estimating the relative helicity of coronal magnetic fields

Solar Phys (2011) 272:243–255DOI 10.1007/s11207-011-9826-2

Estimating the Relative Helicity of Coronal MagneticFields

J.K. Thalmann · B. Inhester · T. Wiegelmann

Received: 13 April 2011 / Accepted: 16 July 2011 / Published online: 1 September 2011© Springer Science+Business Media B.V. 2011

Abstract To quantify changes of the solar coronal field connectivity during eruptive events,one can use magnetic helicity, which is a measure of the shear or twist of a current-carrying(non-potential) field. To find a physically meaningful quantity, a relative measure, givingthe helicity of a current-carrying field with respect to a reference (potential) field, is oftenevaluated. This requires a knowledge of the three-dimensional vector potential. We presenta method to calculate the vector potential for a solenoidal magnetic field as the sum of aLaplacian part and a current-carrying part. The only requirements are the divergence free-ness of the Laplacian and current-carrying magnetic field and the sameness of their normalfield component on the bounding surface of the considered volume.

Keywords Helicity, magnetic · Magnetic fields, corona

1. Introduction

During solar eruptions like flares or coronal mass ejections, part of the previously storedmagnetic energy is transformed into other forms of energy, such as kinetic energy or heat.Along with this energy release, the corona often undergoes changes in the field line con-nectivity. A favorable way of quantifying these topological changes of magnetic fields isto investigate the magnetic helicity, which is a measure of the shear or twist of a current-carrying (non-potential) field. The magnetic helicity is (almost) conserved in (resistive) idealmagnetohydrodynamics since its dissipation time is long compared to the time during whichenergy is dissipated. The major change of coronal helicity is assumed to be due to upwardconvection from below the solar surface and advection into interplanetary space by coronalmass ejections. Hence, the coronal helicity content may be completely determined by theflow of helicity through the photosphere and its loss rate through the solar wind.

J.K. Thalmann (�) · B. Inhester · T. WiegelmannMax-Planck-Institut für Sonnensystemforschung, Max-Planck-Str. 2, 37191 Katlenburg-Lindau,Germanye-mail: [email protected]

mailto:[email protected]

244 J.K. Thalmann et al.

For a volume V which is enclosed by a surface ∂V through which no magnetic fieldpenetrates (a “magnetically closed” volume where n · B = Bn = 0 on ∂V and where n is theunit outward normal), the magnetic helicity is

Hm =∫

VA · B d3x, (1)

where the magnetic helicity density hm = A · B (Elsasser, 1956). Here, A denotes the mag-netic vector potential and B = ∇ × A is the solenoidal (∇ · B = 0) magnetic field, fullycontained in V . So far, A is defined up to an arbitrary gauge field ∇ξ which, however, doesnot affect Equation (1). In fact, for a magnetically closed V , a gauge transform A → A+∇ξ

yields no change of the value of the helicity integral (Equation (1)) so that the helicity Hm

is indeed gauge independent and thus well defined (Berger, 1984, 1999a, 1999b; Bellan,1999). However, the definition of the magnetic helicity density hm is gauge dependent andtherefore without physical meaning.

For the solar corona and many other applications, the considered volume V is magneti-cally open since magnetic flux emerges through the photosphere from below (thus, Bn �= 0on ∂V ). Any change of the gauge then results in a change of Equation (1); i.e., Hm nolonger provides a physically meaningful quantity. Several attempts have been made to solvethis gauge problem, in particular, by restricting the gauge freedom of A (for a review seeDémoulin, 2007).

A useful modification of Equation (1) is the relative helicity of a field B with respect tothat of a reference field B = ∇ × A in the form

Hrel =∫

V(A + A) · (B − B)d3x, (2)

with the condition Bn = Bn on ∂V (Berger and Field, 1984; Jensen and Chu, 1984; Finn andAntonsen, 1985). Since the field difference B − B yields no flux through ∂V , Equation (2)remains invariant if the gauge potentials ∇ξ and ∇ ξ are added to A and A, respectively.

