Outline of Presentation. Magnetic Carpet Small-scale photospheric magnetic field of the quiet Sun. Evolution of Magnetic Carpet 12/07/2012

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1 Outline of Presentation Karen Meyer 1 Duncan Mackay 1 Aad van Ballegooijen 2 Magnetic Carpet 2D Photospheric Model Non-Linear Force-Free Fields 3D Coronal Model Future Work Conclusions 1 University of St Andrews 2 Harvard-Smithsonian Center for Astrophysics Magnetic Carpet Small-scale photospheric magnetic field of the quiet Sun km Surface motions dominated by s/g flows - Convective cells, - Mean diameter 14,000 km. Upflow, radial outflow ~ 0.5 km/s, downflow. Build up of flux concentrations at boundaries. SDO/HMI: 73,000x73,000 km, 6 hours Evolution of Magnetic Carpet SDO/HMI: 101 x 101 arcsecs ~ 73,000 x 73,000 km, 6 hours Emergence - Appearance of pairs or clusters of new magnetic flux features: equal amount of each polarity. Cancellation - Mutual loss of flux when opposite polarity features come into contact. Fragmentation - Breaking apart of a large feature into two or more separate fragments. Coalescence - Two or more same-polarity features join together. Photospheric recycle time 1-2 hours (Hagenaar et al. 2008) Large active region: Mx Flux detection limit: ~ few x Mx e.g. Hinode/SOT, Sunrise SOHO/MDI active region, December days. 1

2 Magnetic Carpet Fluxes: Ephemeral regions and internetwork features: - Emerge within s/g cells as pairs or clusters of opposite polarity flux Motivation for Magnetic Carpet Model No noise or instrumental limitations: total flux known at all times, time averaging not necessary. Complete control: flux does not drift into and out of field of view. Flux balance is maintained. Image: Priest et al Flux: ~ Mx - Diameter: few thousand km - Lifespan: IN mean 2.9 min ER mean 4.4 hr Swept to cell edges by s/g flows, where they form the magnetic network: - Long-lived, slow moving magnetic features Knowledge of processes: know exactly where and when emergence, cancellation, coalescence and fragmentation are occurring. Parameters from observations: make some estimate of, for example, cancellation and fragmentation rates. Stepping stone toward inserting real magnetogram data into coronal code: first understand situation where we have complete control. Hinode/SOT 41x41 arcsecs A two-component model: The Model 2D model: realistic model for photospheric evolution of magnetic carpet. 3D model: continuous evolution of non-linear force-free corona, driven by photospheric boundary motions. Photospheric Model: Ingredients Steady supergranular flow profile: Peak flow 0.5 km/s. Random motions representing granulation. Periodic in x and y. Includes emergence, cancellation, coalescence and fragmentation. Photospheric Model: Ingredients Lower Boundary Treatment Probability distribution for newly emerging bipoles (Thornton and Parnell, 2010): B z analytically specified at each time step given a Gaussian profile: Number of bipoles emerging in the range [ϕ 1, ϕ 2 ] cm -2 day -1 : Not advecting numerically avoids: numerical overshoot, numerical diffusion, pile-up at cancellation point due to forcing numerically. Log-log plot of frequency of emergence against flux emerged for a 5 hour sequence of Hinode/SOT magnetograms. 2

3 Emergence Fragmentation Features grow in flux as Gaussian profiles separate. Initial separation rapid (~ 3 km/s) Later slowing to ~ 0.5 km/s S/g flows take over until another feature is encountered. Similar to method of Parnell (2001). Currently a feature may only split into two other features at once. Cancellation and Coalescence 2D Magnetic Carpet Model Two features defined to be within interaction range will move towards one another. Features shrink as Gaussian profiles overlap. Once centres meet, one or both elements removed. 50 x 50 Mm computational box. 250 hour simulation. 1 min between frames. Range for newly emerging bipoles: Mx Unit flux: φ 0 = Mx Photospheric Model: Results Model reaches steady state after ~ 24 hours. (Emergence rate cancellation rate, Mx cm -2 s -1 ) Formation of a magnetic network. Mean magnetic field ~ 6 7 G. Photospheric recycle time: 1.48 hours (1-2 hours, Hagenaar et al. 2008) Average lifespan of flux element: 10 minutes (1-20 mins, mean 2.9±2.0 mins, Zhou et al. 2010) Max. lifespan of flux element: 3 4 hours (Ephemeral regions, mean lifespan ~ 4.4 hours, Harvey and Martin, 1973) Hinode: spicules on the limb (Y. Suematsu 2009) Modelling the Solar Corona Cannot currently measure coronal magnetic field Low pressure field spreads out, so is weaker Zeeman splitting becomes small. Low density any measurements are integrated over a large volume. In particular cannot measure small-scale structure. Can see effect of magnetic field on surrounding plasma e. g. Coronal loops Spicules Many modelling techniques exist to describe the coronal magnetic field SDO/AIA 171 Å: 1 million K coronal loops. 3

