Magnetic Energy and Helicity Budgets in the Active-Region Solar Corona. I. Linear Force-Free Approximation
aa r X i v : . [ a s t r o - ph ] J un Magnetic Energy and Helicity Budgets in the Active-Region SolarCorona. I. Linear Force-Free Approximation
Manolis K. Georgoulis & Barry J. LaBonte † The Johns Hopkins University Applied Physics Laboratory,11100 Johns Hopkins Rd. Laurel, MD 20723-6099, USA
ABSTRACT
We self-consistently derive the magnetic energy and relative magnetic helic-ity budgets of a three-dimensional linear force-free magnetic structure rooted ina lower boundary plane. For the potential magnetic energy we derive a gen-eral expression that gives results practically equivalent to those of the magneticVirial theorem. All magnetic energy and helicity budgets are formulated in termsof surface integrals applied to the lower boundary, thus avoiding computation-ally intensive three-dimensional magnetic field extrapolations. We analyticallyand numerically connect our derivations with classical expressions for the mag-netic energy and helicity, thus presenting a so-far lacking unified treatment ofthe energy/helicity budgets in the constant-alpha approximation. Applying ourderivations to photospheric vector magnetograms of an eruptive and a nonerup-tive solar active regions, we find that the most profound quantitative differencebetween these regions lies in the estimated free magnetic energy and relativemagnetic helicity budgets. If this result is verified with a large number of activeregions, it will advance our understanding of solar eruptive phenomena. We alsofind that the constant-alpha approximation gives rise to large uncertainties inthe calculation of the free magnetic energy and the relative magnetic helicity.Therefore, care must be exercised when this approximation is applied to photo-spheric magnetic field observations. Despite its shortcomings, the constant-alphaapproximation is adopted here because this study will form the basis of a com-prehensive nonlinear force-free description of the energetics and helicity in theactive-region solar corona, which is our ultimate objective.
Subject headings:
MHD — Sun: atmosphere — Sun: chromosphere — Sun:corona — Sun: magnetic fields — Sun: photosphere † Deceased October 24, 2005
1. Introduction
The magnetic origin of solar eruptions has been established over the past several decadesof solar research. Most eruptions originate in active regions that are, in general, closedmagnetic structures rooted in the solar photosphere. Magnetized plasma motions in thesolar atmosphere prevent a magnetic structure from attaining a minimum-energy, current-free state. Excess magnetic energy in active regions is manifested by the appearance ofelectric currents (Leka et al. 1996). Eruptive and noneruptive manifestations, such ascoronal mass ejections (CMEs) and confined solar flares, respectively, must be fueled fromthis reservoir of free magnetic energy that is thought to be released in intermittent episodesof magnetic reconnection.A popular, early, view of a magnetic energy release event involved a nonpotential pre-event state relaxing into a potential, or nearly potential, post-event state of the magneticconfiguration. The excess (nonpotential) magnetic energy was thought to be released duringthe relaxation. Seminal works on magnetic helicity (Woltjer 1958; Cˇalugˇareanu 1961; Berger& Field 1984; Finn & Antonsen 1985; Berger 1985; 1988; Moffatt & Ricca 1992 and others),however, demonstrated that this view is incomplete or even misleading: magnetic helicityrelates to the linkage of a magnetic structure (twist, torsion, and writhe), which is globally invariant even under resistive processes, such as magnetic reconnection. Helicity is presentwherever electric currents are present. Therefore, an isolated helical magnetic structurecannot relax to a potential state unless its magnetic helicity is bodily removed from it.This provides a plausible interpretation for CMEs (Low 1994; Rust 1994a; 1994b), providedthat the magnetic helicity in the erupting structures is not transferred to other parts ofthe solar atmosphere along preexisting or reconnected magnetic field lines. Indeed, eruptiveactivity seems to be necessary for the Sun where differential rotation and subsurface dynamocontinuously generate helicity in the two solar hemispheres (Berger & Ruzmaikin 2000), witha statistical hemispheric segregation exhibited by magnetic structures of opposite senses ofhelicity (Pevtsov, Canfield, & Metcalf 1995). In confined events the magnetic configurationcannot relax to the potential state but, at best, it may relax to the lowest possible energy statethat preserves the pre-event amount of magnetic helicity. This is known to be a constant-alpha, linear force-free (LFF) state (Woltjer 1958; Taylor 1974; 1986). Several works rely onthe Woltjer-Taylor theorem, although controversy remains over its applicability to the Sun(Kusano et al. 1994, but also Antiochos & DeVore 1999).Whether magnetic helicity per ce is important for solar eruptions is also a subject ofdebate (e.g. Rust 2003; Rust & LaBonte 2005, but also Phillips, MacNeice, & Antiochos2005). Regardless, however, knowledge of the magnetic helicity is essential for a completeassessment of the magnetic complexity present in the solar atmosphere. Berger & Field 3 –(1984) and Finn & Antonsen (1985) derived a gauge-invariant definition of magnetic helicityapplying to open and multiply connected volumes such as the ones assumed for the solaratmosphere. The resulting relative magnetic helicity subtracts the helicity of the reference(potential) field so a nonzero value implies by definition the presence of free magnetic energyin the configuration. The relative magnetic helicity H m has two equivalent forms in theabove works, namely H m = Z V ( A ± A p ) · ( B ∓ B p ) d V , (1)where B p and A p are the potential magnetic field and its generating vector potential, re-spectively, and B , A are the respective quantities of the nonpotential field. The integrationrefers to the open volume V that contains the part of the magnetic structure extending abovea lower boundary. To derive equation (1), both Berger & Field (1984) and Finn & Antonsen(1985) assumed nonlinear force-free (NLFF) magnetic fields. The force-free approximation isprobably necessary for helicity calculations because only in this case one obtains some knowl-edge of the magnetic field vector and the generating vector potential required to evaluateequation (1).Equation (1) cannot be evaluated in the active-region atmosphere, however, becausethe magnetic field vector is unknown above the lower boundary, be it the photosphere orthe low chromosphere. Currently, active-region magnetic fields can only be measured in thisboundary, so the only way to evaluate the relative magnetic helicity through equation (1)is by force-free (preferably NLFF) field extrapolation into the active-region corona usingthe measured magnetic fields as the required boundary condition. However, the NLFFextrapolation of observed solar magnetic fields remains an active research area where even themost successful of the existing techniques (Schrijver et al. 2006 and references therein) are tooslow to fully exploit the spatial resolution of modern (let alone, future) magnetographs. TheNLFF approximation should always be pursued given that the LFF approximation is almostcertainly an oversimplification for most active-region fields and it can even be misleadingin several cases. Even the NLFF approximation is most likely invalid in the photosphere(Georgoulis & LaBonte 2004), although it may hold in and above the chromosphere (Metcalfet al. [1995]; see, however, Socas-Navarro [2005]).Even in case equation (1) is evaluated, however, it does not establish a link betweenthe relative magnetic helicity and the magnetic free energy of the studied configuration.In addition, it might be risky to evaluate a volume integral of extrapolated fields at largeheights above the boundary because numerical effects might settle in and affect the result.One, therefore, envisions a convenient surface-integral representation of the relative magnetichelicity that might alleviate the need for full-fledged three-dimensional extrapolations. Toour knowledge, this has been attempted only in the LFF approximation following either the 4 –theoretical analysis of Berger (1985) or the “twist” helicity of Moffatt & Ricca (1992). In thefirst case (D´emoulin et al. 2002; Green et al. 2002), the employed formula for the relativemagnetic helicity is H m = 2 α n x X l =1 n y X m =1 | b u l ,v m | ( u l + v m ) / , (2)where α is the unique, representative value of the force-free parameter and b u l ,v m is theFourier amplitude of the measured normal magnetic field for the harmonic ( u l , v m ) in atwo-dimensional Fourier space with linear dimensions n x , n y . Equation (2) is a linearizedversion of the actual formula of Berger (1985). Linearization helps avoid | H m | → ∞ when | α | → (2 π/L ), where L is the linear size of the computational domain. Detailed discussionsand an extension of Berger’s (1985) analysis will be given in § H m = 18 αL Φ , (3)where L is the characteristic footpoint separation length of the tube and Φ is the magneticflux carried by the tube. Though useful, equations (2) and (3) also lack a much wanted linkbetween the relative magnetic helicity and the free magnetic energy in the LFF magneticstructure that would enable a complete, self-consistent, description of its energetics. More-over, it is not clear how to generalize equations (2) and (3) into a NLFF calculation that, asshould be always kept in mind, must be the ultimate objective of the calculation.The above difficulties and lack of information in the calculation of the total relativemagnetic helicity prompted alternative lines of research. The lower boundary of a closedmagnetic structure, where all magnetic field lines are supposed to be rooted, acts as the driverof the evolution in the structure either via boundary flows or via the injection of additionalstructure through it. Therefore, magnetic helicity can either be transported to and from thestructure through this boundary or it can be generated by flows on the boundary . Based onthese principles, Berger & Field (1984) derived a surface-integral expression for the temporalvariation ( dH m /dt ) of the relative magnetic helicity in a magnetic configuration. Besides itsdependence on magnetic field vectors and vector potentials, as in equation (1), ( dH m /dt )depends on the boundary flows. The advantage of the Berger & Field (1984) expression for( dH m /dt ) is that it does not explicitly require force-free fields. The calculation of ( dH m /dt )has been attempted by numerous authors over the past few years (see, e.g., Nindos 2006; In the Sun, generation of helicity above the photosphere automatically implies the generation of an equaland opposite amount of helicity below the photosphere, to ensure a zero net helicity. total relative magnetic helicity. The total relative helicity isa focus of this work, so the formula of Berger & Field (1984) will not be discussed further.This study is the first of a series of studies that perform a self-consistent calculation ofthe total magnetic energy and relative magnetic helicity in a closed magnetic configuration.We devise a practical way to calculate magnetic energies and helicities from solar active-region vector magnetograms provided that the observed magnetic configuration is isolatedand flux-balanced on the presumed “plane” of the observations. The final expressions forthe magnetic energy and helicity are derived in the form of surface, rather than volume,integrals. We always assume that the studied magnetic field configuration is in a force-free equilibrium. In this study, we provide the analytical foundation of a constant-alpha,LFF, energy-helicity calculation. A NLFF generalization of the energy-helicity equationswill be the subject of a later study. The LFF analysis of this work relies on the energy-helicity formula of Berger (1988) evaluated via an application and extension of Berger’s(1985) analysis for the magnetic energies and the relative magnetic helicity. Our objective inthis work is the derivation of practical LFF energy and helicity equations that can be readilyapplied to solar vector magnetogram data. The magnetic energy budgets for a constant-alphamagnetic configuration are discussed in §
2. The LFF energy-helicity formula is discussed in §
3. The relative magnetic helicity is derived both as a volume and as a surface integral in §
4. In § §
2. Gauge-invariant definitions and the magnetic energy equation2.1. Magnetic field and the vector potential
Assuming planar geometry, consider a magnetic structure B extending in the half space z ≥ S ( z = 0). Let an open volume V of the half space z ≥ S . If S is not a flux (magnetic) surface, i.e. if B · ˆz | S = 0, then the configuration is analogousto a solar magnetic structure rooted in a small (assumed planar) part S of the photosphereand extending to infinity above S . Here ˆz is the unit vector along the z -axis of a Cartesiancoordinate system with an arbitrary origin on S . In the absence of plasma, this magneticconfiguration can only be the vacuum, current-free magnetic field B p if the configuration isisolated (not interacting with other configurations) and flux-balanced on S . The presence of 6 –plasma dictates a current-carrying magnetic structure B c , such that B = B p + B c . (4)The divergence-free properties of B and B p together with equation (4) ensure that B c isalso divergence-free. As a result, we can define generating vector potentials A p , A , and A c for B p , B , and B c , respectively. By definition, ∇ × A p = B p , (5a) ∇ × A = B . (5b)In addition, the Coulomb gauge is adopted for both A p and A to provide ∇ · A p = 0 , (6a) ∇ · A = 0 . (6b)Since S is not a flux surface, however, both the definition of B p and some topologicalproperties of the field, most notably the ones present in its magnetic helicity integral, arenot unique (gauge-invariant) and hence lack a physical meaning (e.g. Dixon et al. 1989;Berger 1999). Berger (1988) addressed the problem by providing gauge conditions for A and A p such that both B p and the magnetic helicity can be uniquely defined. These conditionsare A p · ˆn | ∂ V = 0 and A p × ˆn | ∂ V = A × ˆn | ∂ V and were formulated for a volume V boundedby a surface ∂ V , where ˆn is the unit vector normal to ∂ V and oriented outward from V .If V extends to infinity, Berger (1988) stresses that A p and A must additionally vanish atinfinity. This restricts the above gauge conditions to the lower boundary S , so that A p · ˆz | S = 0 , (7a) A p × ˆz | S = A × ˆz | S , (7b)where ˆz = − ˆn . From equation (4) and the conditions of equations (5)-(7) we can deriveadditional conditions for the vector potential A c . Writing equation (4) in terms of vectorpotentials, we obtain A = A p + A c + ∇ φ , where φ is an arbitrary scalar. Choosing thegauge such that ∇ φ = 0, one obtains A = A p + A c , (8)If one now takes the dot and cross products of equation (8) with ˆz , then one obtains A c · ˆz | S = A · ˆz | S , (9a) A c × ˆz | S = 0 , (9b) 7 –where we have used equations (7a) and (7b) to reach equations (9a) and (9b), respectively.For a volume V bounded by a surface ∂ V , another condition for the uniqueness (gauge-invariance) of B p and the magnetic helicity is that B and B p share the same normal compo-nent on ∂V , i.e. B · ˆn | ∂ V = B p · ˆn | ∂ V . In our case, where V is only bounded by S and extendsto infinity above S ( z > A p , A , B p , B vanishing at infinity, the condition refersonly to the boundary S , i.e. B · ˆz | S = B p · ˆz | S . (10)It is not necessary to independently pose equation (10), however, as it stems directly fromequations (5) and (7b). Take the normal (vertical) components of equation (5a) and (5b) on S to obtain ∇ h · ( A p × ˆz ) | S = B p · ˆz | S , ∇ h · ( A × ˆz ) | S = B · ˆz | S , (11)respectively, where ∇ h denotes differentiation on the horizontal plane, i.e. the boundary S .Equation (10) immediately follows from combining equations (7b) and (11). In addition,from equations (4) and (10) we obtain B c · ˆz | S = 0, so B c is a closed , toroidal magnetic fieldon S . In fact, B c is purely toroidal in any cross-section S ′ of V . This has been concludedby Kusano et al. (1994) and Berger (1999) who argued that the net toroidal flux of B p and B should be the same along any cross-section of V . The potential field B p being purely poloidal , on the other hand, one expects B p · B c = 0. A construction of B p and B c bypoloidal and toroidal components, respectively, can also be found in Berger (1985). From the general equation (4) it is clear that the total magnetic energy E = [1 / (8 π )] R V B d V of a closed magnetic structure is simply the sum of the potential magnetic energy E p =[1 / (8 π )] R V B p d V and the nonpotential magnetic energy E c = [1 / (8 π )] R V B c d V stored in theconfiguration in the form of electric currents: E = E p + E c . (12)Our objective will be to derive a convenient expression for each of the terms in equation(12). In this section we provide general energy expressions enabled by the gauge invariantdefinitions of the vector potentials A , A p , and A c . Equations for the potential energy E p can be directly applied to solar magnetic field measurements. Applicable expressions for thetotal energy E of the magnetic structure are given in the following sections, where the LFFapproximation is adopted. 8 –From the definition of A p , equation (5a), the potential magnetic energy of the configu-ration is given by E p = 18 π Z ∂ V A p × B p · ˆn dσ , (13)where dσ is the surface element on ∂ V . To reach equation (13) we have used Gauss’s theoremand the current-free condition, ∇ × B p = 0. Since A p and B p , together with A and B , allvanish at infinity, however, the above surface integral applies only to the lower boundary S .The potential energy E p becomes, therefore, E p = 18 π Z S B p × A p · ˆz d S . (14)Similarly, from the definition of A , equation (5b), the total magnetic energy of theconfiguration is given by (see also Berger 1988) E = 18 π Z S B × A p · ˆz d S + 18 π Z V A · ∇ × B d V . (15)The total energy from equation (15) naturally tends to the potential energy in case themagnetic field vector B tends to its current-free limit B p . Decomposing B into B p and B c , one may derive equation (12), where the potential energy is given by equation (14) and E c = [1 / (8 π )] R V B c d V .In view of the definitions and conditions for A and A p , equations (5)-(7), on the otherhand, the relative magnetic helicity of equation (1) can be written as (e.g., Berger 1999) H m = Z V A · B d V , (16)Equation (14) enables the calculation of the potential energy E p for a flux-balancedmagnetic configuration B , regardless of whether the magnetic field vector B is fully knownon S . What is needed is the boundary condition for the vertical field B z = B · ˆz on S ,that uniquely determines the potential magnetic field B p and its vector potential A p onthe boundary. In particular, assuming B p = −∇ ψ , where ψ is a smooth scalar, ∇ ψ = 0,Schmidt (1964) showed that ψ ( r , z ) = 12 π Z Z B z ( r ′ ) dx ′ dy ′ p ( r − r ′ ) + z , (17)where r = x ˆx + y ˆy , r ′ = x ′ ˆx + y ′ ˆy are vector positions on S , defined for a given Cartesiancoordinate system centered on S . For the vector potential A p one similarly obtains (see alsoDeVore 2000) A p ( r , z ) = ∇ × ˆz Z ∞ z ψ ( r , z ′ ) dz ′ . (18) 9 –Although exact, equations (17) and (18) are computationally extensive. Much faster alter-natives are provided by means of Fourier transforms. Alissandrakis (1981), in particular,showed that B p ( r ) = F − ( − iu √ u + v b u,v ) ˆx + F − ( − iv √ u + v b u,v ) ˆy + B z ˆz , (19)while Chae (2001) showed that A p ( r ) = F − ( ivu + v b u,v ) ˆx + F − ( − iuu + v b u,v ) ˆy , (20)where b u,v = P L x l =1 P L y m =1 B z m,l exp [ − i ( ul + vm )] is the Fourier amplitude of B z , u = (2 πl/L x ), v = (2 πm/L y ), L x , L y are the linear dimensions of S , and F − ( g ) denotes the inverse Fouriertransform of a function g . Albeit much faster, however, equations (19) and (20) assumeperiodic boundary conditions for B p and A p which contradicts the assumption of A and B vanishing at infinity. This problem is also well known. To mitigate the effects of the periodicboundary conditions assumed when Fourier transforms are used, one typically surroundsthe initial flux concentration with a region of zero flux. In our calculations in § E p can be readily calculated for flux-balanced photospheric or chro-mospheric (not necessarily vector) magnetograms of solar active regions. If a vector magne-togram is available, the vertical field B z on S is provided by rotating the measured magneticfield components to the local, heliographic, reference system (Gary & Hagyard 1990). Al-ternatively, the line-of-sight component can be used instead of B z provided that the studiedactive region is located sufficiently close to the center of the solar disk. This requirementtypically minimizes the impact of viewing projection effects caused by the curvature of thesolar surface.Unlike the potential magnetic energy E p , the total magnetic energy E , equation (15),cannot be calculated without additional assumptions or by using line-of-sight magnetograms.This is because A and ∇ × B are generally unknown on and above the (photospheric orchromospheric) boundary S .
