1. ALFVEN WAVE ENERGY TRANSPORT IN SOLAR FLARES Lyndsay Fletcher University of Glasgow, UK. RAS Discussion Meeting, 8 Jan 2010 1

2. Flare ‘cartoon’ 2 Unconnected, stressed field Energy flux Post-reconnection, relaxing field - shrinking and untwisting 1) Field reconfigures and magnetic energy is liberated via magnetic reconnection. 2) Energy transmitted to the chromosphere, where most of the flare energy is radiated (optical-UV). How does the energy transport happen? relaxed field – ‘flare loops’ It is clear that the energy for a solar flare is stored in stressed coronal magnetic field (currents). Footpoint radiation, fast electrons, ions

3. Particle beams or waves? 3 1)Pre-flare energy storage => twisted field, so energy release => untwisting – i.e. an Alfvenic pulse. Consequences? 2)Earth’s magnetosphere provides an example of efficient particle acceleration by Alfven waves, generated in substorms. 1)Since the (1970s) it is clear that the corona contains insufficient electrons to explain chromospheric HXRs (Hoyng et al. 1973, Brown 1976). Overall flare beam/return currents electrodynamics in a realistic geometry is far from understood. In the ‘standard’ flare model an electron beam accelerated in the corona transports energy to the chromosphere. Here we propose a wave-based alternative, motivated by the following:

4. Flare energy requirements Flare energy = 6 x 10 32 ergs over ~ 1000 sec. Woods et al 2005 Power directly measured in the optical can be up to 10 29 erg s -1 Power in fast electrons inferred from hard X-rays is around the same. flare Flare energy radiated from a small area: HXR footpoints ≈10 17 cm 2, (WL footpoints can be smaller.) Power per unit area ≈ 10 11-12 erg cm -2 4 G-band (CH molecule) Fe (stokes I) Fe (Stokes V) Isobe et al 2007 Source FWHM = 5 x 10 7 cm Flare total irradiance

5. Flare electrons Flares are very good at accelerating and heating electrons. Radiation from non-thermal electrons is observed in the corona and chromosphere. So a wave model must also accelerate electrons. Krucker et al. 2008 5 Coronal X-rays imply ≈ 1-10% of electrons are accelerated and decay approx. collisionally (e.g. Krucker et al 2008). Chromospheric X-rays require a ‘non-thermal emission measure’ (e.g. Brown et al 2009) cm -3

6. Wave speed and Poynting flux 10,000 km The flare corona is quite extreme…. Coronal |B| deduced from gyrosynchrotron: Active region magnetic field strength at 10,000 km altitude (≈ filament height): ≈ 500 G average ≈ 1kG above a sunspot Brosius & White 2006 Coronal density ~ 10 9 m -3  v A ≈ 0.1 - 0.3c Transit time through corona = 0.1 – 0.3 s ‘Poynting Flux’ So flare power ≈ 10 11 erg cm -2 needs  B ≈ 50 G (though note, reflection coeff ~ 0.7 initially) Gyrosynchrotron emission (contours) above a sunspot 6 e.g. Lee et al (98) Brosius et al (02)

7. MHD simulations of Alfven pulse propagation 3D MHD simulations of reconnection/wave propagation Diffusion region assumed small Track Poynting flux and enthalpy flux. Sheared low-  coronal field, erupting y=0 plane: ‘Poynting flux’ in x direction y = 0 plane ‘Poynting flux’ in z direction Photospheric projection: Temperature (grey) Poynting flux (red) x z (Birn et al. 2009) 7 Inwards Poynting flux Downwards Poynting flux Time development of energy fluxes

8. Wave propagation in low-  plasma In corona & upper chromosphere, v A ≈ v th,e i.e.,  ≈ m e /m p The wave has an E II and can damp by electron acceleration (e.g. Bian talk) T = 4  10 6 K T = 3  10 6 K T = 2  10 6 K T = 10 6 K Case of  << m e /m p (‘inertial’ regime) requires k  large - i.e.  ≈ 3m to get acceleration to 10s of keV. (McClements & Fletcher 2009)  = m e /m p VAL-C 1.0 0.5 Accelerated fraction 1.03.05.0 

9. Wave propagation in  ≥ m e /m p plasma 9 Case of  ≥ m e /m p (kinetic regime): Wave can damp for larger transverse scales – order of  s = c/  pi Damping by Landau resonance (electron acceleration, Bian & Kontar 2010) – damping rate (s -1 ) is: So, still need to generate quite small transverse scales by phase mixing/turbulent cascade (Bian) Tnene B  cross Req’d  Corona10 6 K10 9 cm -3 500 G0.1s  = 0.3 km Chrom.10 4 K10 11 cm -3 1 kG1s  = 0.7 km Sample values, assuming || = 100km

10. Heating & acceleration in the chromosphere Electron acceleration needs  acc <  e-e In chromosphere, electron heating first (c.f. Yohkoh SXT & EIS impulsive footpoints @ 10 7 K, Mrozek & Tomczak 2004, Milligan & Dennis 2009) electrons heat,  scattering increases, and non-thermal tail produced. Electron acceleration timescale is that on which large k  is generated, e.g. by turbulent cascade: Take max =10km,  B/B = 10%, v A = 5000 km/s then  turb ≈ 0.02s e.g. @10 7 K, 1% of electrons have E > 5keV. at 10 11 cm -3, 10 7 K, 5keV electrons have  e-e = 0.02s => acceleration. e.g. Lazarian 04

11. Electron number estimates 11 Look at upper/mid VAL-C chromosphere: heating of chromosphere within 1/  (kinetic) = 1s T increases, tail becomes collisionless – within 1/  turb ~ 0.02s Non-thermal emission measure in chromosphere Accelerated fraction f ~ 0.01 n e ~ 10 11 cm -3 n h ~ 10 12 cm -3 (ionisation fraction ~ 10%) So volume V = 10 25 cm 3 If h = 1000km, needs A = 10 17 cm 2 - similar to HXR footpoint sizes. A h Chromospheric accelerating volume

12. Conclusions During a flare, magnetic energy is transported through corona and efficiently converted to KE of fast particles in chromosphere. Proposal – do this with an Alfven wave pulse in a very low  plasma Small amount of coronal electron acceleration in wave E field Perpendicular cascade in chromosphere & local acceleration 12 RAS Discussion Meeting, 8 Jan 2010 12 Overall energetics and electron numbers look plausible Many interesting questions concerning propagation & damping of these non-ideal (dispersive) waves in ~ collisionless plasmas.