Magnetic Helicity Reversal in the Corona at Small Plasma Beta

Home , Magnetic helicity

The Astrophysical Journal, 869:2 (9pp), 2018 December 10 https://doi.org/10.3847/1538-4357/aae97a Preprint typeset using LATEX style emulateapj pab v. 21/02/18

Philippe-A. Bourdin1, , Nishant K. Singh2,3, , Axel Brandenburg3,4,5,6, 1 Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, A-8042 Graz, Austria 2 Max-Planck-Institut f¨urSonnensystemforschung, Justus-von-Liebig-Weg 3, D-37077 G¨ottingen,Germany 3 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden 4 JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80303, USA 5 Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden and 6 Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA Received 2018 April 11; revised 2018 October 17; accepted 2018 October 17; published 2018 December 5

Abstract Solar and stellar dynamos shed small-scale and large-scale magnetic helicity of opposite signs. However, solar wind observations and simulations have shown that some distance above the dynamo both the small- scale and large-scale magnetic helicities have reversed signs. With realistic simulations of the solar corona above an active region now being available, we have access to the magnetic field and current density along coronal loops. We show that a sign reversal in the horizontal averages of the magnetic helicity occurs when the local maximum of the plasma beta drops below unity and the field becomes nearly fully force free. Hence, this reversal is expected to occur well within the solar corona and would not directly be accessible to in-situ measurements with the Parker Solar Probe or SolarOrbiter. We also show that the reversal is associated with subtle changes in the relative dominance of structures with positive and negative magnetic helicity. Keywords: Sun: corona — Sun: magnetic fields — solar wind — dynamo — magnetohydrodynamics (MHD) — methods: numerical

1. Introduction coronal exterior (Warnecke et al. 2011, 2012). This unex- Magnetic helicity is an important invariant in ideal and pected behavior is what is referred to as magnetic helicity nearly ideal magnetohydrodynamics (MHD); see Biskamp reversal. Such reversals have also been found in analytic so- (2003) and the original work of Woltjer(1958). It plays lutions of simple dynamo models with a force-free exterior a crucial role in characterizing the topological complexity (Bonanno 2016) and in mean-field models with magnetic of coronal magnetic fields (Berger & Field 1984), and it helicity fluxes included (Brandenburg et al. 2009). is also responsible for the possibility of premature quench- Numerical simulations are currently the best way of test- ing of the underlying dynamo (Gruzinov & Diamond 1994). ing and studying in detail the idea of the magnetic helicity An obvious remedy to the dynamo problem is to let excess reversal. Here, we consider the magnetic helicity that was magnetic helicity escape through the boundaries (Black- found to emerge from the MHD simulations of Bourdin, man & Field 2000; Kleeorin et al. 2000), especially through Bingert, & Peter(2013, hereafter BBP), who used a so- coronal mass ejections (Blackman & Brandenburg 2003) lar magnetogram of an active region (AR) as a boundary and the solar differential rotation, which also acts on open condition. Reconnection and the associated coronal heat- field lines rooted in magnetically quiet regions of the pho- ing were driven by random footpoint motions, as envisaged in the early work of Parker(1972). The simulations of tosphere (DeVore 2000); see Berger & Ruzmaikin(2000) 1 for estimates of the relative importance of different contri- BBP used the Pencil Code and covered a larger domain 2 3 butions to the total magnetic helicity flux. Much of the (235 × 156 Mm ) compared to earlier ones with the Stag- magnetic helicity transported by differential rotation out ger Code (Gudiksen & Nordlund 2002, 2005a,b). through the surface has entered through the equator; see The AR model of BBP is observationally driven by Brandenburg & Sandin(2004). Its contribution to the net line-of-sight magnetograms taken from Hinode/SOT-NFI magnetic helicity loss may therefore be subdominant. Nev- (Kosugi et al. 2007; Tsuneta et al. 2008). The model pro- ertheless, estimates for the Sun invariably result in a total vides a sufficient amount of energy to the corona (Bourdin arXiv:1804.04153v2 [astro-ph.SR] 8 Dec 2018 loss of ∓1046 Mx2 per 11 year cycle (Berger & Ruzmaikin et al. 2015). It also compares well with various coronal ob- 2000; DeVore 2000; Brandenburg & Sandin 2004; Branden- servations (BBP) and shows similarities to coronal scaling burg 2009) in the northern and southern hemispheres, re- laws, e.g., for the temperature along loops that were de- spectively. One would therefore expect to see that the mag- rived from earlier observational and theoretical works (cf. netic helicity shed at the solar surface agrees with what is Bourdin et al. 2016). passing through the solar wind at larger distances. How- The coronal EUV emission is synthesized from the MHD ever, this does not seem to be the case because in the so- model using the Chianti atomic database (Dere et al. lar wind, the magnetic helicity was found to have mostly 1997; Young et al. 2003) using the method of Peter et al. a positive sign in the northern heliosphere (Brandenburg (2004, 2006). The 3D structure of the AR loop sys- et al. 2011), whereas at the solar surface it is mostly neg- tem matches the reconstruction from Stereo observations. ative in the north (Seehafer 1990). A similar result was Also, the plasma flow dynamics along those loops matches also obtained in numerical simulations of dynamos with a the Doppler shift pattern observed by Hinode/EIS (Cul- hane et al. 2007) in the coronal Fe XII emission line.