3. The energy-helicity formula in the linear force-free approximation
In the force-free approximation, ( ∇ × B ) × B = 0, the total magnetic energy E of astructure extending in V is provided by the magnetic Virial theorem (Molodensky 1974; Aly1984) E = 14 π Z ∂ V [ 12 B R − ( B · R ) B ] · ˆn dσ , (21) 10 –where R is a vector position with arbitrary origin in V . For planar geometry, V extending toinfinity and being bounded only by S at z = 0, and under the assumption that the magneticfield strength B vanishes with distance more rapidly than R − / , the Virial theorem reducesto its well-known form E = 14 π Z S r · B B z d S , (22)where r = x ˆx + y ˆy is a vector position with arbitrary origin on S . Equation (22) hasbeen applied to solar active regions (Metcalf, Leka, & Mickey 2005; Wheatland & Metcalf2006) assuming potential or force-free (not necessarily linear) magnetic fields. The explicitdependence of equation (22) on the coordinate system leads to inconsistencies if the employedmagnetic field vector is not force-free (for a detailed discussion of problems related to themagnetic Virial theorem see Klimchuk, Canfield, & Rhoads 1992). Although well-known andparticularly useful, the Virial theorem does not link the magnetic energy budgets with therelative magnetic helicity in a self-consistent way. For this reason, we will hereafter followour alternative formulation for the potential magnetic energy (equation (14)) and the totalmagnetic energy in the LFF approximation. As shown in Figure 8 and explained in § ∇ × B = α B , the total magnetic energyfrom equation (15) gives E = 18 π Z S B × A p · ˆz d S + 18 π Z V α A · B d V , (23)and corresponds to the energy-helicity formula of Berger (1988). In case of the LFF approx-imation, where the force-free parameter α is constant in V , the dependence between E andthe relative magnetic helicity H m becomes explicit. Substituting equation (16) into equation(23) for constant α , we obtain E = 18 π Z S B × A p · ˆz d S + α π H m . (24)Since the relative magnetic helicity depends entirely on the presence of electric currents sothat H m = 0 for B = B p , the first term in the rhs of equation (24) must correspond tothe magnetic energy that does not include the energy stored in electric currents for any nonzero α and H m . This ground-state energy can only be the potential energy E p , so theenergy-helicity formula in the LFF approximation reads E = E p + α π H m . (25)A proof of equation (25) is provided in Appendix A. 11 –In a constant-alpha magnetic structure the sense (sign) of the relative magnetic helicity H m is dictated by the chirality (sign) of the unique value of the force-free parameter α .Therefore, αH m > α and H m are nonzero.If αH m = 0, then both α and H m are zero by definition. In this case, E = E p fromequation (25). By means of equation (12), moreover, the LFF approximation implies alinear dependence between the free magnetic energy E c and the relative magnetic helicity H m , namely, E c = α π H m . (26)The monotonic dependence of E c on α and H m can be understood if one considers that both anonzero relative helicity and a nonzero free energy depend on, and are directly proportionalto, the existence of electric currents ( α = 0). Of course, equation (26) is valid only forconstant α because the sense of helicity is the same throughout the magnetic structure. Fora non-constant α and in case of equal and opposite amounts of helicity being present in thestructure, the net relative helicity H m becomes zero. This would give E c = 0 for α = 0 inequation (26), which is not true.Given that the potential energy is readily calculated (equation (14)), it is evident fromequation (25) that knowledge of the relative magnetic helicity H m is sufficient to fully eval-uate the LFF energy-helicity formula. However, H m cannot be evaluated from the generalequation (16). This is because the vector potential A is unknown in V , although the LFFmagnetic field B can, in principle, be calculated everywhere in V . In the next section wederive convenient expressions for A and H m in the LFF approximation.