An alternative has been proposed by Berger and Field (1984) as

H ′rel =

∫V∪U

(A · B − A · B)d3x, (3)

where the magnetic field B is extended from volume V beyond that part of ∂V which ispenetrated by magnetic flux into an adjacent volume U such that Bn = 0 on the surface ofV ∪ U . Consequently, H ′

rel is gauge independent. The form of B in the extended volume Uhas no influence on H ′

rel if the reference field obeys B = B in U . Moreover, if the vectorpotential of the reference field is chosen such that

n × A = n × A (4)

on the surface of V , the extended volume U makes no contribution to H ′rel at all. For this

reason, this gauge restriction on A and A is often applied in theoretical calculations. Foridentical reference fields B in V , Hrel (Equation (2)) and H ′

rel (Equation (3)) are equivalent(Finn and Antonsen, 1985).

Whenever time-dependent magnetic flux penetrates the walls of a volume V , helicitytransport becomes an issue. For an electric field given by Maxwell’s equations and B = ∇ψ

Estimating the Relative Helicity of Coronal Magnetic Fields 245

chosen as a potential field, Berger and Field (1984) derived an expression for the helicityflux through ∂V :

dtHrel = −2∫

VE · B d3x + 2

∫∂V

(A × E − ∂tψA) · n dS, (5)

where ∇ · A = 0 in V is assumed. The volume integral measures the internal helicity dissi-pation and the surface integral quantifies the flow of relative helicity across ∂V . Note thatin coronal processes the former is negligible, and so the helicity transport is largely deter-mined by the latter, i.e., only requiring the knowledge of A, the current-carrying field, andthe plasma velocity on ∂V . In addition, the last term in the integrand of Equation (5) vanishesif

n · A = 0 (6)

is required on ∂V . The convenience of the resulting expression must be paid by giving upthe remaining gauge freedom: given B, B is determined from Bn = Bn on ∂V , A is uniquelyfixed by An = 0 (Equation (6)) and A by the requirement At = At (Equation (4)).

Although the above-mentioned gauge constraints (Equations (4) and (6)) are often usedin theoretical considerations, they seem to be a burden in practical cases where the helicityhas to be calculated numerically from a given field, e.g., extrapolated from observed surfacefields. In this paper, we give a recipe for the calculation of the vector potentials A and Aobeying the above-described constraints so that Equations (3) and (5) can also be used forpractical computations in their simplified form.

2. Practical Computation of the Relative Helicity

To compute the relative helicity in the solar corona, we follow Kusano, Suzuki, andNishikawa (1995) and decompose the magnetic field as B = B + B. By assumption, B isa closed current-carrying field fully contained in V , while B is an open current-free (poten-tial) field. The magnetic field is assumed to be solenoidal, B has the same distribution as Bon ∂V (i.e., Bn = Bn), and Bn = 0.

We assume the Coulomb gauge for A and A and solve

�A = −μ0j in V, (7)

∇ · A = 0 in V, (8)

n · ∇ × A = Bn on ∂V, (9)

where μ0 is the permeability of free space and j is the electric current density.The computation is performed in two steps. We first solve the inhomogeneous problem

for A = A − A, i.e.,

�A = −μ0 j in V, (10)

∇ · A = 0 in V, (11)

n × A = 0 on ∂V, (12)

where the boundary condition (12) is a direct consequence of constraint (4). We wish tofind an inhomogeneous solution to Equation (10), taking the electric currents into account.


A gauge A = A + ∇ ξ requires �ξ = 0 in V with n × ∇ ξ = 0 on ∂V from Equation (12). Itimmediately follows that ξ is constant on ∂V and hence the solution of Equations (10) – (12)is unique.

