2009 Laboratory, Space, and Astrophysical Plasma Workshop 20 February 2009 A An Overview of O i f Solar Eruption Theories Solar Eruption Theories G. S. Choe G. S. Choe School of Space Research Kyung Hee University i i Yongin, Korea
Contents 1 1. Overview of solar eruptive phenomena Overview of solar eruptive phenomena 2. Current sheets – A necessary condition f for magnetic reconnection i i 3. 3. MHS equilibria with different field S equ b a w t d e e t e d topologies 4 4. B i f Brief review of dynamic theories i f d i th i 5. On the field opening p g
Solar Activity and Magnetic Field • A longitudianl magnetogram shows only line-of-sight component. l i di l h l li f i h • Polarity inversion line – a border between different polarities
Solar Eruptive Phenomena p • Solar Flares – EM radiation in all wavelengths – 1-4 hour duration – 10 26 -10 32 erg of EM energy released • Prominence Eruption Prominence Eruption – Sudden rise (and disappearance) of a prominence – Often accompanied by a flare and/or a CME Often accompanied b a flare and/or a CME • Coronal Mass Ejections (CMEs) – Density-enhanced mass motion in the corona – 10 14 -10 17 g, 100-2000 km s -1 g – 10 28 -10 33 erg of kinetic energy
Solar Flare on 2000 July 14 (Bastille day event)
Prominence Eruption Prominence Eruption
Coronal Mass Ejection Coronal Mass Ejection SOHO LASCO C3 observation of the corona for a month
X-ray Emission of a Flare • Soft X-ray: along the loop; thermal • Hard X-ray: H d X - Loop-top - Footpoins: more non-thermal
Standard Eruptive Flare Model − CSHKP Models − CSHKP Models Shibata et al., 1995
Solar Eruption and Magnetic Shear • Magnetic Shear — Tilt of a magnetic field vector from the direction of the corresponding potential field • Essential condition for solar flares, formation and eruption of , p prominences and coronal mass ejections
Solar Plasmas Are Highly Conductive! Solar Plasmas Are Highly Conductive! The plasma is frozen in the magnetic field. Definition of the general magnetic reconnection Definition of the general magnetic reconnection by Schindler and Hesse (1988).
Magnetic Reconnection − A More Conventional Picture A M C ti l Pi t • A thin current layer with high current density is a necessary condition for magnetic reconnection. • A singular current sheet would be a sufficient condition for magnetic reconnection.
Why Current Sheets? Why Current Sheets? in vacuum in vacuum � � � � � � magnetic ideal MHD reconnection plasma p
Current Sheet Formation Is Natural for Emerging Fluxes Emerging flux model of solar flares by Heyvaerts et al. (1977)
Flaring regions are more or less bipolar. Shibata et al., 1995
Is This Transition Possible? Sorry! y I adopted a figure of the Earth’s magnetotail. If the static configuration changes its field topology by a continuous change of a physical boundary condition, the transition is possible.
Solar Eruption and Magnetic Shear The magnetic field component in the direction of the polarity inversion line (hereafter the toroidal component) polarity inversion line (hereafter the toroidal component) is important!
Force-free Field Approximation pp Magnetohydrostatic (MHS) Equilibrium × − ∇ − ρ ∇ = J B p Φ 0 g Force-free field × B = J 0 Justified in the solar corona by Justified in the solar corona by p p β β = = = = << << plasma plasma 1 1 . 1 1 π p B 2 /( 8 ) magnetic << << h h H H 2 2 . photospher e - corona corona
2.5D MHS Equilibrium – Grad-Shafranov Equation ⎛ B ⎛ ⎞ ⎞ d d B 2 2 ⎜ ⎟ = ∇ ψ = − + J z p 2 ⎜ ⎟ z ψ d ⎝ ⎠ 2
Problem Setting of 2.5D Equilibrium g q I. Generating Function Method Arbitraril Arbitraril y y impose impose ψ ψ ) ) ) ) B z B p( ψ p( ψ ( ( and and ψ ψ r r and and solve solve for for ( ( ) ).
An Analytic Solution by B. C. Low (1977)
An Analytic Solution by B. C. Low (1977) • Increasing the toroidal current density can create a current density can create a magnetic island. • The toroidal current density • The toroidal current density, however, is not a physically controllable quantity controllable quantity. • Thus, the solution sequence i is not physically continuous. h i ll i
Problem Setting of 2.5D Equilibrium g q II. Imposing the footpoint displacement Imposing 1 footpoint displacement 1. footpoint displacement 2. entropy per flux
N merical Sol tions of BVP2 Numerical Solutions of BVP2 Choe & Lee 1996 Magnetofrictional method: • Start with any configuration with a desired field topology y g p gy • Remove kinetic energy after every time step
Dynamic Resistive MHD Simulations Dynamic Resistive MHD Simulations Mikic et al 1988 Mikic et al. 1988 Biskamp (1988) criticizes that the reconnection is initiated p ( ) by squeezing of the adjacent arcades.