4. The relative magnetic helicity in the linear force-free approximation4.1. Volume-integral representation
It is straightforward to obtain a volume-integral expression for the relative magnetichelicity H m from the energy-helicity formula, equation (25), using the definitions of thepotential and the total magnetic energies: H m = 1 α Z V ( B − B p ) d V . (27)This expression was used by Hagino & Sakurai (2004) who assumed A = 0 on S . Despiteits simplicity, however, equation (27) cannot be directly compared to the general equation(16) because the form of A is not obvious. To make this conceptual step, we decompose B in equation (27) into its potential and nonpotential components to find, after some algebra, 12 –that H m = 1 α Z V B c · B d V , (28)where the condition B p · B c = 0 has been used. Equation (28) is identical to equation (16)for A = (1 /α ) B c , or A = 1 α ( B − B p ) . (29)One may verify that the definition of A in the LFF approximation, equation (29),complies with all the conditions of equations (5)-(7). Moreover, it is clear that A | S = 0,contrary to what Hagino & Sakurai (2004) assumed. This being said, A does not havea vertical component on S because B and B p share the same vertical component on S ,equation (10). Adopting A · ˆz | S = 0, however, equations (9) will give A c × ˆz | S = A c · ˆz | S = 0 . (30)From equations (30), then, A c | S = 0, rather than A | S = 0. Since A c | S = 0, A p and A coincide on S via equation (8), namely A | S = A p | S . (31)Of course, equation (31) does not preclude ∇ × A | S = ∇ × A p | S . This is because these curlsinclude vertical gradient terms ( ∂/∂z ) and A = A p above S .Equation (28) for the relative magnetic helicity satisfies all the requirements of the LFFapproximation. Moreover, it is more complete than the equation H m = (1 /α ) R V B d V ofPevtsov, Canfield, & Metcalf (1995). Clearly, from the above expression lim α → | H m | = E p lim α → (1 / | α | ), which tends to infinity while it should tend to zero. Given that B inequation (28) corresponds to an LFF magnetic field, the integrand B c · B = B − B · B p is known at any location in V if B z is known and flux-balanced on S . Therefore, the totalrelative magnetic helicity of a closed and flux-balanced magnetic structure with constant α in V can be calculated using equation (28). By extension, equation (28) can be used tocalculate the total relative magnetic helicity of an isolated solar active region for which theconstant-alpha approximation is assumed valid and for which flux-balanced photospheric orchromospheric vector magnetic field measurements exist. A representative, unique value ofthe force-free parameter α can be calculated by an array of techniques (Leka & Skumanich1999; Leka 1999) with an alternative technique described in § a priori the height above the photosphere where integration should stop. There aresome partial remedies for both of the above problems: the use of a much larger computationvolume than that required to contain the magnetic structure may limit periodic effects within V , while the maximum integration height can be either equal to the linear size of the surface S or determined by the contribution to the relative magnetic helicity H m . If integrationabove a certain height makes insignificant contributions to H m then integration stops at thisheight. In any case, calculating H m from equation (28) is a very time-consuming task. Forthis reason, we derive in the next section a first-principles surface integral for H m in the LFFapproximation. Regardless of the force-free approximation, the total magnetic energy E of a closedmagnetic structure extending into V and rooted in S consists of the potential magneticenergy E p of the structure and the nonpotential (free) magnetic energy E c due to electriccurrents (equation (12)). Expressing E c in terms of E p , one may write E = (1 + f ) E p , (32)where E c = f E p and f is generally a positive and dimensionless variable. The constant-alpha approximation readily provides a condition for f , namely lim α → f = 0. In addition, f must be a function of α and increasing | α | should increase f monotonically giving rise to asymmetric profile of f ( α ) with respect to α = 0, i.e. f ( | α | ) = f ( −| α | ). The form of f can bederived analytically in the LFF approximation if one uses the formulation of Berger (1985).The details of the derivation are given in Appendix B. Here we provide two expressions forthe variable f . The first is the exact analytical formula, while the second is a linearized,with respect to α , version f l of it, useful to keep the free energy and relative helicity finitewhen | α | d → (2 π/L ), where d is the elementary size on the boundary S and L is the linearsize of the magnetic structure on S . For an observed magnetogram, d corresponds to thelinear size of a pixel expressed in physical units. In particular, f = F , and f l = F l d α , (33)where F = P n x l =1 P n y m =1 | b u l ,v m | ( u l + v m ) / − ( u l + v m − α d ) / ( u l + v m ) / ( u l + v m − α d ) / P n x l =1 P n y m =1 | b ul,vm | ( u l + v m ) / , (34a) 14 –and F l = 12 P n x l =1 P n y m =1 | b ul,vm | ( u l + v m ) / P n x l =1 P n y m =1 | b ul,vm | ( u l + v m ) / , (34b)respectively. In equations (34), b u l ,v m is the Fourier amplitude of the vertical magnetic field B z for the harmonic ( u l , v m ) in a Fourier space with dimensions n x , n y . The linearization f l implies a minimum value for f that results in the estimation of a minimum free magneticenergy E c and relative magnetic helicity | H m | in the LFF approximation. The underestima-tion of E c and H m is negligible for small | α | and increases as α increases (see § | α | /d ) → (2 π/L ) is a well-known problem of the LFF magnetic fields that are not fullydescribed by the boundary condition on S in this case (e.g., Alissandrakis 1981).The quadratic dependence of f on α in both the exact and the linearized case guaranteesa symmetric profile of f ( α ) and a vanishing f for α →
0. This dependence has also beendemonstrated graphically by Sakurai (1981) in analytical force-free fields.From equations (32) and (33) we can now parameterize all terms of the energy-helicityformula (equation (25)) with respect to one of these terms. As the free parameter we choosethe potential magnetic energy E p for which the general expression of equation (14) exists.Then, the exact and the linearized surface-integral expressions for the total magnetic energyin a constant-alpha magnetic structure read E = (1 + F ) E p , (35a)and E = (1 + F l d α ) E p , (35b)respectively. For the free magnetic energy we obtain E c = F E p , (36a)and E c = F l d α E p , (36b)respectively, while for the relative magnetic helicity we find H m = 8 πα F E p , (37a)and H m = 8 π F l d αE p , (37b) 15 –respectively. Notice that the exact formula for the relative helicity, equation (37a), stillyields | H m | → | α | → F ∝ α tends to zero faster than α . Equation (37b)gives values that are a factor of four smaller than those of the linearized relative helicityof D´emoulin et al. (2002) and Green et al. (2002) (equation (2)). This can be seen fromequations (37b) and (B3), and by setting d = 1 / (2 π ), which corresponds to Berger’s (1985)unit length assuming a computational box with linear size L equal to unity. Part of thediscrepancy has been corrected by D´emoulin (2006) who admits that the original expressionof equation (2) was a factor of two too high. In Appendix B we show that the linearizationintroduces an additional (1 / §§ δE , δE c , and δH m of thetotal magnetic energy, the free magnetic energy, and the total magnetic helicity, respectively.Uncertainties of the potential energy, equation (14), stem from the uncertainties δB z of thenormal (vertical) magnetic field component B z . Although the values of δB z are generallyknown for a given magnetogram, it is difficult to propagate them into the potential energybecause of the extrapolations required to infer the potential magnetic field and its vectorpotential. In our case the extrapolations are performed using Fast Fourier transforms. Forthis reason we will ignore the uncertainties δE p of the potential energy, although we expectthat these uncertainties should not be very significant, given that the vertical magnetic fieldcomponent is the least uncertain measured component of the magnetic field vector, especiallyfor active regions located close to the center of the solar disk. For the same reason we will alsoignore the uncertainties δ F l of F l . Excluding δE p and δ F l , the only source of uncertaintiesis the uncertainties δα in the inference of the force-free parameter α . These uncertaintiesgive rise to a nonzero δ F in the value of F (equation (34a)). From equations (35) - (37),then, we obtain the following uncertainty expressions:For the exact and linearized total magnetic energy, δEE = E c E δ FF , (38a)and δEE = 2 E c E δα | α | , (38b)respectively. For the free magnetic energy, δE c E c = δ FF , (39a) 16 –and δE c E c = 2 δα | α | , (39b)respectively. For the relative magnetic helicity, δH m | H m | ≤ r ( δαα ) + ( δ FF ) , (40a)and δH m | H m | = δα | α | , (40b)respectively. The “ ≤ ” symbol in equation (40a) is due to the fact that F and α areinterrelated. Given that it is also difficult to propagate the uncertainties of | α | into δ F , wewill hereafter use the linearized expressions of the uncertainties, equations (38b) - (40b). At this point we have derived two types of expressions for the relative magnetic helicity inthe LFF approximation, namely the volume integral of equation (28) and the surface integralsof equations (37). To ensure consistency, these expressions must provide nearly identicalresults for small values | α | of the force-free parameter, while the linearized expression ofequation (37b) must provide a lower limit of the relative magnetic helicity as | α | increases.To avoid errors due to observational uncertainties and to make a safer evaluation of thevolume integral of equation (28) we use semi-analytical models of magnetic structures, ratherthan observed solar magnetograms. For a simple representation of the twist present in themagnetic configurations we use dipolar magnetic field models. For a given dipole withfootpoint separation L sep , we define the dimensionless quantity N = αL sep . This quantity isgenerally a dimensionless measure of α . In the particular case of field lines winding about anaxis (not necessarily assumed here), N is a measure of the total end-to-end number of turnsof the dipole. Our dipoles are characterized according to their N -values. We first createthe analytical distribution for the vertical magnetic field B z normal to the horizontal plane S and then we apply LFF extrapolations in the volume V using the same B z -distributionas boundary condition and assuming different α -values stemming from different N -values ineach extrapolation. Extrapolations are performed using the Fast Fourier transform methodof Alissandrakis (1981). For this test we use positive α -values which results in right-handedhelicities. Using equal and opposite α -values would only change the sense of twist and hencethe sign, but not the magnitude, of the calculated magnetic helicity. The magnetic energybudgets, equations (35) - (36), are insensitive to the sign of α . 17 –Our model dipoles have a fixed footpoint separation length L sep = 100 (in arbitraryunits) and are embedded in a boundary plane S with linear dimensions L x = L y = 200,assuming an elementary length d = 1 as the unit length. The separation length L sep repre-sents the distance between the positive- and the negative-polarity centers of the dipole on S . We use an array of N -values, where N ∈ [0 . , α = ( N/L sep ). To make sure that equations (35) - (37) do not depend on the details of aparticular model, we use three different models of B z on S . For each model, the two polaritycenters are placed at vector positions r and r , respectively, on S , such that | r − r | = L sep .The number of harmonics used for the Fourier-transform calculation of F and F l , equations(37a) and (37b), respectively, is kept fixed in all cases and is equal to n x = n y = 256. Weuse the following models:(1) A Gold-Hoyle dipole solution (Gold & Hoyle 1960), i.e., B z ( r ) = B [ 11 + q ( r − r ) −
11 + q ( r − r ) ] , (41)where r is the vector position of a given location on S and B , q are positive constants.In this test we have used a fixed B = 10 and an array of q -values, q ∈ [0 . , q -value has been applied to the full array of N -values.(2) A solenoidal dipole solution (Sakurai & Uchida 1977), i.e., B z ( r ) = B π (16+ π ¯ L sep ) / X i =1 { s i q ρ i ) + π ¯ L sep [ I ( k i )+ 16(1 − ¯ ρ i ) − π ¯ L sep − ¯ ρ i ) + π ¯ L sep I ( k i )] } , (42)where s = 1, s = − q is a positive constant, ¯ ρ i = ( ρ i /q ); i ≡ { , } , are thenormalized, with respect to q , distances of a location r on S from r and r ( ρ i = | r − r i | ; i ≡ { , } ), ¯ L sep = L sep /q is the normalized, with respect to q , separation length, k i = 64 ¯ ρ i / [16(1 + ¯ ρ i ) + π ¯ L sep ], and I ( k i ), I ( k i ) are the complete elliptic integrals ofthe first and second kind, respectively, i.e., I ( k i ) = Z π/ dθ p − k i sin θ and I ( k i ) = Z π/ q − k i sin θdθ . (43)Here we have used a fixed B = 10 and an array of q -values, q ∈ [0 . , . N -values.(3) A submerged poles dipole solution (Longcope 2005 and references therein), i.e., B z ( r ) = B q { r − r ) + q ] / − r − r ) + q ] / } , (44) 18 –where B and q are positive constants. The constant q , in particular, represents thedepth below S in which the two magnetic monopoles are placed. The depth of eachmonopole can, in principle, be different than that of the other(s), but here we use afixed depth, as well as a fixed magnetic field strength B for each monopole, to createa flux-balanced magnetic configuration on S . Here we use B = 10 and an arrayof depths q , q ∈ [0 . , N -values.Comparing the volume-integral expression, equation (28), with the surface-integral ex-pressions, equations (37), for the relative magnetic helicity H m gives the expected resultsfor all q - and N -values. Three of these results, one for each model, are given in Figure1a. The q -values for each model in Figure 1a were selected with the sole purpose of givingrise to well-separated helicity values, for convenience in the visual comparison. These selec-tions are q = 0 . , .