Subsequently, we solve the homogeneous problem

�A = 0 in V, (13)

∇ · A = 0 in V, (14)

n · ∇ × A = Bn on ∂V. (15)

We search the set of solutions to the homogeneous problem (Equation (13)), accounting forthe magnetic flux Bn = Bn through the boundaries. A gauge A = A + ∇ ξ requires �ξ = 0in V . However, the gauge field ξ is not affected by boundary condition (15); thus, ξ isstill ambiguous on ∂V and we can use this freedom to fix A on ∂V with the boundaryrequirement (6).

2.1. Numerical Implementation in a Cube

First, we solve the inhomogeneous problem (Equation (10)) in a cubic box and considerthree scalar Poisson problems for each Cartesian component i = (x, y, z) of the magneticvector potential

�Ai = −μ0ji, (16)

subject to the mixed Dirichlet/Neumann boundary conditions

Ai = 0, where ei ⊥n, (17)

∂iAi = 0, where ei ‖ n (18)

on ∂V . Here, Equation (18) guarantees that ∇ · A = 0 on ∂V and thus in the entire domain.For a vanishing current density (j = 0), A also vanishes, so that the vector potential foundup to now yields the current-carrying magnetic field B = B − B only.

We then solve the homogeneous problem (Equation (13))

�Ai = 0, (19)

subject to the Dirichlet boundary condition

Ai = given, where ei ⊥n, (20)

so that n · ∇ × A = Bn and

∂iAi = 0, where ei ‖ n (21)

on ∂V . The yet unknown tangential boundary values in Equation (20) are chosen so that∇t · At = 0. With this choice, Equation (21) yields ∇ · A = 0 on the boundary and hence onthe whole of V . The solution of Equation (19) is now well defined up to an arbitrary additiveconstant. In order to guarantee the vanishing tangential divergence on ∂V , we write At interms of a two-dimensional (2D) stream function of a Laplacian field φ, defined only on thesurface ∂V ,

At = −n × ∇φ. (22)


By insertion into Equation (15) we obtain

�tφ = −Bn. (23)

We solve Equation (23) separately on each face of our computational domain and denoteeach solution by φf , where f = (1, . . . ,6) stands for the 2D face of our cubic computationalvolume. We impose the Neumann boundary condition

∂nφf = σf eAe, (24)

where Ae are constant components of A along the edges e of each face to be determined.The unique values of A along each edge ensure that A is continuous on the entire surface.However, we have to introduce factors σf e = {+1,−1} to account for the different orienta-tions an edge has with respect to the two faces to which it is connected. The parameters σf e

ensure that the sum over all four values of σf eAe on the edges of a face yields the correctcirculation around the face. This way, the circulation on each face gives the correct outflow:

�f =∫

f

B · dS =∮

∂f

A · dl =4∑

e=1

σf eAe Le. (25)

Here �f denotes the net flux through a face and Le represents the length of the edge. For ourcubic computational box, we have 12 edges with unknown values Ae . On the other hand,Equation (25) gives only six flux-balance equations to determine the 12 unknown Ae . Tofind a unique solution, we supplement Equation (25) by requiring

∑12e=1 A2

e to be minimal.Once these values are found, we can supply the boundary condition (24) to Equation (23)and find φf on each face and thus At from Equation (22).

To summarize, the total computational work load includes the solution of:

– one algebraic equation to find the 12 unknowns Ae (Equation (25))– six 2D Laplace problems to determine φf ((Equation 23))– three 3D Laplace problems to find A (Equation (19))– three 3D Poisson problems to find A (Equation (16)).

The vector potential A = A + A calculated so far satisfies Equation (4). The additionalcondition (6) can be introduced by solving another Laplace problem of the form �ξ = 0subject to ∂nξ = −An. The condition A′

n = 0 then holds for the new vector potential A′ =A + ∇ ξ .