4 Considering static equilibria, so neglect explicit time dependence. Length scales considered << coronal pressure scale height, so neglect gravity. β 1 within the solar corona, so neglect the pressure gradient. Force-free Condition: Ampère s law: Force-free Condition: Ampère s law: α - describes the twist of the field - is a function of position - must be constant along field lines 3 cases: α=0 α=constant α= α(r) Potential Field Linear Force-Free Field Non-Linear Force-Free Field (Image: Yeates 2008) Non-Linear Force-Free Fields: see Schrijver et al for a comparison of methods. 4

5 Magnetofrictional Method Coronal field evolves through a series of non-linear force-free equilibria in response to lower boundary motions. 2D magnetic carpet model is lower boundary. Initially potential coronal field. Magnetofrictional Method The Lorentz Force acts against an artificial friction to cause the system to relax towards a non-linear force-free equilibrium. Magnetofrictional velocity: Artificial friction term Global solar coronal magnetic field (eg. Yeates et al., 2008) Decaying active region observed by SOHO/MDI (Mackay et al., 2011) Coronal interactions of small-scale magnetic fields (Meyer et al., 2012) Coronal field induction equation: Hyperdiffusion: Magnetofrictional Method The Lorentz Force acts against an artificial friction to cause the system to relax towards a non-linear force-free equilibrium. Magnetofrictional velocity: Artificial friction term Why Non-Linear Force-Free Fields? Allows for electric currents and free magnetic energy. arying twist regions of high and low helicity, and varying sign of α. Much less computationally intensive than full MHD: Can model more complex simulations, relatively fast. Magnetofriction? Coronal field induction equation: Hyperdiffusion: Our method (van Ballegooijen, Priest & Mackay, 2000) produces a time evolution: Other methods produce independent extrapolations of the coronal field at each frame. Continuous from one frame to the next in our model a memory of previous currents and connectivity is maintained. 3D Coronal Model: Setup Free Magnetic Energy 48 hr series of synthetic magnetograms hr - model has reached a steady state 1 min between magnetograms 50 x 50 x 25 Mm (resolution: 512 x 512 x 256) 500 relaxation steps between each frame (0.12 s) Four strengths of overlying field: 0 G, 1 G, 3 G and 10 G Total magnetic energy within the volume: Free magnetic energy = W nlfff W p.f. Green 0 G Black 1 G Blue 3 G Red 10 G B 2 W = d 8π 5

6 Free Magnetic Energy Rate of change of total magnetic energy: Coronal field induction equation: dw dt = 1 4π d B 2 dt 2 d A = v B + ε t Rate of change of total magnetic energy: With: I 1 4π dw dt = S IdS Qd v B + ε B + η 4B 2 α α Energy injected or removed through the lower boundary surface. d B 2 dt 2 = B B = B v B + ε t dw dt = S IdS Qd Q B2 4π υ v 2 + η 4 α 2. Energy dissipated within the coronal volume due to magnetofriction and hyperdiffusion. Description of Q in Meyer, Mackay and van Ballegooijen, Q B2 4π υ v 2 + η 4 α 2 Qd t Qddt Energy dissipation Q as a function of height at t=128 h Q integrated in the line of sight Q as a function of height 6

7 Future Work 2D Model Successfully produced a realistic model for the photospheric evolution of the solar magnetic carpet, that reproduces many observational properties. Meyer, Mackay, van Ballegooijen & Parnell, Solar Phys., 272, 29 (2011). Improvements: Evolving s/g flow profile e.g. taken from observations. More complex flow profile inclusion of vorticity would have implications for coronal evolution. Fragmentation believed to occur due to underlying convective flows. Q in the x-y plane, at z=3 Mm Q in the y-z plane, integrated in the line of sight Play with parameters switching off emergence, varying fragmentation. Improvements due to new observational results. Balltrack method for tracking flow fields, by Hugh Potts, University of Glasgow (Potts et al. 2004, Future Work 3D Model Apply magnetofrictional code to observed magnetograms: Do regions of interest in 3D model (e.g. stored or dissipated energy, electric currents) correspond to regions of interest in coronal images? Conclusions 2D model successfully reproduces many observed properties of magnetic carpet. Meyer, Mackay, van Ballegooijen & Parnell, Solar Phys., 272, 29 (2011). SDO/HMI: 101 x 101 arcsecs ~ 73,000 x 73,000 km, 6 hours Feature tracking how to events at the photosphere affect the corona? (Both simulated and real magnetograms) New coronal remap time? (Currently 1.4 hr, Close et al. 2004) Continuous evolution of the coronal magnetic field means that connectivity is maintained from one frame to the next. Free magnetic energy may be built up in non-potential field, and stored along closed field lines. Energy dissipation is greatest where magnetic field is strong (i.e. near the sources) and at sites of changing magnetic topology. Meyer, Mackay & van Ballegooijen, Solar Phys. (2012). Thank You! 7