2, and 10 for the Gold-Hoyle (GH), the Sakurai-Uchida (SU) and thesubmerged-poles (SP) model, respectively. Figure 1a gives rise to the following conclusions:(1) All helicity expressions give very similar results for a given model, which suggests thatthe LFF equations (35) - (37) are model-independent.(2) For N = αL sep →
0, all expressions give H m →
0. Therefore, H m from equations(28) and (37) corresponds to the gauge-invariant relative magnetic helicity discussedin §§ α increases for a fixed L sep , Figure 1a shows the quadratic increaseof the magnetic helicity in the dipoles, for a fixed boundary condition B z on S .(3) Clearly, all expressions for H m give nearly identical results for small α . As α increases,the linearized surface-integral expression (equation (37b); dotted curves) consistentlyprovides a lower H m (the exact surface-integral expression of equation (40a) is repre-sented by solid curves and rectangles), as expected. The volume-integral expressionfor H m (equation (28); dashed curves and triangles) gives slightly higher values thanboth surface-integral expressions.Tests with N > H m increasesexponentially after some maximum N -value, while the surface-integral expressions continueto increase quadratically . This maximum N -value is model-dependent and, in case of theSP model, it changes even with varying model parameters. This inability to predict the Of course, at the vicinity of | α | d ≃ (2 π/L ), which in our parameter selection corresponds to N = π , theexact surface-integral H m increases abruptly to become infinite for | α | d = (2 π/L )
19 –behavior of the volume-integral H m for large N , combined with spurious results of theFourier-transform extrapolations in these cases, enhances one’s impression that the volume-integral helicity is less reliable than the surface-integral expressions and that, among otherproblems, it is susceptible to artifacts incurred by Fourier-transform extrapolations at largeheights above the boundary S .Figure 1b shows the underestimation factor caused by the use of the linearized surface-integral expression of equation (37b) for the three cases depicted in Figure 1a. In particular,the dashed curves show the ratio between the volume-integral and the linearized surface-integral helicities, while the solid curves show the ratio between the exact surface-integraland the linearized surface-integral helicities. Evidently, the underestimation factor is nearlymodel-independent when the two surface-integral expressions are compared, while the situ-ation is less predictable when the volume-integral and the linearized surface-integral expres-sions are compared. From the comparison between the surface-integral helicity expressions,one sees that, even for large N -values, underestimation does not exceed a factor of ∼ . α ∼ − M m − and L sep ∼ M m , we obtain N ∼
1. The expected underesti-mation factor for this case is . .
1, which is very modest compared to the errors expectedfrom other assumptions, and especially the use of the constant-alpha approximation itself.In summary, the results shown in Figure 1 demonstrate that the surface-integral expres-sions of equations (37) lead to reliable estimates of the total relative magnetic helicity in aconstant-alpha magnetic structure. By extension, the surface integrals of equations (35) and(36) provide reliable estimates of the total magnetic energy and the free magnetic energy,respectively, in the structure.
5. Application to observed solar active region magnetic fields5.1. Data selection and determination of basic parameters
In this section we apply the results of the previous analysis to vector magnetograms ofsolar active regions. In particular, we calculate the LFF magnetic energy and helicity bud-gets (equations (14) and (35) - (37)) using photospheric vector magnetogram data obtainedby the Imaging Vector Magnetogram (IVM; Mickey et al. 1996; LaBonte, Mickey, & Leka1999) of the University of Hawaii’s Mees Solar Observatory. IVM’s photospheric magnetog- 20 –raphy consists of recording the complete Stokes vector at each of 30 spectral points throughthe Fe I . o in the orientation of thetransverse magnetic field component. Azimuth disambiguation of the employed IVM mag-netograms was performed by means of the nonpotential magnetic field calculation (NPFC)method of Georgoulis (2005a) - see also Metcalf et al. (2006) for a comparative evaluationof the method with respect to other disambiguation methods. Figures 2a and 2b depict twodisambiguated vector magnetograms of NOAA ARs 8844 and 9165, respectively. Only partof the IVM field of view is shown in both images, to exemplify the magnetic structure of thetwo ARs. Shown are the heliographic magnetic field components on the heliographic plane.The relative isolation of the two ARs on the solar disk at the time of the IVM observations(2000 January 25 and September 15 for ARs 8844 and 9165, respectively), as well as theARs’ very different records of eruptive activity prompted us to use these data in this firsttest of our LFF energy / helicity calculations.The timeseries of the magnetic flux Φ during the IVM observing interval for both ARs areshown in Figure 3. For both cases, we notice that the IVM field of view encloses fairly well-balanced magnetic flux distributions. NOAA AR 8844 is more flux-balanced than NOAAAR 9165, with an imbalance always kept below 5%. The maximum imbalance of NOAAAR 9165 is around 10%. Our derivations require flux-balanced magnetic structures andthe above slight imbalances are not expected to significantly impact our results. The firstnoticeable difference between the two ARs is in their respective amounts of magnetic flux:on average, the magnetic flux in NOAA AR 9165 (Φ ∼ . × M x ) is a factor of ∼ . Recently, the IVM focused on the chromospheric magnetically sensitive line Na I (5896 ˚A). These ob-servations have started providing chromospheric vector magnetograms of solar active regions.