3. Testing the Method

3.1. Force-Free Test Solutions

For the atmospheric layers from the mid-chromosphere to the mid-corona, the solar at-mosphere is regarded as being almost entirely force-free (Gary and Alexander, 1999;Gary, 2001). Then a proportionality between the electric current density j and the mag-netic field B is found in the form ∇ × B = μ0j = αB, where α is the force-free parameterand is generally a function of space. Simpler approximations to the coronal magnetic fieldare potential fields (α = 0) and linear force free fields (α = const, assuming j to be linearlyproportional to B). The most general approximations are nonlinear force-free fields where


the proportionality between j and B is no longer linear (α = α(x)). Except for a few existinganalytical solutions of nonlinear force-free magnetic fields (see, e.g., Low, 1988), the relatedset of equations must be solved numerically.

In this work we use as an example the semi-analytic set of separable solutions of Low andLou (1990) describing nonlinear force-free fields in axisymmetric geometry, associated witha point source placed at the origin. These solutions contain free parameters for the locationof the point source in Cartesian coordinates somewhere outside the considered volume Vand the orientation of the axis of symmetry with respect to the edges of V . We calculatesemi-analytic equilibria in cubic boxes with four different grid resolutions. The smallestcubic domain V1 is of dimension Lx × Ly × Lz = 20 × 20 × 16 pixels with a grid spacingof, say, h1. Here, Lx , Ly , and Lz denote the length of the cubic box along the x-, y-, andz-directions in pixels, respectively, where the origin of the Cartesian coordinate system isplaced at (x, y, z) = (0,0,0). We enhance the grid spacing by a factor of two so that thenext larger volume V2 is of dimension 40 × 40 × 32 pixels and with a grid spacing ofh2 = h1/2. Correspondingly, the volumes V3 and V4 are of dimensions 80 × 80 × 64 pixelsand 160 × 160 × 128 pixels, respectively, with a grid spacing of h3 = h1/4 and h4 = h1/8,respectively. The origin of the analytic model field is placed at (x, y, z) = (0,0,−0.3Lz)

and has its symmetry axis oriented at � = π/2 with respect to the physical z-axis. We useBn on ∂V of these 3D force-free magnetic fields B to calculate the corresponding potentialfield B = ∇ψ , where ψ is the solution to the Laplace equation with the Neumann boundarycondition ∂nψ = Bn. Thus, Bn = Bn on ∂V as required for our relative helicity computations.

3.2. Evaluation of the Method

After solving for the total vector potential A, we have to test if its curl correctly reproducesthe input magnetic field B and the associated electric currents j. Therefore, we calculateB� = ∇ × A and j� = ∇ × B� and check how well they agree with the input model fields Band j, respectively (ideally, B� = B and j� = j).

To quantify the degree of agreement between a given vector field X and a field X� to betested, we use some of the metrics introduced by Schrijver et al. (2006). First, we test theagreement by calculating the vector correlation Cvec, which is defined as

Cvec(X,X�) =∑

i Xi · X�i√∑

i |Xi |2 ∑i |X�

i |2, (26)

where X�i and Xi are two arbitrary 3D vectors fields at each point i. For two identical vector

fields, Cvec = 1. To quantify how well the energy contained in X is reproduced, we normalizethe magnetic energy of the field X� to be tested to that of X. As a global measure, we use

Cerg(X,X�) =∑

i |X�i |2∑

i |Xi |2 , (27)

where, ideally, Cerg = 1 for identical energy contents.As can be seen in Table 1, the vector alignment of the magnetic fields and the electric

current density (as denoted by Cvec(B,B�) and Cvec(j, j�), respectively) increases with in-creasing grid resolution. The relative error of Cvec(B,B�) decreases from ≈1% in V1 to≈0.1% in V4, and the corresponding values of Cvec(j, j�) drop from ≈3% in V1 to ≈0.2%in V4 (see the third and fourth columns in Table 1, respectively). Also the match of the en-ergy content Cerg(B,B�) clearly depends on the grid resolution, even more so than does the


Table 1 Vector correlation Cvec and energy correlation Cerg (as defined in Equations (26) and (27), respec-tively) of the model fields (B, j) and the fields B� and j� as calculated from the vector potential A, whereB� = ∇ × A and j� = ∇ × B� . V1, V2, V3, and V4 are boxes with subsequently finer resolutions of h1, h1/2,h1/4, and h1/8, respectively, and of dimensions 20 × 20 × 16 pixels, 40 × 40 × 32 pixels, 80 × 80 × 64pixels, and 160 × 160 × 128 pixels.

