Sept. 13, 2007 A Global Effect of Local Time- Dependent Reconnection In Collaboration with Dana Longcope Eric Priest

Sept. 13, Introduction - Reconnection on Sun (a) Solar Flare:

Sept. 13, 2007 An Effect: seismi cflare wave In photosphere (MDI) (8-100 km/s 70 mins)

Sept. 13, 2007 Chromospheric flare wave:

Sept. 13, 2007 Yohkoh- glowing in x-rays Coronal Heating:

Sept. 13, 2007 Hinode X-ray Telescope (1 arcsec):

Sept. 13, 2007 Close-up How heated?

Sept. 13, 2007 Coronal Tectonics Model  --> web separatrix surfaces (updated version of Parker nanoflare/topological dissipation)  Corona filled w. myriads of J sheets, heating impulsively (see talk by Haynes and by Wilmot-Smith) Each ‘Coronal Loop’ --> surface in many B sources  As sources move --> J sheets on surfaces --> Reconnect --> Heat

Sept. 13, there are many nulls in corona ?? Some j sheets at nulls/separators (see Pontin talk) “X marks the spot” Many sheets also at QSLs-non-nulls

Sept. 13, 2007 If reconnection heating corona at many sheets, 1. How does energy spread out ? -- conduction along B -- waves across B 2. If reconnection time-dependent, how much energy liberated locally/globally? Simple model problem

Sept. 13, Magnetic fields in 2D contours are field lines X-point: A = const current: A - flux function j B x =B ’ y, B y =B ’ x B y +iB x =B ’ (x+iy)=B ’ w

Sept. 13, 2007 branch cut branch point A Current Sheet (Green 1965) Current I 0 j ? large r

Sept. 13, 2007 At large r X-point field

Sept. 13, 2007 At large r perturbation: line current Most of perturbation energy to distance L Lots of energy far from CS ~ ~

Sept. 13, 2007 Suppose sheet reconnects Local process but has global consequences: Decrease I --> B must change at great distance How ??

Sept. 13, Model for effect of reconnection Linearize about X-point B 0 :  is “turned on” current diffuses i.e. reconnection Assume a small current sheet, i.e.(natural diffusion length) so that sheet rapidly diffuses to circle and dynamics can be approximated by linearising about X Assume B t=0 is due to current sheet

Sept. 13, 2007 Equations Use only axisymmetric (m=0) part natural variable

Sept. 13, 2007 Natural New Variables r I r (r)  I 0 Expect:

Sept. 13, 2007 Combine: Governing equations EVEV EVEV II I III wave diffusive Natural length-scale: Induction Motion

Sept. 13, 2007 Nondimensionalise -----> EVEV EVEV EVEV EVEV I I I I I EVEV EVEV I ? large r and small r

Sept. 13, Limits (i) Large r (wave) limit: when telegraphers equations R EVEV EVEV EVEV I I I II0I0 I0I0 >>

Sept. 13, Limits (i) Large r (wave) limit: when telegraphers equations rightward leftward (Fast Magneto- Sonic waves) EVEV EVEV EVEV EVEV I I I II0I0 R t I0I0 I0I0 I >>

Sept. 13, 2007 (ii) Small r (diffusive) limit: when Current density: classic diffusion: (from t=0) EVEV EVEV EVEV << I0I0 r I0I0 I0I0 I I I I IdId I0I0 I (i.e. r small) jdjd r jdjd

Sept. 13, 2007 (iii) Numerical Solution I E I E I E I E I E I E R R=20 R=-7 E=0 I=0 E=0 I’=0 Use a uniform staggered grid in R, advancing I and E alternately EVEV EVEV I II (11 orders of magnitude in r),

Sept. 13, Numerical Solution I E E I E I E I E I E R R=20 R=-7 E=0 I=0 E=0 I ’=0 initial t=0.001/  A Diff n term advanced implicitly in an operator splitting method EVEV EVEV EVEV I I II I I0I0 I0I0 I0I0

Sept. 13, 2007 Numerical Solution for I ( diffusive solution) (Departs slightly from diffusive solution at small r) I log r

Sept. 13, 2007 Numerical Solution Diffusive solution I(r) R t Transition: diffusive to wave solution Wave solution

Sept. 13, 2007 EVEV EVEV EVEV I II II Wave solution: Numerical solutions for I and E u

Sept. 13, 2007 I Magnetic Field: Sheath of Current propagates out leaving I = 0 behind

Sept. 13, 2007 E v --> v EVEV EVEV I But flow near X does not disappear -- it slowly increases ! In wake of sheath - a flow

Sept. 13, 2007 (diffusive limit) Flux function A? A -->X-point Electric field I Numerical solution has E const (driven by outer wave soln) Diffusive soln has E decreasing in time log r t A(0,t)A(r,t)

Sept. 13, 2007 I I 5. Resolving the Paradox -- a 3rd regime -- at later times - eg in j(r,t): new regime ~ I Diffusive solution j t log r jdjd j d (0,t)

Sept. 13, 2007 Resolving the Paradox j EVEV EVEV I I I But what is j(r,t)? At large t advection - diffusion I0I0 j a/d (r,t)

Sept. 13, 2007 New Regime I0I0 EVEV I0I0 I I j a/d (r,t) Peak in j at X-point remains -- produces a steady E (independent of )

Sept. 13, 2007 Energetics EVEV I Magnetic energy Kinetic energy Poynting flux Ohmic heating EVEV For an annular region (in or out) (negative)

Sept. 13, 2007 Energetics Heat (integrated from t=1/ ) (i) K.E Heat (ii) Large t K.E Diffusion --> Magnetic energy into Ohmic heat Magnetic energy into K.E

Sept. 13, Summary Response to enhanced  in current sheet (CS): (i) Diffusion spreads CS out (ii) Wave (Lorentz force) carries current out at v A - as sheath Most magnetic energy is converted into kinetic energy in wave -- Coronal heating -- reconnection + wave may later dissipate. (iii) Peak in j at X remains --> steady E independent of i.e. fast

Sept. 13, 2007