[email protected] 1 https://github.com/pencil-code 2 Bourdin, Singh, & Brandenburg

The magnetograms for driving these simulations from the s = cV ln P − cP ln ρ + s0 is the specific entropy, s0 is a con- bottom boundary give just the line-of-sight magnetic field, stant, cP and cV are the specific heats at constant pressure or Bz near disk center. During the first hour of solar time, and constant volume, respectively, χij = χ0δij +χSpitzBîBˆj we do not yet apply any large-scale driving motions de- is the thermal diffusivity, Bˆ is the unit vector of the mag- rived from the observed movements of magnetic patches in netic field, ν is the kinematic viscosity, η is the magnetic the photosphere. We only apply the horizontal small-scale diffusivity, χ is the Spitzer field-aligned heat conduc- velocities that mimic granulation. Therefore, these photo- Spitz tivity, χ0 is an isotropic contribution, g is the gravitational spheric horizontal motions are purely stochastic and sta- acceleration, and S is the traceless rate-of-strain tensor with tistically mirrorsymmetric, and there is no obvious mecha- 1 1 the components Sij = 2 (ui,j + uj,i) − 3 δijuk,k. Here, we nism to break the statistical mirrorsymmetry of the model. solve for A because then B is automatically divergence free. In particular, there is no Coriolis force or differential rota- Instead of solving for s, we use the logarithmic temperature tion. Nonetheless, it turns out that helicity emerges readily ln T , which is directly related to s and ln ρ. Using the log- within the initial phase of our model. arithmic density ln ρ, we are able to capture many orders Although there is no direct injection of helicity, the model of magnitude in the density stratification that our model can still produce magnetic helicity through a complex ar- atmosphere covers. rangement of multipolar spots (Bourdin & Brandenburg 2018). We use a magnetogram of a small and stable AR 2.2. Physical details about the simulations observed during 2007 November 14 in the southern hemi- For the radiative cooling function ρΛ(T ), we use a real- sphere. Indeed, we find helicity, as is readily demonstrated istic tabulation of Λ provided by Cook et al.(1989). In by looking at the vertical profile of the mean current helic- that work, the important emission peak from highly ion- ity density, hJ · Bi , where J = ∇ × B/µ is the current xy 0 ized iron lines is included, which efficiently cools the model density, B is the magnetic field, µ is the vacuum perme- 0 corona. The characteristic half-time of this cooling is below ability, and h...i denotes horizontal averages. We find xy 30 minutes. that the profiles generally show a sign reversal within the During the first 35 minutes of physical time, however, first 5–15 Mm above the surface. This is equally remarkable this loss term is turned off together with the heat conduc- because the upper regions are topologically connected with tion along the field. This is to prevent excessive cooling the lower ones through the same large-scale structures. Any of the corona during the initial phase in which the gran- small-scale magnetic fields seem to be interspersed within ular motions cause magnetic disturbances that still need other structures and are still associated with the large-scale to propagate from the photosphere into the corona. The magnetic loops extending from one footpoint to the other. simulation is then continued for another about 35 minutes The purpose of this work is to quantify the magnetic after all physical terms in the equations are turned on. Fur- helicity reversal in detail and to associate it with coronal ther details about the switching on can be found in Bourdin heating along EUV-emissive loops. We further characterize et al.