Cube Resolution Cvec(B,B�) Cerg(B,B�) Cvec(j, j�)

V1 h1 0.98768 0.89011 0.96664

V2 h1/2 0.99447 0.98817 0.98134

V3 h1/4 0.99979 1.00377 0.99395

V4 h1/8 0.99996 1.00363 0.99836

Figure 1 Normalized root-mean-square deviation Erms (vertical axis) as function of the grid resolution h

(horizontal axis). Black (gray) filled circles represent the values of Erms(B,B�) and Erms(j, j�), respectively,and the dotted lines are corresponding power-law fits.

vector correlation. The relative error in the energy content Cerg(B,B�) drops from ≈10% inV1 to ≈0.3% in V4 (see the fifth column in Table 1).

Then, we estimate the normalized root-mean-square deviation Erms. This means that, tomeasure the average distance between two fields, we employ the root-mean-square deviationand divide it by the range of values of the reference field X in the form

Erms(X,X�) = E′rms(X,X�)

Xmax − Xmin, (28)

with Xmax and Xmin denoting the maximum and minimum values of X, respectively, and

E′rms(X,X�) =

√1N

∑N

i=1(Xi − X�i )

2

∑i |Xi | , (29)

where N is the number of elements of X. If the two fields X and X� are identical, Erms = 0.As expected, Erms(B,B�) and Erms(j, j�) decrease with increasing grid resolution (see

Figure 1). After fitting a power law, we find that the normalized root-mean-square deviationof the magnetic field scales as Erms(B,B�) ∝ h1.6. The corresponding values of the currentdensity scale as Erms(j, j�) ∝ h1.0, i.e., decrease linearly with the grid resolution.

To give a visual impression of the quality of our method, in Figures 2 and 3 we compareB and B� = ∇ × A for V4. As described above, this is the volume with the finest grid resolu-


Figure 2 Comparison of B (first row) and B� = ∇ × A (second row) at (x, y, z = 0), i.e., the bottom bound-ary of V4, which is a cubic box of dimension Lx × Ly × Lz = 160 × 160 × 128 pixels. The third row showsthe difference B−B� . The left, middle, and right columns show the x-, y-, and z-components of the magneticfields, respectively.

tion, with dimension 160 × 160 × 128 pixels. ‘In Figures 4 and 5 the corresponding electriccurrent densities j = ∇ ×B and j� = ∇ ×B� are shown. The bottom boundary of the compu-tational box at (x, y, z = 0), where the magnetic field and the electric currents are strongest,is shown in Figures 2 and 4, respectively. A vertical cut through the cubic computationaldomain at (x, y = Ly/2, z) is shown in Figures 3 and 5. As already found quantitatively, themagnetic fields B and B� show a very good vector alignment, as do the current densities jand j�.

4. Discussion

To evaluate the magnetic helicity content of the solar corona, a knowledge of the magneticvector potential and the associated magnetic field vector is needed. The tools for extrapo-


Figure 3 Comparison of B (first row) and B� = ∇ × A (second row) at (x, y = Ly/2, z), i.e., a vertical cutthrough V4, which is a cubic box of dimension Lx × Ly × Lz = 160 × 160 × 128 pixels. The third rowshows the difference B − B� . The left, middle, and right columns show the x-, y-, and z-components of themagnetic fields, respectively. For better visibility, the fields are displayed between z = 0 and z = Lz/2.

lating the force-free magnetic field in the outer solar atmosphere, for which routine mea-surements do not exist, are well established. We presented a method to calculate the vectorpotential for a given solenoidal magnetic field as the sum of a Laplacian part and a current-carrying part. We demonstrated that our method to calculate the magnetic vector potential isaccurate to within a few percent concerning the reproduction of the vector orientation andenergy content of a given magnetic field. Although tested with the help of a nonlinear force-free field model, we emphasize that our method is not restricted to magnetic fields that areconsidered to be force-free. The knowledge of the vector potential allows us to calculate therelative magnetic helicity as well as its flow across boundaries for realistic solar cases as ameaningful quantity.

