theory of the measurements of this quantity is still partly an exploratory field of research. HeliSol shall develop, improve and observationally implement diverse methods to properly estimate magnetic helicity in the solar environment. This lead to the second of the problematic addressed by the HeliSol project: How to efficiently measure magnetic helicity and its evolution in the different layers of the Sun?

Two methods are frequently used to estimate the magnetic helicity in the Sun. The first method, the volume method, aims at estimating the magnetic helicity in the coronal volume of the solar atmosphere. Since magnetic field measurements are only made at the solar surface, it is necessary to use a model in order to reconstruct the 3D coronal magnetic field. Such 3D reconstructions are episodically done for given active regions on the Sun. The service Fromage (FRench Online MAGnetic Extrapolations), administrated by G. Aulanier, partner of this project, of the Bass2000 data base, “Service d’Observation” center of the Conseil National des Astronomes et Physiciens (CNAP), is an example of service producing 3D reconstruction of observed magnetic field and open to the French and International community (http://www.lesia.obspm.fr/fromage/). State of the art methods for extrapolations are also developed by G. Valori, collaborator to the HeliSol project (e.g. Valori et al 11, 13).

From the 3D magnetic field atmospheric reconstruction, but also more directly from 3D magnetic field data sets of numerical simulations, the team involved in the HeliSol project recently developed a method and algorithm that enable the proper estimation of magnetic helicity (Valori et al. 12). Within the HeliSol project this method will be applied to reconstructed observations of the solar magnetic field and to numerical simulations of solar phenomena in which magnetic helicity is present. One of the direct problems that will be tested is the sensibility of the volume helicity estimation to the different models of the reconstruction of the 3D coronal magnetic field. This will enable a better understanding of the validity of the estimation method in the solar corona, in this way asserting the precision of helicity measurement in the solar atmosphere.

The second method to estimate the helicity in the solar corona relies on the estimation of the flux of helicity through the solar surface. By integrating in time the flux of helicity it is possible to estimate the accumulation of helicity in the solar atmosphere. This now most widely used method has first been applied to solar active regions by Chae et al. 01. The estimation method only requires time series of line-of-sight magnetograms. From the magnetogram time series it is possible to extract the transverse flow velocity of the magnetic field (called flux transport velocity) using local correlation tracking methods (see review of Démoulin & Pariat 09). Pariat et al. 05, 06 provided a series of methods that could be used to properly determine the helicity flux which improved the existing methods of estimation. In the pipeline to compute helicity flux numerous approximation are done at different stages that can influence the helicity estimation (e.g. magnetic field measurements, velocity field extraction, helicity proxy). One of the subgoals of the HeliSol project is to better determine how sensitive magnetic helicity flux estimations are to each of these stages. To this purpose, we will test how helicity measure depends on the magnetic-field inversion method (Bommier et al. 07), on the flux transport velocity parameters, and on the helicity estimation method.

The two helicity estimation methods are thus complementary and provide different views of the helicity property of active regions. The helicity flux accumulation method is only providing the injection of helicity through the bottom boundary of active regions and is hence not able to track helicity that would be ejected during a solar eruption via the twisted flux rope that form a CME. On the other hand the volume methods is strongly reconstruction-model dependant and is more effort consuming (time, resource and expertise) than the helicity flux method.

The combined analysis of both these methods allows however a more complete description of the helicity evolution of the studied system. Early comparative benchmarking of these methods have been performed (Lim et al 07) and are being carried by an ISSI team (cf Section 1.6) which demonstrated a good level of coherency between these 2 methods. The HeliSol project will push forward these comparisons by systematically using these combined methods to analyze the evolution of helicity in the case of observed active region.

In addition, the magnetic helicity flux measures can be used in conjunction with the coronal reconstruction to determine the 3D distribution of the injection of magnetic helicity in the solar corona (Pariat 05, Dalmasse et al. 14). Dalmasse et al. 13 have used this new method to properly map in 3D the distribution of magnetic helicity within solar region (cf. Figure 3). It is thus possible to determine not only the amount of helicity that is being injected in active region but to determine where and how this helicity is being injected within active regions.

Strong of this recent proof of concept, HeliSol will analyze regions which display simultaneous injection helicity of opposite sign. This type of regions is believed to produce more energetic flares since helicity annihilation by the interaction of opposite sign helicity is believed to allow the release of a larger amount of magnetic energy (cf. 1.5). Through the HeliSol project we will thus compare the eruptivity and strength of active region in regards of their helicity content.


Figure 2: Helicity conservation in a numerical simulation of a solar active event (Pariat et al. 15). Right: snapshots of the magnetic field lines configuration (Helioviewer). Left: comparison of the helicity in the domain and its flux through the boundaries: helicity conservation is smaller than 2%.

Magnetic helicity and solar flares/eruptions

The central mechanism acting in solar flares/eruptions is magnetic reconnection. This universal mechanism of highly conducting plasmas (stellar atmospheres, Earth magnetosphere, controlled nuclear fusion laboratory devices) allows the release of the stored magnetic energy and its transformation in kinetic and thermal energies. Magnetic reconnection allows the dynamic and rapid reconfiguration of the magnetic field which is otherwise and most of the time locked in specific configuration.