(2014) and Bourdin(2014). We use here data from a the magnetic helicity reversal in spectral space and demon- fully developed state at 63 minutes physical time. strate that it occurs in different wavenumber intervals at Unlike ideal models of MHD, where η = ν = 0 and the same height. dissipation is modeled by highly nonlinear diffusion opera- 2. Our approach tors that cannot easily be stated in concise form and effective Reynolds or Lundquist numbers are difficult to spec- In the present work, we use a snapshot from the simula- ify, we use in our model constant values of ν, η, and also tions of BBP to analyze the production and vertical vari- 10 2 −1 χ0. The value of the magnetic diffusivity η = 10 m s ation of magnetic and current helicity densities as well as (= 1014 cm2 s−1) is about eight orders of magnitude big- their spectra. Before discussing those aspects in detail, we ger than that estimated for the solar corona. This choice begin with the basic equations solved in BBP and present is required for numerical stability and for having a grid a brief summary of the physical properties of those simula- Reynolds number near unity. On the other hand, we use tions. a realistic value for the viscosity, ν = 1010 m2 s−1, which results in a Prandtl number of unity because η = ν. How- 2.1. Basic equations ever, as pointed out by Rempel(2017), the relative impor- BBP solved the continuity equation, the equation of mo- tance of Ohmic and viscous heating changes toward the lat- tion, the induction equation, and an energy equation, which ter when realistically large values of the magnetic Prandtl includes the necessary energy sinks to get realistic and self- numbers are taken into account; see Brandenburg(2014) consistent coronal heating and cooling terms: for the relation between the dissipation ratio and the magnetic Prandtl number. The isotropic heat conduction is set D ln ρ 8 2 −1 12 2 −1 = −∇ · u, (1) to χ0 = 5 × 10 m s (= 5 × 10 cm s ). We use a re- Dt −10 5/2 Du alistic coronal value of κSpitz = 1.8 · 10 T / ln ΛC with ρ = −∇P + ρg + J × B + ∇ · (2νρS), (2) ln ΛC = 20 being the Coulomb logarithm. We obtain the Dt field-aligned Spitzer conductivity as Ds 2 2 2 ρ T = −∇ · F − ρ Λ(T ) + µ0ηJ + 2ρνS , (3) 1 5/2 Dt χSpitz = κSpitzT /ρ. (5) ∂A cP = u × B − µ ηJ, (4) ∂t 0 2.3. Boundary conditions where P = (R/µ) ρT is the gas pressure, R is the universal The model is periodic in the horizontal directions and gas constant, µ = 0.67 is the mean atomic mass, T is the employs a potential-field extrapolation on the top bound- temperature, Fi = −ρcP χij∇jT is the conductive heat flux, ary. To formulate the potential-field boundary condition, Magnetic helicity reversal in the corona 3

2 we deﬁne the Fourier-transformed magnetic vector poten- 0 < C >xy / xy [1 / Mm] tial as -0.004 -0.002 0.000 0.002 0.004 Z A˜(k , k , z, t) = A(x, y, z, t) eik·rd2r, (6) x y M,rel

