Phys 777 Plasma Physcis and Magnetohydrodynamics 2004 Fall Instructor: Dr. Haimin Wang

Lecture 8

Solar Flares Solar Flares • Flares are most violent form energy release. They always occur at magnetic neutral lines. They are observed in Hα, microwares, EUV, X-ray and Hard X-ray. • X-ray classification: A,B,C, M,X (A,B: subflare)

Morphology: • Single loop --- compact, short lived • Two Ribbon --- long duration events, usually big

2 Basic Theory: Magnetic Reconnection

• Current sheets may be found in neutral lines and reconnect due to: (1) resistive instability, such as tearing mode (2) neutral line collapses due to separate flux systems are pushed together (3) sudden enhancement of resistivity • Steady reconnection ρprocess is governed by: r r r Ohm’s law: E + v × B =η∇ × B B Equation of motion: (vr ⋅∇)vr = −∇p + ∇ × B × , ∇ ⋅ B = 0 µ Continuity Equation: ∇ ⋅(ρvr) = 0

3 4 5 6 7 Unidirectional Fields x y v = −v , v = −v , B = B(x)yˆ x 0 a y 0 a 1 B2 p = const −ρ v2 − 2 2 v x dB µ E − 0 B =η a dx

• If diffusion is absent, η=0 B= Ea/v0 x • If diffusion is present, then inside diffusion region field can slip through the plasma • The width of the diffusion region is derived by

v0 x dB η 1/ 2 L B =η , l = ( ) a = 1/ 2 a dx av0 Rm 8 In Fig 10.3,ρ Bi are carried in, B0 are carried out reconnection. ρ B0 < Bi Some energy is lost by heating and kinetic energy.

• Parker derived * Bi η v0 = vA = 1/ 2 , vi = (µρc ) l * ρ ivi L = cv0l l << L if vi << vA ( c / ρi )

• Pressure balance: ρ 2 c =ρi +Bi /2µ ρ ρc Ti = (1+ βi ) i Tc

In reality. We have to consider the ratio between radiation and convection 9 10 11 12 13 Simple Loop Flares • It is most likely due to new flux emergence (Fig10.7). • There are 3 phases (1) Preflare phase: emerging flux reconnects with outlines fields, shock waves radiate from a current sheet and heat the plasma. (2) Impulsive phase: on set of turbulence in current sheet caused rapid expansion: E field accelerates particles

Downward ---- Hْalpha, Hard X-ray • • Upward ---- radio Type IV bursts (3) Main phase: both heat and particles are conducted down to lower chromosphere.

14 • Condition for onset of flare: 2 2 16 Tc > Tturb = 1.8 × 10 Bi/Vi When current sheet reaches a critical height current density reaches a certain value to trigger to flare Priest et. al. shows variation of critical height as a

function of Bi and Vi

• Next, treat flare by two models: Thermal Nonequilibrium and Instabilities

15 16 17 18 Thermal Nonequilibrium

• Normally, magnetic fields are considered as the source of flare energy, but thermal non-equilibrium could occur in the core of an active region to cause flux. 104 K heating 107 K plasma It may be a simple loop flare or a trigger for two-ribbon events (pre-flare heating) • This phase is metal stable • A simple static equation of equilibrium:

19 d  5/ 2 dT  2 k0T  = ne Q(T ) − H ds  ds 

Conduction optically thin radiation Heating Fig 10-10 shows that when H reaches a critical value heating occurs along the dashed line.

Kink Instability • Magnetic fields are unstable when twisted. The dominant stability effect is line-tying of ends of loop in dense photosphere. So we introduce the value critical twist Φc

20 21 22 • Consider a cylindrical flux tube with length 2L. φ ξ 2 2 2 Force balance: ξ dp d B + Bz Bφ 0 = + ( µ ) + φ dr dr ξ 2 µr B B z Suppose solution = [ δR (R),−i z 0 (R),i ξ 0 (R)]cos ei(mφ+kz) B ξ B π 2L µ R ∞ 1 d 2 2 The W = F( ) − GξRdR ∫0 dR

F.G are function of R, p, Bφ , Bz, k defined in p262 Solving Euler-Lagrange equation, subject to ξ ξd R R = 1 = 0 at R = 0, for m = 1 dR ξ dξ R R = 0 = 1 at R = 0, for m ≠ 1 dR 23 • Fig 7.8 gave a sample solution

