Review of Magnetic Helicity and Field Topology 1. 2. 3. Helicity: basic properties, open geometries Observations of helicity and helicity flux Twist and Writhe Mitchell Berger L=5 Open volumes True Field Equivalently, we set the helicity of the potential field to zero (and assume helicity is bilinear). Reference Field Helicity Dissipation Helicity Dissipation Reconnection of two flux tubes converts mutual helicity to self (twist + writhe) helicity Linton, Dahlburg and Antiochos 2001 Relaxation (Nandy, Hahn, Canfield, & Longcope 2003) Relaxation But … relaxation to linear force-free state requires 1. Confinement (The only linear force-free field in infinite space outside a sphere is a potential field) e.g. relaxation in one flux tube, or under one helmet streamer 2. Killing all subhelicities (through widespread reconnection – but does line-tying make this less likely than in laboratory plasmas?) Decompositions of Magnetic Helicity 1. Fourier Spectra 2. Poloidal-Toroidal 3. Regions of Space 4. Self and Mutual Helicity 5. Twist and Writhe 1. Fourier Spectra Physical Meaning: Represent field as sum of circularly polarized modes. Each mode has self linking, but there is no net linking between modes. Fourier helicity spectra do not always detect helical structure … Helicity spectrum is identically zero! Example – twisted ring of flux Asgari-Targhi & B 2009 2. Poloidal - Toroidal Physical Meaning: Poloidal and Toroidal Fields link each other, but not themselves. B85, Low 2010 3. Self and Mutual Helicity Suppose we divide the coronal magnetic field into two pieces. In each piece, the field lines begin and end at the photosphere. We can write the helicity as a sum of self helicities H1 and H2, and mutual helicities H12 : H = H1 + H2 + 2H12. H1 = 1.5 H2 = -1.2 H12 = 0.22 1 2 2 2 1 2 Individual twist helicity may be difficult to observe. The total mutual helicity of a collection of threads may be easier. Example: a set of sheared loops (each of unit flux) H = 6.1 A simple prominence model Self Helicities Mutual Helicities barbs spine Hb = 6.1 Hs = -5.1 arcade Ha = -2.2 Hbs = -2.9 Hba = -5.5 Hsa = -12.9 Here the self helicity of the barbs is of opposite sign to the helicities of the spine and overlying arcade Self Helicity of one flux tube The self helicity arises as a sum of twist within the tube and writhe of the tube axis. Choosing the volume to just contain the flux tube gives only the twist helicity. Implications for onset of kink instability for fat flux tubes. Longcope & Malanushenko 2008; Malanushenko et al 2009 Helicity Flow through the photosphere Helicity Observations Current helicity jz/Bz from vector magnetograms (Abramenko et al 1997; Pevtsov & Latushko 2002) Effects of differential rotation on active regions (van Ballegooijen et al 1998; Devore 2000; Green et al 2002; Nindos et al 2003) Helicity flow through photosphere (Kusano et al 2002; Tian 2003; Démoulin and Berger 2003; Chae et al 2004; Kusano et al 2005; Longcope, Ravindra, & Barnes 2007; Kazachenko et al 2010) Magnetograms plus best-fit force-free extrapolations (Démoulin et al 2002; Aulanier et al 2002; Georgoulis & LaBonte 2007) Helicity Flow through the photosphere A combination of LCT and time evolution of vector magnetograms can reconstruct the velocity field, giving the best observation of helicity flow (Welsh et al 2004; Longcope 2004; Kusano et al 2005) There is a gauge-invariant and physically meaningful way of mapping a helicity flow density (Pariat, Démoulin &B 2005; Pariat, Nindos, Démoulin & B 2006) – leads to much more coherent maps. 3. Helicity flow into active regions more coherent than previously thought. Helicity Flow through the photosphere On the largest scales, differential rotation injects helicity into each hemisphere. Helicity Flow through the photosphere Helicity Flow through the photosphere Getting the helicity flux right Some non-helical motions can have large fluctuations in helicity flux, creating noise Helicity Flux negative Helicity Flux positive Single footpoint moving in a straight line When practical, a different formula for helicity flux is much less noisy! The θij braiding terms measure braiding of the tubes connected to foot points i and j (unless these belong to same tube). The θii spinning terms measure twisting of the tube connected to foot point i. (Longcope, Ravindra, & Barnes 2007) Field line helicity flux can be mapped! (In practice, each of the two footpoints must be treated separately) Twist and Writhe Universal language for describing tubes ribbons, and curves Biology: DNA and proteins (Fuller 1971, Ricca & Maggioni 2006) Engineering: elastic rods (van der Heijden & Thompson 2000) Fluid Mechanics: magnetic tubes and vortex filaments (Ricca 1995) W=-0.72 Tw=0 L=-1 W=-0.72 Tw=6 L=5 Supercoiled DNA (R. Friddle) Helicity Decomposition Magnetic Helicity for (thin) flux tube with axial flux Φ: Writhe can be determined by subtracting twist from helicity Writhe Helix with three turns: Writhe = 2.68 Writhe = 0.46 Writhe of Magnetic Fields Kink instability: internal twist converted to writhe (Ricca & Moffatt 1995, Rust 1996, Turok, Berger and Kliem 2010) Stretch-Twist-Fold Dynamos: large scale positive writhe helicity, small scale negative twist helicity (Gilbert 2003) Outer Convection Zone: coriolis force on rising tubes creates large scale positive writhe helicity, small scale negative twist helicity (in North) ‘bihelical fields’ (Blackman & Brandenburg) effect (Longcope & Pevtsov): helicity source in active regions? Winding Number method A simpler and more efficient method for calculating writhe divides the writhe into local and nonlocal terms (Berger & Prior 2006): This methods divides up a curve into pieces at its maxima and minima, then computes the “local writhe” of each piece, and the “nonlocal writhes” between pieces. local writhe = -0.45 Nonlocal = winding number between the two pieces = 0.02 local writhe = -1.36 Vertical Helical Tube with n turns Helix Shape 2.5 Writhe 2 n n n 3 2 1 1.5 1 0.5 Writhe = 2.68 Writhe = 0.46 0 0 0.2 0.4 0.6 0.8 radius units of footpoint separation 1 Kinked Loop: Sine Height profile Two loops with identical Writhe = -0.2 You need height to calculate writhe! Following the writhe of an erupting filament.