21 –larger than the flux in NOAA AR 8844 (Φ ∼ . × M x ). One might also notice a veryslight increasing trend in the evolution of Φ in NOAA AR 8844 (from ∼ . × M x to ∼ . × M x ) within the 2 hr of the IVM observations, implying that the magneticstructure is growing. This is typical of emerging flux regions.After disambiguation, we need to calculate a unique force-free parameter α for eachmagnetogram. To do so, we calculate the slope in the scatter plot between the vertical curl( ∇ × B ) z of the magnetic field B and the vertical field B z for strong-field locations of themagnetograms. By strong-field locations we mean locations with magnetic field componentsexceeding the 1 σ threshold, where we have taken 1 σ to correspond to a vertical magneticfield of 100 G and a horizontal magnetic field of 200 G , typical of the IVM. Because theLFF approximation is a gross simplification of the photospheric active-region magnetic fields,however, the uncertainty in the value of the slope is often larger than the slope itself due tothe substantial scatter in the pairs of [( ∇ × B ) z , B z ]-values. To restrict the uncertainty inthe calculation of α we have developed the following procedure: we obtain several α -valuesfrom the slope of the scatter plot, each calculated using a different significance threshold, asshown in Figure 4. Let α k be the value of α for a given significance threshold σ k = kσ ; k ≥ k be the respective unsigned magnetic flux used in the calculation. Then, the adoptedunique value of α and its uncertainty δα are obtained by the flux-weighted averages α = P k α k Φ k P k Φ k and δα = P k | α − α k | Φ k P k Φ k , (45)respectively. An example of this calculation is shown in Figure 4. The flux-weighted average α is indicated by the solid line and the surrounding shaded area indicates the extent of itsuncertainty δα . The above process provides a maximum-likelihood α -value with a reasonableuncertainly and is repeated for every vector magnetogram of the timeseries to obtain therespective timeseries for α . Using other methods to calculate α (see, e.g., Leka & Skumanich1999 and Leka 1999) we verified that the timeseries of α obtained by equations (45) are moresmooth (less spiky) and with smaller uncertainties for each α -value, than the α -timeseriesstemming from the other methods. The timeseries of the force-free parameter α for both tested ARs are shown in Figure 5.As we discussed in § α -values are generally consistent with each other, givingrise to fairly well-defined averages in both cases. The overall twist for NOAA AR 8844 isright-handed ( α > α < α appears to decrease, in absolute value, in the course of time. Coincidentally, 22 –the average absolute values | ¯ α | of α for both ARs are almost identical ( ¯ α = 0 . ± . M m − and ¯ α = − . ± . M m − for NOAA ARs 8844 and 9165, respectively). We note inpassing that the value of ¯ α for NOAA AR 8844 is in excellent agreement with the valueof 0 . M m − , calculated by Pariat et al. (2004). The latter α -value was inferred bycombining the best LFF match of the active-region corona using simultaneous EUV imagesfrom TRACE with a best LFF fit of the observed horizontal magnetic field. The magneticfield vector in Pariat et al. (2004) was acquired by the high-resolution vector magnetographonboard the balloon-borne Flare Genesis Experiment (FGE; Bernasconi et al. 2001).Although the two studied ARs happen to have almost the same α -values, albeit withdifferent signs, the much larger magnetic flux carried by the eruptive NOAA AR 9165 isexpected to lead to much larger energy / helicity budgets than the respective budgets of thenoneruptive NOAA AR 8844. The relative magnetic helicity and the respective magneticenergies for ARs 8844 and 9165 are plotted in Figures 6 and 7, respectively. There weshow both the linearized (equations (35b)-(37b); red curves) and the exact (equations (35a)-(37a); blue curves) surface-integral expressions for energy and helicity. Before discussingand comparing individual values, we note that the linearized expressions generally provideslightly lower magnitudes of energy and helicity. This is clearly the case for NOAA AR8844 (Figure 6), while in cases where the linearized values are larger than the exact valuesfor NOAA AR 9165 (Figure 7), the difference is within error bars. That the linearizedexpressions provide lower limits of energy and helicity was concluded from the analysis inAppendix B and verified using semi-analytical models in § δα of α . This being said, one notices the close correspondence ofthe α -value timeseries of Figures 5 with the timeseries of the linearized helicity of Figures 6and 7. Clearly, the success of the LFF energy/helicity estimations depends on the reliabilityof the inference of α . This is a key feature that one should keep when trying to generalizethe LFF energy/helicity formulas into NLFF ones, valid for a variable α within the field ofview.The average energy/helicity values from Figures 6 and 7 are summarized and comparedin Tables 1 and 2. Table 1 shows the comparison between average magnetic fluxes, α -values,and helicities, while Table 2 focuses on the comparison between the various average energybudgets from the two ARs. It is quite useful that the average α -values are nearly identical 23 –for the two ARs. We then notice that the eruptive NOAA AR 9165, with a factor of ∼ . ∼ . − .
5) larger than those of NOAA AR8844. The average relative magnetic helicity and free magnetic energy of the eruptive AR,however, are ∼ . − . E c / ¯ E ), normalized by the total magnetic energy. For the noneruptive NOAA AR8844, the free energy is ∼ .
7% - 6% of the total magnetic energy. For the eruptive NOAAAR 9165, the free energy corresponds to ∼ .
3% - 13 .
1% of the total energy, which is afactor of ∼ . − . ∼
44% (a few hours before a major X10 flare) to ∼ −
80% (in the course of, and shortly after, the flare), of the total energy. These ratiosappear extraordinarily high, at least in view of eruption models that predict the eruptiononset when the free energy exceeds 10 −
15% of the total energy (see, for example, DeVore& Antiochos 2005). Of course, NOAA AR 10486 was an extraordinary AR, which mightaccount for its unusual behavior.Despite the large difference of energy and helicity budgets between the two ARs, noticethat significant magnetic helicity is present even in the noneruptive NOAA AR 8844. Indeed,the average relative helicity of the AR is ¯ H m ≃ (1 . ± . × M x , with the helicitybudget of a typical CME estimated at ∼ × M x (DeVore 2000). With a minor helicityincrease, therefore, the AR should be capable of producing a typical CME before relaxing tothe potential state. Interestingly, a faint halo CME occurred above the AR on 2000 January26 at ∼ hr later, on 2000 January 27(Schmieder et al. 2004). As the AR was still growing during the IVM observations, it is 24 –likely that its magnetic helicity was further increased by January 26. No significant flaringactivity was associated to the CME.NOAA AR 9165, on the other hand, gave an eruptive M2 flare a few hours before theIVM observations on 2000 September 15, as well as two even stronger eruptive flares (M5.9and M3.3) on the next day. Its relative magnetic helicity, ∼ ( − ± × M x wasenough to launch nearly seven typical CMEs. Perhaps not surprisingly, the AR survived forseveral more days and could clearly be followed until it crossed the western solar limb.Back in our analysis, the compromise brought by the LFF approximation is reflectedon the uncertainties accompanying the estimations of the free magnetic energy in both ARs.Obviously, the free magnetic energy is a crucial parameter in assessing the eruptive potentialof a given AR (see, e.g., Metcalf, Leka, & Mickey 2005). With average linearized free energiesof (0 . ± . × erg and (1 . ± . × erg for NOAA ARs 8844 and 9165, respectively,the lowest uncertainties are estimated at ∼
67% and ∼ α -value for this magnetogram. Estimates of the Virial theorem are represented bydashed curves and triangles. Solid curves and rectangles refer to the linearized expressions,while dotted curves refer to the exact expressions. For a convenient comparison, the scalingfor the potential energy (blue curves) is different than the scaling for the total energy (redcurves). From the plots in Figure 8, we first notice that our potential-energy expression,equation (14), gives almost identical results with the Virial theorem for both ARs. Theaverage fractional differences | E p − E p ( V irial ) | / ( E p + E p ( V irial ) ) are ∼ .
7% and ∼ .
2% forARs 8844 and 9165, respectively. The difference is larger for the total energies. On average,the fractional difference is ∼ .
6% ( ∼ . ∼ .
7% ( ∼ . and helicity budgets, and (ii)they are physically intuitive, derived from first principles, and, hopefully, capable of beinggeneralized for NLFF magnetic fields.
6. Summary and discussion
The reliable calculation of the magnetic energy and helicity budgets in the active-regionsolar corona is an essential step toward the quantitative understanding of solar eruptions andhas profound space-weather applications. Our goal is to derive a practical set of equationsthat are applicable to solar vector magnetograms and can evaluate the magnetic energy andrelative magnetic helicity budgets in a physically intuitive, self-consistent manner. Here weprovide expressions for the magnetic energy and relative helicity budgets in case of a constant-alpha, flux-balanced, magnetic structure, thus implementing the LFF approximation. Theseequations are to be generalized into magnetic structures with non-constant alpha values, thusimplementing the NLFF approximation. This objective will be pursued in a later study.To perform our LFF analysis we separately derive each of the terms present in theenergy-helicity formula of Berger (1988), namely the total magnetic energy, the potentialmagnetic energy, and the relative magnetic helicity related to the free magnetic energy, to-gether with their uncertainties. Our analysis unifies numerous expressions for the relativehelicity and links several virtually unconnected studies into a self-consistent energy-helicitydescription that is practical enough to be applied to vector magnetograms of solar active re-gions. For the ground-state, potential, magnetic energy we provide a general surface-integralexpression, equation (14). This expression gives results practically identical to those of themagnetic Virial theorem. The potential magnetic energy is then used as a free parameterto explicitly determine the total and free magnetic energy, as well as the relative magnetichelicity. The variable relating the potential energy to the free energy and the relative helicityhas been calculated in two ways - an exact and a linearized one - by using and extendingthe analysis of Berger (1985). As a result, the magnetic energy and helicity budgets arecalculated self-consistently as surface integrals , equations (35) - (37). This developmentreduces significantly the required computations. Reliability and computational speed areessential elements of a future real-time or near real-time calculation of the magnetic energyand helicity budgets in solar active regions.To test our derivations we used three different types of semi-analytical LFF magneticdipoles ( § π/L ) and hence lead to infiniteenergies and helicities, as is well-known for LFF magnetic structures.Two series of solar vector magnetograms, one for an eruptive and another for a nonerup-tive active region, were thereafter subjected to our analysis ( § . The crucial point, however, is the reliable calculation offree energies and helicities. The LFF approximation is certainly not very reliable, as can beseen from the large error bars accompanying our free energy and helicity estimates ( ∼ For an alternative criterion, based on the magnetic connectivity in solar active regions, see Georgoulis& Rust (2007).