While reconnection transform the magnetic energy, Taylor 1974, in order to interpret the dynamics of laboratory plasma devices, conjectured that magnetic helicity would also be quasi-conserved even if non-ideal MHD processes are acting (as in magnetic reconnection). Because of the inherent difficulty to measure magnetic helicity, even on numerical dataset (cf. Section 1.3.2), this conjecture has been little tested, both experimentally and numerically, over the years and only with limited precision (e.g. Barnes et al. 86, Ji et al. 95, Zhang 13).

Thanks our recent advance in magnetic helicity estimation methods, the team of the HeliSol project has recently demonstrated that magnetic helicity was indeed extremely well conserved in an solar active-like event (Pariat et al. 15, cf. Figure 2). This fundamental result, to this day the clearest confirmation of Taylor’s conjecture, indicates that magnetic helicity can be an extremely useful quantity to describe solar eruptions. However this results needs to be further tested and confirmed with different codes and on different test-cases. Within the HeliSol project, we indeed to further constrain the precision and the range of validity of the conservative property of magnetic helicity in non-ideal MHD.

The underlying reason of the conservation of magnetic helicity is related to its inverse cascade property. While magnetic energy tends to cascade from larger to smaller spatial scales where it is dissipated, magnetic helicity is noted to evolve from smaller to larger scale, hence dissipated much less efficiently (Frisch et al. 1975). During magnetic reconnection, as the system is reconfiguring, the magnetic helicity that was present in the field lines is redistributed. Magnetic helicity is transferred between different subdomain. Magnetic helicity is in particular used to form the large twisted flux ropes which constitute the magnetic backbone of the erupting CMEs. The precise redistribution of magnetic helicity during solar/flare and eruption is still unclear and the HeliSol project aims to answers the following question: How does magnetic helicity evolve during the violent solar eruptions?

In order to tackle this problem, we will use different state-of-the art numerical simulations of solar flare/eruptions (e.g. Aulanier et al. 10, Masson et al. 13) to track the evolution of magnetic helicity during the generation of eruptive flares. Using our existing methods to measure magnetic helicity we will follow the evolution of magnetic helicity and its ejection within the CMEs. We will measure the condition of formation of the flux rope and the accumulation of helicity as they are formed and erupt. This will also requires the development of new methods and algorithm to track and measure helicity in subdomains and/or in different geometry (e.g. spherical geometry). These new methods will then be readily available to the community for further research beyond HeliSol.

Furthermore, it has also been observed that magnetic helicity directly impacts the very dynamics of magnetic reconnection. Magnetic helicity provide a lower bound to the minimum energy that can be stored in an isolated system (Woltjer 1958). For non isolated systems, the ejection of magnetic helicity, the annihilation of helicity by the interaction of subdomains of helicity having different sign, can lead to very different dynamic of the magnetic reconnection and the release of different amount of magnetic energy. The numerical simulations of Linton et al. 01 and Del Sordo et al. 10 show that when systems of opposite helicity interact by magnetic reconnection, a larger amount of magnetic energy can be dissipated. This heuristic result encourages us to look at observed active regions of opposite magnetic helicity (cf. Section 1.3.2 and Figure 3).

In order to better understand the dynamic of helicity during magnetic reconnection, HeliSol will study the transfer of helicity during reconnection in the previous numerical simulations as well as in new simulation set-up to study the annihilation of magnetic helicity. In order to carry out this project we will implement the computation of a promising quantity, field line helicity (Russel et al. 15, Yeates & Hornig 14) which enable the study of helicity per elementary magnetic element. This quantity will enable the precise tracking of magnetic helicity as it is redistributed by magnetic reconnection. The HeliSol project should provide the first opportunity to follow this quantity in numerical simulations of active like events.

The final goal of the HeliSol project is to push forward the prediction of solar eruptions. Based on the insights gained from answering the previous questions, we aim at answering the question: Is it possible to determine a proxy based on magnetic helicity that would enable effective solar flare/eruption prediction?

The existence of a threshold for eruption is a theoretically debated issue (e.g. Amari et al. 03, Jacobs et al. 06, Zhang & Low 06). Ideal instability to trigger eruption such as the kink and torus instabilities involve a threshold in twist and/or electric currents (cf. Démoulin & Aulanier 10). While in the eruption context both these quantities are linked to magnetic helicity, the theoretical description of these instabilities has never been really performed. From the numerical simulation side, because, until-now, no method was properly able to compute helicity in 3D datasets, no clear result has been obtained. From the observational sides, preliminary results seem to indicate that a threshold amount of magnetic helicity is a necessary condition for the generation of flares/CMEs (Nindos et al. 04, Tziotziou et al. 12).

Thanks to our state-of-the art method to compute helicity, the HeliSol project will unable a careful study of the pre- and post-eruption conditions in term of magnetic helicity. By studying helicity accumulation before and its evolution during solar eruption, the HeliSol project will derive several proxy based on magnetic helicity and determine if one can be used as a predictive tool for flares/eruptions. Magnetic helicity can indeed be decomposed in multiple ways: e.g. self and mutual helicity, twist and writhe helicity; non-potential and cross helicity. By tracking these quantities both in the simulated and observed cases we will determine whether a helicity based proxy can be used as a good deterministic proxy of solar eruptions, hence improving the existing prediction scores of space weather centers.

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