0 M 4 where k = (kx, ky) and r = (x, y). On the lower and upper C z boundaries, we thus have ˜ ∂A ˜ = −|k|A(kx, ky, z∗, t), (7) 0 ∂z 3 ] where z∗ denotes the locations of the boundaries. Apart m M from this, the top boundary is closed for any plasma ﬂows [

and is thermally insulating. h 0 g i

At the bottom boundary, realistic atmospheric tempera- 2 e ture and density are imposed. We also adopt Equation (7) h at the bottom boundary, but keep the minus sign on the 13.3 Mm right-hand side. This corresponds to an inverted potential-

field extrapolation, whereby contrasts in the magnetic field 0 are increased, instead of smearing them out like for the top 1 boundary. This mimics the effect of flux tubes becoming 5.5 Mm narrower when entering below the photosphere. Because this increase of contrast would quickly lead to artifacts, 0 like wiggles in the Bz component, this increase of contrast is limited to about one-third of the pressure scale height -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 < > [10 - 6 T2 Mm] some 100 km below the photosphere so that artifacts in Bz M,rel xy are avoided. With this method, we obtain the ghost zones Figure 1. Horizontal averages of the gauge-dependent magnetic he- for all three components of A just beneath the lower pho- licity, the gauge-independent relative helicity, and the current helicity tospheric boundary. normalized to B2 versus height. 2.4. Gauge dependence of magnetic helicity To compare a vertical profile of HM and HM,rel, we com- In general, the local magnetic helicity density pute the horizontal averages of both quantities. To get HM = A · B (8) hHM(z)ixy of the magnetic helicity density HM(z), we simply average over horizontal slices from the height z and with depends on the gauge of the vector potential A. On the a thickness of ∆z equal to our vertical grid spacing. For the lower and upper boundaries of the simulation domain, our profile of HM,rel(z), we simply subtract the volume integrals magnetic field is driven toward a potential state. The re- above the heights z and z + ∆z. Each integral uses differ- sulting A at these boundaries is in the Weyl gauge. How- ent potential fields, A, B and A, B , which we ever, the gauge could in principle still drift because we have pot,z pot,z+∆z extrapolate from the known states Bz(z) and Bz(z + ∆z), no boundary restrictions other than periodicity along x and respectively. The average relative magnetic helicity density y. Therefore, we must check if such a gauge drift occurs and contained in this xy layer is then just the difference if it significantly changes our simulation results. We use the relative helicity from Equation (5) of Finn & HM,rel(z) − HM,rel(z + ∆z) Antonsen, Jr.(1985), similar to the formulation of Berger hHM,relixy(z) = , (10) Vxy∆z & Field(1984), to obtain a gauge-independent helicity as ZZZ ∞ which we normalize to the volume of the layer Vxy∆z to be 1 comparable to hH (z)i ; see Figure1. HM,rel(z) = (A+Apot)·(B−Bpot) dz dy dx (9) M xy z 2 We find that the horizontal averages of our magnetic helicity density HM and the relative helicity density HM,rel are Apot and Bpot are the nonhelical potential fields extrap- olated from the known state of B (z) at the height z. On very similar. Both magnetic helicities do show sign reversals z that are located roughly at the same height, like the max- the upper boundary z = Lz, our magnetic field is already almost potential and very close to the nonhelical extrapo- ima and minima; see the dotted and red lines in Figure1. The vertical profile of the current helicity, HC = µ0j · B, lation Bpot(Lz). The magnetic helicity therefore vanishes toward the top of the domain and we may omit the vol- shows a qualitatively similar trend also with a sign reversal in the corona, albeit higher up; see the blue dashed line. ume from Lz to ∞ in the integrals of Hrel. Our factor of one-half compensates for the addition of the two similar We show that our magnetic helicity density is therefore not significantly influenced by the gauge drift along the periodic quantities A and Apot in Equation (9), which allows the relative magnetic helicity to be quantitatively similar to the directions. Therefore, we may continue to use the gauge- dependent magnetic helicity HM as a good proxy of the magnetic helicity HM. Because the components of B and gauge-independent relative helicity HM,rel. Bpot are normal to all nonperiodic boundaries (here, the top and bottom of the simulation domain), they are either identical by construction or their differences are negligible. 2.5. Magnetic and current helicity spectra Equation (9) gives us therefore a gauge-independent rela- Through most of this work, we show both current tive helicity (Berger & Field 1984; Finn & Antonsen, Jr. and magnetic helicity. In particular, we consider two- 1985). dimensional current and magnetic helicity spectra defined 4 Bourdin, Singh, & Brandenburg 2