27 –al. 2004) which, however, require immensely time-consuming calculations. 3D MHD modelsare certainly capable of advancing our physical understanding of solar eruptions but, becauseof their intense computations, they cannot contribute to a real-time, or near real-time, spaceweather forecasting capability. Alternatively, non-force-free energy and helicity estimatescan be obtained if an active region is continuously observed from its formation and there-after. In this case, total energies and helicities can be calculated by temporally integratingthe Poynting flux and magnetic helicity injection rate, respectively. If the birth of an activeregion is not observed, then the initial energy and helicity can only be assumed. In anycase, both the Poynting flux and the helicity injection rate require the flow velocity of themagnetized plasma on the boundary of the magnetic field measurements. Inferring a reliable flow velocity is a completely independent, as well as highly nontrivial, problem (for a review,see Welsch et al. 2007).Our force-free equations are physically better suited to apply to chromospheric, ratherthan photospheric, vector magnetograms. It has yet to be established whether the NLFFapproximation holds for the active-region chromosphere (see Metcalf et al. [1995] in con-junction with Socas-Navarro [2005]) but it is almost certainly more valid there than in thephotosphere. The first high-quality chromospheric vector magnetograms have already beenobtained (the above authors as well as Leka & Metcalf 2003; Metcalf, Leka, & Mickey 2005;Wheatland & Metcalf 2006) but a routine acquisition and reduction of such data may stillbe a task for the future. In brief, force-free equations may be applied to photospheric vectormagnetograms as a zero-order (LFF) or first-order (NLFF) approximation, but one expectslarger uncertainties in the values of energy and helicity budgets, than when chromosphericvector magnetograms are used.Concluding, we emphasize that the present analysis cannot fully uncover the importanceof magnetic helicity in solar eruptions. Here we only show two examples that appear to pointto this direction but an answer would require large numbers of active regions and NLFFenergy/helicity equations, as already said. Our objective here was to calculate the relativemagnetic helicity in active regions as an integral part of the energetics and complexity of thestudied magnetic structures. It would be an important leap forward if it was convincinglyshown that flare- and CME-prolific active regions exhibit significant quantitative differencesin their free magnetic energy and/or total relative helicity (large free magnetic energy doesnot necessarily imply a large total relative magnetic helicity because roughly equal andopposite amounts of helicity may be simultaneously present - see Phillips, MacNeice, &Antiochos [2005]) compared to quiescent ARs. Intriguing clues to this direction stem fromthe study of the structural magnetic complexity in solar active regions (e.g. Georgoulis2005b; Abramenko 2005) or the calculation of the free magnetic energy in active regionswith exceptional flare and CME records (Metcalf, Leka, & Mickey 2005) but the role of 28 –helicity is yet to be uncovered. Some pieces of evidence suggesting the importance of helicityin solar eruptions stem from the frequent presence of sigmoids in eruptive active regions(Rust & Kumar 1996; Canfield, Hudson, & McKenzie 1999), apparently due to significantamounts of helicity with a prevailing sign, the presence of large and highly variable alphavalues in eruptive active regions (Nindos & Andrews 2004), and the statistical correlationbetween large helicity injection rates and X-class flares/CMEs (LaBonte, Georgoulis, & Rust2007). Our forthcoming NLFF analysis will be well suited to address the role of helicity insolar eruptions and we intend to carry out this study in the future.This work is dedicated to the memory of its co-author, Barry J. LaBonte. Barry isremembered as a deeply knowledgeable, distinguished colleague and an inspiring mentor. Iam grateful to D. M. Rust for our continuous interaction on magnetic helicity in the Sun andfor a critical reading of the manuscript. I also thank A. Nindos and S. R´egnier for clarifyingdiscussions on helicity issues and an anonymous referee whose numerous critical commentsand suggestions resulted in substantial improvements in the paper. Partial support for thiswork has been received by NASA Grants NAG5-13504 and NNG05-GM47G.
A. Equivalence of equations (24) and (25) for the energy-helicity formula inthe linear force-free approximation
To show that equations (24) and (25) are equivalent in the LFF approximation, it issufficient to show that the potential energy E p is given by E p = 18 π Z S B × A p · ˆz d S , (A1)for any LFF magnetic field B = B p .We first decompose B in equation (A1) into its potential (poloidal) and nonpotential(toroidal) components, B p and B c . Then, equation (A1) becomes E p = 18 π Z S B p × A p · ˆz d S + 18 π Z S B c × A p · ˆz d S . (A2)The first integral of equation (A2) is already the potential energy as shown in equation (14).To prove equation (A1), therefore, it is sufficient to show that Z S B c × A p · ˆz d S = 0 . (A3) 29 –From B p · B c = 0, we construct the volume integral R V B p · B c d V = 0. Substituting thedefinition of A p from equation (5a) into this volume integral, we find after some analysisthat Z ∂ V A p × B c · ˆn dσ = − Z V A p · ∇ × B c d V . (A4)Taking into account that (i) A p vanishes at infinity, and (ii) ∇ × B c = ∇ × B , because ∇ × B p = 0, equation (A4) further reduces to Z S B c × A p · ˆz d S = − Z V A p · ∇ × B d V . (A5)In the LFF approximation, however, ∇ × B = α B , with α constant, so equation (A5) gives Z S B c × A p · ˆz d S = − α Z V A p · B d V . (A6)Given the gauge-invariant definition of the relative magnetic helicity, however, it can beshown (Berger 1988; 1999) that Z V A p · B d V = 0 . (A7)Combining equations (A6) and (A7) we obtain equation (A3). Therefore, equation (A1) istrue and hence equations (24) and (25) in § B. Derivation of the variable linking the potential and the total magneticenergy in the LFF approximation
Here we will derive the form of the dimensionless variable f in equation (32). Thisvariable links the total and the free magnetic energies in a constant-alpha magnetic structure.We will use and extend the analysis performed in Appendix AII of Berger (1985). Assumingplanar geometry, Berger (1985) utilized Chandrasekhar’s (1956; 1961) decomposition of anarbitrary magnetic field vector into a poloidal and a toroidal components and, in view of theLFF approximation, he derived the total magnetic energy and the relative magnetic helicity.Following Berger (1985), the total magnetic energy of the structure is given by E = π n x X l =1 n y X m =1 | b u l ,v m | k l,m , (B1)where b u l ,v m is the Fourier amplitude of the vertical magnetic field B z for the harmonic( u l , v m ) in a two-dimensional Fourier space with linear dimensions n x , n y . In addition, we 30 –have k l,m = u l + v m − α ′ . The force-free parameter α ′ is expressed in inverse length units(i.e., 1 /x , where x is the number of unit lengths required for α ′ = 1) and not in physical units.This is why it is represented by α ′ , while the α used so far refers to the force-free parameterexpressed in physical units. Typically, α ′ = αd , where d is the unit length expressed inphysical units. Berger (1985) assumes periodic boundary conditions and a length unit of[ L/ (2 π )], where L is the linear dimension of the magnetic structure on the boundary S .Moreover, u l = (2 πl/L ) and v m = (2 πm/L ). Then, the direct and inverse Fourier transformof B z can be performed on S only so that one can write B z ( x, y ) | S = n x X l =1 n y X m =1 b u l ,v m e i ( u l x + v m y ) . (B2)The required boundary conditions for b u l ,v m in order to have a real and finite magnetic energyand helicity is b u l ,v m = 0 for √ u + v ≤ | α ′ | (see also Alissandrakis 1981).Assuming that the magnetic structure does not include electric currents ( α ′ = 0), thenequation (B1) provides the potential magnetic energy of the structure, namely E p = π n x X l =1 n y X m =1 | b u l ,v m | q l,m , (B3)where q l,m = u l + v m .From equation (32), the variable f is given by the dimensionless ratio f = E − E p E p . (B4)Substituting equations (B1) and (B3) into equation (B4) we obtain f = P u P v | b u,v | q − kkq P u P v | b u,v | q , (B5)where we have denoted P n x l =1 P n y m =1 by P u P v for simplicity. The ratio of sums in equation(B5) depends on α ′ because of its dependence on k . This dependence can cause problemswhen | α ′ | → (2 π/L ) because k → l = m = 1 in this case and f becomes infinite. Thisproblem is not new; that LFF fields sometimes give solutions that are not fully specified bythe boundary condition and may include infinite energy has been explicitly acknowledgedby Alissandrakis (1981), but also by Chiu & Hilton (1977), using a different analyticalframework. Clearly, this is a caveat of the LFF approximation and restricts its applicability.To avoid infinite energy values when | α ′ | → (2 π/L ), equation (B5) can be linearized with 31 –respect to α ′ , assuming small values of α ′ . We first write ( q − k ) / ( kq ) = ( q − kq ) / ( kq ).Expanding q − kq in a MacLaurin series, one finds q − kq ≃ ( q − k ) / α ′ /