In several studies, the magnetic helicity of coronal magnetic fields has been investigatedand partly compared to the helicity injection through the solar photosphere. In these studies,


Figure 4 Comparison of j = ∇ × B (first row) and j� = ∇ × B� = ∇ × ∇ × A (second row) at (x, y, z = 0),i.e., the bottom boundary of V4, which is a cubic box of dimension Lx × Ly × Lz = 160 × 160 × 128pixels. The third row shows the difference j − j� . The left, middle, and right columns show the x-, y-, andz-components of the electric current densities, respectively.

the current-carrying coronal magnetic field is usually approximated using linear or nonlin-ear force-free fields. For instance, Démoulin et al. (2002) used a linear force-free approachto show that the sign of the relative magnetic helicity associated with a simple bipolar andisolated active region corresponded to the hemisphere where the active region emerged.Nindos and Andrews (2004) found that the pre-flare coronal helicity tends to be smallerwhen flares without associated coronal mass ejection occur. Georgoulis and LaBonte (2007)concluded that comparing the helicity budget of solar active regions might be a safe wayto distinguish between eruptive and non-eruptive regions. However, Régnier, Amari, andCanfield (2005) explicitly showed that the relative helicity can be very different in absolutevalue and sign when estimated from a linear or nonlinear force-free field, matched fromthe same boundary conditions. In fact, a linear force-free approach generally approximatesshorter (longer) field lines less (more) sheared than they actually are. Thus, the amount of


Figure 5 Comparison of j = ∇ × B (first row) and j� = ∇ × B� = ∇ × ∇ × A (secondrow) at (x, y = Ly/2, z), i.e., a vertical cut through V4, which is a cubic box of dimensionLx × Ly × Lz = 160 × 160 × 128 pixels. The left, middle, and right columns show the x-, y-, andz-components of the electric current densities, respectively. For better visibility, the fields are displayed be-tween z = 0 and z = Lz/2.

helicity present in the solar corona is likely to be underestimated, and the more general non-linear force-free approximation is necessary to improve the match of the coronal helicitycontent (Démoulin, 2007). Nonlinear force-free field models have been used in several stud-ies finding, e.g., support of the hemispheric sign trend (Régnier, Amari, and Kersalé, 2002;Régnier and Priest, 2007), the injection of magnetic helicity before a flare onset (Régnierand Canfield, 2006), and a good correlation between the time profile of the coronal helicityand the accumulation by injection through the photosphere (Park et al., 2010).

Until now, the investigation of active-region magnetic fields related to large flares hasbeen restricted due to the limited availability of vector magnetograph data necessary for thenonlinear force-free extrapolations. This drawback will be compensated as soon as data fromthe Solar Dynamics Observatory Helioseismic and Magnetic Imager (Graham et al., 2003)


is available. With its high temporal and spacial resolution we will be able to study the rela-tive magnetic helicity and the topology of the pre- and post-eruptive fields in unprecedenteddetail. The application of our method to estimate the helicity content above solar active re-gions will enable us to compare it to the amount of helicity injected through the photosphereand its contribution to the coronal helicity content. Our nonlinear force-free models enableus to understand the evolution and transport of the helicity density. The magnetic helicityis known to dissipate more slowly than the magnetic energy, and helicity injection by linetying is also a relatively slow process. If we find a significant change of the helicity contentin our magnetic field models, this could mean that the corresponding part has been ejectedinto interplanetary space.

Acknowledgements J.K. Thalmann was supported by DFG grants WI 3211/1-1 and WI 3211/2-1. B. In-hester and T. Wiegelmann were supported by DLR grant 50 OC 0904.

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estimating the relative helicity of coronal magnetic fields

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