a) full FOV b) full FOV 0 c) 1 0 AR AR 4 QS QS 0 1 3 ] ] 0 1 m m

M M 4.9 Mm [ [

t t h h 0 g g i i 2 e e h h 0 0 1 0 1 AR 4.9 Mm 4.9 Mm QS minima / maxima 1 - 0 0 -8 -6 -4 -2 0 2 4 6 8 -0.2 0.0 0.2 1 10-6 10-4 10-2 100 102 104 - 6 2 2 < M >xy [10 T Mm] < M >xy / xy [Mm] plasma beta Figure 2. (a) Magnetic helicity for the full FOV, the AR core, and only the QS area as a proﬁle of horizontal averages versus height. (b) Magnetic helicity normalized to B2. (c) Plasma beta value ranges for AR and QS. 2 as 0 1 1 X ˜ ˜ ∗ ˜∗ ˜ HC(k) = 2 J · B + J · B , (11) k−<|k|≤k+ 1 X ˜ ˜ ∗ ˜∗ ˜ ] HM(k) = 2 A · B + A · B , (12) m J B almost force free J B 1 M 0 [

k−<|k|≤k+ 1 t h g i where k± = k ± δk/2 and δk = 2π/L with L = 235 Mm be- e ing the size of the magnetograms and tildes denote, again, h

Fourier transformation. Under horizontally isotropic con- AR < J B >xy ditions, we have AR < J B >xy

0 QS 0

2 1 µ0HC(k) = k HM(k), (13) i.e., the current helicity spectrum is directly related to the 0.0 0.2 0.4 0.6 0.8 1.0 magnetic helicity spectrum, but weighted with a k2 factor, Figure 3. Force-free parameters κJ·B and κJ×B for the AR core so high wavenumber contributions in HM(k) get enhanced. area and the complementary QS area. At z = 10 Mm, the magnetic It is sometimes convenient to define the magnetic and field is nearly fully force free, so κJ×B → 0 and κJ·B → 1. current helicity densities as Figure2(a). To assess this quantitatively, we plot in Fig- HM = A · B, HC = J · B. (14) ure3 vertical profiles of the characteristic nondimensional wavenumbers κJ·B and κJ×B defined through (Warnecke Note, in particular, that at each value of z, we have & Brandenburg 2010) Z Z 2 2 HM dk = hHMixy, HC dk = hHCixy. (15) 2 h(J · B) ixy 2 h(J × B) ixy κJ·B ≡ 2 2 , κJ×B ≡ 2 2 . (16) hJ B ixy hJ B ixy In the following, however, we often retain the more explicit 2 2 notation in terms of A · B and J · B. Note that κJ·B + κJ×B = 1, so the two are complementary in the sense that when κJ×B → 0, we have κJ·B → 1, and 3. Results vice versa. Looking at Figure3, we see that in 5 Mm . z . 50 Mm, the magnetic field is indeed nearly force free 3.1. Magnetic helicity reversal and κJ·B reaches values close to unity, the largest possible To set the stage, we show in Figure2 vertical pro- value. Consequently, κJ×B is very small in this range for 2 files of hHMixy, hHMixy/hB ixy, and the plasma beta, AR and QS. 2 2µ0hP ixy/hB ixy (cf. Bourdin 2017). For comparison, we also plot the corresponding profiles for averages over the 3.2. Magnetic helicity reversal within a flux rope AR core and the complementary quiet-Sun (QS) area. For In Figure4, we show a visualization of A · B. We see the plasma beta, we also show minimum and maximum val- that the magnetic helicity density changes in a horizontal ues (dotted). In the lower part, for z . 5 Mm, hA · Bixy plane at approximately 5 Mm. We also show the magnetic is positive, while for z & 5 Mm it is negative. In fact, for helicity density in several yz planes through two partic- the full field of view (FOV), and in a small z interval very ularly prominent magnetic field lines labeled as CL1 and close to the lower boundary, the sign of hA · Bixy changes CL2, where CL1 is a EUV-emissive loop in the core of the once again. We return to this aspect again later. AR. While one of the two field lines passes through regions The main focus of this paper is instead the sign reversal where A · B is positive (red) throughout, the other field of hA · Bixy at z ≈ 5 Mm. This happens at a height where line traverses yz planes in which A · B is positive near the the magnetic field begins to become almost force free; see apex of the line (denoted by CL2) and negative in the xy Magnetic helicity reversal in the corona 5 z y x