Dynamic Resistive MHD Simulations Dynamic Resistive MHD Simulations A magnetic island is found to form in an found to form in an isolated sheared arcade. Mikic & Linker 1994
Flux Rope Formation by Reconnection of Looped Field Lines Looped Field Lines Van Ballegooijen & Martens, 1989 A 2.5D magnetic island is a line-tied flux rope in 3D.
Dynamics of a Line Current in the Corona (Van Tend & Kuperus, SP 59 , 115, 1978) Assumption • The solar surface is a rigid perfect conductor. (infinite inductance assumption) • Thus, the coronal current is closed on the photosphere. • This can be modeled by an image current below the surface. Thi b d l d b i t b l th f • No equilibrium position. The ambient potential field and • The current will go to infinity. gravity exert a downward force.
Dynamics of a Line Current in the Corona (Van Tend & Kuperus, SP 59 , 115, 1978) log B log A, B, D stable B(G) (I/c ) C unstable
Flux Rope Catastrophe Model (F (Forbes, Priest, Isenberg, Lin) b P i t I b Li ) • A flux rope with a finite cross- A fl ith fi it section replaces the line current. current. • Photospheric magnetic reconnection resembling flux g cancellation transfers photospheric flux to the flux rope. • The equilibrium position of th fl the flux rope is sought as a i ht function of the poloidal flux in the flux rope the flux rope.
Flux Rope Catastrophe Model p p Forbes & Isenberg, ApJ 373 , 294, 1991 b b
Flux Rope Catastrophe Model Forbes & Isenberg, ApJ 373 , 294, 1991 Priest & Forbes, AARv 10 , 313, 2002
Tether-Cutting Model Sturrock, SP 121 , 387, 1989 Moore et al., ApJ 552 , 833, 2001
Current Driven Instability of Flux Ropes Current Driven Instability of Flux Ropes (Chen, 1989) Assumptions • Coronal current is closed beneath the solar surface. • Coronal current is governed by the subsurface condition. • The solar surface is not a rigid • The solar surface is not a rigid perfect conductor, but it can respond to the coronal evolution. p For a strong enough toroidal current the flux rope is current, the flux rope is unstable to radial expansion if the photospheric inductance is the photospheric inductance is low enough.
Comparison of Two Flux Rope Models Comparison of Two Flux Rope Models Forbes Chen • Photospheric inductance • Photospheric inductance infinite finite • Poloidal flux of the flux • Flux rope poloidal flux p p rope increased by flux injected from below the cancellation surface • Ignorable photospheric • Measurable photospheric motion during eruption motions (horizontal and/or vertical) during eruption
Ejection of Self-closed Field Structures Ejection of Self closed Field Structures (Gibson and Low 1998) • A self closed field structure • A self-closed field structure similar to a torus is embedded in an open field and pinned in an open field and pinned down at one point. • It expands in a self-similar manner manner.
Ejection of Self-closed Field Structures Ejection of Self closed Field Structures • Inertia-dominant Inertia dominant solar interior is not considered considered. • Photospheric field Photospheric field (both vertical and horizontal components) horizontal components) changes a lot during eruption. eruption.
Kink Instability • A twisted flux rope tends to increase the pitch by self- p y braiding. • The instability threshold depends on the BC (e.g., line-tying), aspect ratio l / a , etc. t • The instability by itself does not seem sufficient to not seem sufficient to account for solar eruption. lB φ ( r ( ) ) φ Φ = rB z r ( ) Rust & LaBonte, ApJ 622 , 2 π ⎧ ⎧ ( (Rust & Kumar, , 1996) ) L69, 2005 L69, 2005 ⎪ ⎪ Φ = π ⎨ 2.5 (Hood & Priest, 1981) C ⎪ π ⎩ 5 (Mikic et al., 1990)
Simulations of Kink Instability Φ = π 2 . 1 A magnetic loop equilibrium model by Φ = π Ti Titov & Demoulin & D li 4 . 9 (1999) is used as the Φ Φ C ≈ C ≈ π π 3 3 . 5 5 initial condition initial condition. Torok et al., AA 413 , L27, 2004
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