• Another example uniform axial field, Bz=B0 2LB Φ Φ(R) = φ = 0 twist : 2 2 RBz 1+ R / a

2 2  Φa  B0 p(R) = p∞ +    2L  2µ • Fig 10.11 gives a summary of instability. In the absence of pressure gradient and line-tying the critical twist: 1 2L Φ(0) = − k 2 a This is called Kruskal-Shafranov limit

24 25 Resistive Kink Instability In most of cases, resistive term in the linearised induction equation: ∂B v 1 = ∇×(vv × B ) − ∇×(η∇× B ) ∂t 1 0 1 is negligible because Rm is large but in a layer v v ∇ × (v1 × B0 ) = 0 v ∂B can no longer be balanced by v v ∇×(v1 × B0 ) ∂t

26 For incompressible medium, ∇ ⋅ vv = 0 the locations of the 1 v v singular layers are given by v v if v = e i k ⋅ r . k ⋅ B0 = 0 1 The simplest force free equilibrium in cylindrical geometry is constant α field: Bφ = B0 J 1 (αR ) Bz = B0 J 0 (αR) Its stability may be investigated by seeking solution B = B ( R ) exp( ω t + i(m φ + kz )) ω 1 1 Assuming that there is no line-tying --- resistive internal kink mode, the fasted growing perturbations have long wavelength k ⋅ R s < < 1 and m = 1 grow rate: 2 / 3 τ 2 v v ' − 1 / 3 − 2 / 3 ≈ []R s ( k ⋅ B ) s / B 0 d τ A

2 τ d = Rs / n diffusion time τ A = Rs /vA Alfven travel time 27 Two-Ribbon Flares • A magnetic arcade responds to the slow photospheric motion of its foot point by evolving through a series of force-free equilibria. A some critical amount of shear, the configuration becomes unstable and erupts upwards (Fig 10.10).

• We also have two approaches: multiplicity of force-free equilibria and instability.

28 Existence and Multiplicity of Force-Free Equilibria • Evolution through a set of equilibria, it may erupt if a neighbouring equilibrium ceases to exist or a new equilibrium with low energy becomes possible. • Coronal arcade can be treated as 2-D force-free fields, independent of longitudinal coordinate. Say with components: ∂A ∂A B (x, y) = B (x,y)=− B (A) x ∂Z z ∂x y d 1 eq. 3.55 ∇2 A+ ( B2 ) = 0 dA 2 y Using boundary condition, we can solve this equation and obtain the evolution of fields. • Fig10.15 gave a sample solution 29 30 31 32 33 Eruptive Instability

Fig 10.17(a) shows preflare condition--- a weakly twisted flux tube anchored at its end. A flare loop is at distance d above photosphere, its length is 2L. The field can be represented by uniform twisted force-free field: B B =B (R/a)/(1+R2 /a2) B = 0 R << d θ 0 z 1 + R 2 / a 2

Sufficient conditions for instability are obtained by integrating an Euler-Lanrange Equation

34 d↑ easier to be unstable Fig10.17(b) Φ↑ easier to be unstable E.g. d=3b Φ≥4.2π , unstable Φ≤2π, flux tube is table to all kink perturbations.

Post-flare loops Kopp-Pneuman Model ↓ flare ribbons

Eruption Field line reconnection open one at a time 35 36 37 38 ρ Governing Equations: α α 2 Dvs Dvs 1 Dp GMΘ D vn vnvs +vs =− − 2 cos +vn + cosα + Dt Ds Ds r Dt r Rc ρ D D ρvn A ( A) + (ρvs A) − = 0 Dt Ds Rc

Rc: radius of curvature of field line. A: cross section. α: inclination to magnetic fields Solution:

3 3 1+ 2(r1 /r) 1−(r1 /r) Br = B0 3 sinθ Bθ =−B0 3 cosθ 1+ 2(r1 / RΘ) 1+2(r1 /RΘ)

39 neutral point location

−ωt r1(t) = (1.5−0.5e )RΘ

−5 −1 dr1 ω = 5.7×10 s at r1 = RΘ = 20km / s dt

t = ∞ r1 =1.5RΘ when a tube closed (reconnects), a shock propagates downward, plasma cools and falls to give Hα loops.

40 41 42 43 Homework

Assume that reconnection between two magnetic dipoles causes a flare. Construct a reasonable model to (1) Estimate energy release due to the reconnection; (2) Plot electric current distribution before and after the flare.

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