CL2 SL3 SL1

(–) CL1 (+)

Figure 4. Visualization of coronal loops (field lines) with EUV emission (orange–green volume rendering) above an AR magnetogram (grayscale at the bottom). The semi-transparent layer at 5.5 Mm shows the magnetic helicity density with positive to negative values color-coded from red to blue, saturated at ±0.2 · 10−3 T2Mm. The black circles mark where the field lines cross this horizontal layer. Three vertical opaque planes cut through the cross section of the loops in the core of the AR and also indicate the magnetic helicity density, but saturated at ±0.1 · 10−3 T2Mm; see Section 3.2. plane through 5 Mm (denoted by CL1). This shows that at one would not see a clear one-to-one correlation of helicity least one magnetic helicity reversal is possible right in the and the coronal heating or EUV emission. middle of a field line or loop. The loops SL1–3 connect from one of the main polarities 3.3. Spectral magnetic helicity reversal to the periphery of the AR. SL1–3 show strongly asymmet- ric heating and EUV emissivity. We find that the coronal The study of magnetic helicity spectra has revealed im- heating is particularly strong on that side, where SL1–3 are portant insights about the nature of the turbulent dynamo; rooted in strong negative magnetic helicity (blue in Fig- see Brandenburg & Subramanian(2005) for a review. Ow- ure4). The other end of these side loops connects to low- ing to magnetic helicity conservation, the α effect in mean- helicity areas and there we also see less heating and EUV field electrodynamics (Moffatt 1978; Krause & Rädler 1980) emissivity. Note that under the assumption of horizontal can only produce positive and negative magnetic helicities isotropy leading to Equation (13), both the current helic- to equal amounts, but at different length scales (Seehafer ity and magnetic helicity spectra are related. In particular, 1996; Ji 1999). This leads to a bihelical magnetic field since the spectral magnetic helicity reflects the large-scale (Blackman & Brandenburg 2003; Yousef & Brandenburg properties of current helicity, it corresponds to the integral 2003) with one sign at the scale of the energy carrying ed- over all large-scale patches of the current helicity density. dies (referred to as “small scale”) and another sign at the We also note that the most strongly heated core-loop CL1 scale of the domain (referred to as “large scale”). In the is rooted in two strong positive helicity regions (red in Fig- solar wind, the spectrum is also found to be bihelical, but ure4), and at the same time, we find a negative helicity the signs at both small and large scales are reversed (Bran- (blue) near the loop apex. While we do not want to claim denburg et al. 2011). In the MHD model, the photospheric a direct relation of magnetic helicity and the coronal Ohmic structures are “small scale” and smear out when reaching dissipation of currents that heats our model loops, we need higher atmospheric layers. Coronal loops then define the to point out that local injection of helicity is a way of trans- “large scale” structures. The basic question is now whether porting magnetic energy to the corona and induce currents this apparent swap in sign at small and large scales hap- there. Nonetheless, this could only tell about the volumet- pens abruptly at one particular height and across all scales, ric heating and the EUV emissivity is of course strongly or gradually through an effective shift of the spectrum in modulated by density variations. In particular, when the wavenumber, as perhaps suggested by the idea of an inverse density is low, the heating per particle is high. Therefore, cascade behavior, where the height in the domain plays the role of time in a decaying MHD simulation; see Christens- 6 Bourdin, Singh, & Brandenburg

- 6 2 - 8 2 2 2 2 M [10 T Mm] C [10 T / Mm] k [T ] 0 [T ] 2 1 0

z=1.8 Mm 0 ] - c 3 0 e s c r 1 0 a 0 0 [ 0