2. Moreover,for | α ′ | ≪ √ π/L , one finds kq ≃ q . Then, the linearized equation (B5) becomes f l = α ′ P u P v | b u,v | q P u P v | b u,v | q . (B6)Berger (1985) goes further on to derive a linearized expression for the total relativemagnetic helicity in the volume V above S , namely H m = 4 π α ′ X u X v | b u,v | kq . (B7)Although not explicitly mentioned in Berger’s (1985) analysis, equation (B7) appears tooccur by assuming that q − kq ≃ α ′ , instead of q − kq ≃ α ′ /
2, that we have assumedin equation (B6). In our formulation, therefore, Berger’s (1985) equation (B7) is a factor oftwo too high. If no linearization is performed, then our equation (B5) is in agreement withBerger’s (1985) analysis.In an observed vector magnetogram, the best α -value is inferred in physical units ofinverse length. The scaled value α ′ of the force-free parameter relates to α via the equation α ′ = αd where d is the elementary length in the magnetogram, expressed in physical units. Asin any discrete parameter distribution with a well-defined (preferably fixed) length element,the length d in the magnetogram can be naturally represented by the linear size of themagnetogram’s pixel. From this understanding and using the definition q = u + v , thelinearized expression f l for f can be written as f l = F l d α where F l = 12 P u P v | b u,v | ( u + v ) / P u P v | b u,v | ( u + v ) / . (B8)It is important to emphasize that the linearization f l as shown in equation (B8) providesa lower limit of f , and hence a lower limit of the free magnetic energy E c and the relativemagnetic helicity H m in the LFF approximation (equations (35) - (37)). Green et al. (2002)and D´emoulin et al. (2002) reached the same conclusion when deriving the linearized helicityexpression of equation (2). Since the LFF magnetic energy is the minimum energy for a givenrelative helicity (a consequence of the Woltjer-Taylor theorem), the linearized energy/helicityexpressions are underestimations of the actual energy/helicity values. The underestimationof E c and H m is negligible for small values of | α ′ | and increases as | α ′ | → (2 π/L ). This,however, does not invalidate the linearized energy/helicity expressions for larger | α ′ | . As 32 –we see in Figure 1b, the underestimation factor is reasonable even for large | α ′ | , at least inview of other sources of uncertainties that are expected for observed magnetic configurations,and especially the use of the LFF approximation itself. D´emoulin (2006), also provides apractical explanation of the underestimation effect based on well-known properties of theLFF magnetic fields. REFERENCES
Abbett, W. P., 2003, Fall AGU Meeting, abstract
This preprint was prepared with the AAS L A TEX macros v5.0.
36 –Fig. 1.— Comparison between the volume- and the surface-integral expressions of the relativemagnetic helicity H m in three LFF dipole models with analytical boundary conditions for thevertical magnetic field component. Boundary conditions are taken by the submerged poles(SP) model solution, the Sakurai-Uchida (SU) solenoidal model solution, and the Gold-Hoyle (GH) model solution (see § L sep of the dipoles has been kept fixed. 37 – NOAA AR 884401/25/00, 19:02 UT NOAA AR 916509/15/00, 17:48 UT (a) (b)
Fig. 2.— Parts of the magnetic configuration of the two studied solar ARs. Shown are theheliographic magnetic field components of the ARs on the heliographic plane. (a) Disam-biguated photospheric vector magnetogram of NOAA AR 8844 as obtained by the IVM on2000 January 25 at 19:02 UT. A vector length equal to the tick mark separation correspondsto a horizontal magnetic field of 2300 G . (b) Disambiguated photospheric vector magne-togram of NOAA AR 9165 as obtained by the IVM on 2000 September 15 at 17:48 UT. Avector length equal to the tick mark separation corresponds to a horizontal magnetic fieldof 1760 G . Tic mark separation in both images is 10 ′′ . North is up; west is to the right. 38 –Fig. 3.— Timeseries of the magnetic flux and flux imbalance in the two studied ARs.Negative- (positive-) polarity fluxes are indicated by dashed (solid) curves, with readings onthe left ordinate, while the red line corresponds to the relative magnetic flux imbalance inthe ARs, with readings on the right ordinate. (a) NOAA AR 8844 (b) NOAA AR 9165. 39 –Fig. 4.— Example calculation of a unique α -value in NOAA AR 8844 for a vector magne-togram obtained at 18:18 UT on 2000 January 25. (a) Various α -values obtained for varioussignificance thresholds σ (see text). The straight solid line indicates the flux-weighted aver-age of these α -values and the shaded area indicates the uncertainty in the calculation of thisaverage. (b) Estimates of the total unsigned magnetic flux in the AR for various significancethresholds σ . A threshold of 1 σ corresponds to a vertical magnetic field of 100 G and ahorizontal magnetic field of 200 G . 40 –Fig. 5.— Timeseries of the constant force-free parameter α in the two studied ARs. Thedashed line and the surrounding shaded area correspond to the estimated average α -valueand its uncertainties, respectively. (a) NOAA AR 8844 (b) NOAA AR 9165. 41 –Fig. 6.— Magnetic energy and helicity budgets in NOAA AR 8844. Red (blue) curvescorrespond to the linearized (exact) surface-integral expressions. The error bars have beencalculated from the linearized expressions. (a) Timeseries of the total relative magnetichelicity H m . The dashed line and the surrounding shaded area correspond to the linearizedaverage value and its uncertainties, respectively. (b) Timeseries of the magnetic energybudgets in the AR. The potential magnetic energy is shown by the green curve. The total(free) energy and its uncertainties are shown by the solid (dashed) curves. 42 –Fig. 7.— Same as Figure 6, but for NOAA AR 9165. 43 –Fig. 8.— Comparison between the potential and the total magnetic energy estimates pro-vided by our analysis (equations (14) and (38a), (38b)) and the Virial theorem (equation (22)for the two studied ARs. Shown with blue (red) curves are the potential (total) magneticenergy estimates. The linearized (exact) estimates for the total energy are represented bysolid curves and rectangles (dotted curves). The Virial-theorem estimates are representedby dashed curves and triangles. The error bars correspond to the linearized expression forthe total energy. For clarity in comparing the different energy values, we have applied adifferent scaling for the total energy (with readings on the left ordinate) than the scaling forthe potential energy (with readings on the right ordinate). (a) NOAA AR 8844. (b) NOAAAR 9165. 44 –NOAA AR ¯Φ ( × M x ) ¯ α ( M m − ) ¯ H m ( × M x )Exact Linearized8844...... 5 . ± . . ± .
006 1 . ± . . ± . . ± . − . ± . − . ± . − . ± . . ± .
06 1 7 . ± . . ± . Table 1: Synopsis of the average magnetic flux, α -value, and relative magnetic helicity bud-gets for NOAA ARs 8844 and 9165. The third row refers to the ratio | P /P | between agiven parameter P of NOAA AR 9165 and the respective parameter P of NOAA AR8844.NOAA AR E p ( × erg ) ¯ E c ( × erg ) ¯ E ( × erg )Exact Linearized Exact Linearized8844...... 2 . ± . . ± .
13 0 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
06 7 . ± . ± . . ± . . ± . Table 2: Synopsis of the average potential, free, and total magnetic energy budgets, respec-tively, for NOAA ARs 8844 and 9165. The third row refers to the ratio ( P /P ) betweena given parameter P of NOAA AR 9165 and the respective parameter P8844