Solar flares, current sheets, and finite-time singularities in magnetohydrodynamics

Yuri Litvinenko

Department of Mathematics, University of Waikato, New Zealand

Flares and FTS – p.1/44 Flares and FTS – p.2/44 Outline

Solar flares: observations and theories

Reconnecting current sheets: basic properties

Self-similar solutions for current sheet formation in 2D magnetohydrodynamics (MHD) and Hall MHD

Finite-time singularities in Hall MHD

Flares and FTS – p.3/44 GOES X-ray image of a solar flare (9/9/2005)

Flares and FTS – p.4/44 TRACE close-up of the same flare

Flares and FTS – p.5/44 The flaring solar corona (Yohkoh X-ray image)

Flares and FTS – p.6/44 SDO EUV image of an eruption (31/08/2012)

Flares and FTS – p.7/44 Why study solar flares?

1. Solar flares have a direct effect on the Earth.

Energetic particles, accelerated in solar flares, are dangerous to astronauts and to electronic instruments in space. The atmosphere expands due to heating by the flare radiation, increasing the drag on the satellites orbiting around the Earth. Plasma density variations in the ionosphere can impact high-frequency radio communication and GPS navigation systems. Power grid vulnerability, pipeline corrosion, ...

2. Flare-like energy release processes take place on other stars and in accretion disks.

Flares and FTS – p.8/44 Energy release in flares

Flare: impulsive release of magnetic energy as radiation, heat, fast flows and particles in the lower corona of the Sun.

Total flare energy up to 1032 erg: E ≃ B2 B = 100 G,L = 1010 cm = L3 ⇒ 8π ≥ E Flare time t 103 s: f ≃ L B tf 100 tA, tA = , vA = ≤ vA √4πρ

Flares and FTS – p.9/44 Resistive energy release rate

L W = I2R = I2 η S I BL, S L2 = W η ∼ ≃ ⇒ ∼ Problem: η T −3/2 = W is too small. ∼ ⇒ Solution: flatten I into a sheet. η l L, S lL = W ≪ ≃ ⇒ ∼ l The famous Sweet–Parker (1957) scaling:

l W η1/2 ∼ ∼

Flares and FTS – p.10/44 Reconnecting current sheet

Flares and FTS – p.11/44 Solar flare geometry

(Sturrock, 1968)

Flares and FTS – p.12/44 Observational evidence for reconnection

(McKenzie, 2001)

Flares and FTS – p.13/44 RHESSI X-ray images of a current sheet

(Sui and Holman, 2003)

Flares and FTS – p.14/44 Flare geometry in three dimensions

(Machado et al., 1983)

Flares and FTS – p.15/44 Hugh Hudson’s archive of flare cartoons

Flares and FTS – p.16/44 More flare cartoons

Flares and FTS – p.17/44 Even more ...

Flares and FTS – p.18/44 A recent flare cartoon

(Janvier, 2014)

Flares and FTS – p.19/44 Current sheet formation at a neutral line

Flares and FTS – p.20/44 Parameters of a solar active region

Typical values:

L = 109.5 cm,T = 106 K, ρ = 10−15 g cm−3,

B B = 102 G, v = 109 cm s−1 A √4πρ ≃ Dimensionless resistivity:

c2 η = 10−14.5 4πvALσ ≃ Dimensionless ion inertial length:

c −6.5 1/2 di = 10 >η Lωpi ≃

Flares and FTS – p.21/44 Dimensionless equations of Hall MHD

Momentum equation:

∂ v +(v )v = p + J B t · ∇ −∇ ×

Ohm’s law: E + v B = d (J B p ) × i × − ∇ e

Incompressibility: v = 0 ∇ · Maxwell’s equations:

E = ∂ B, B = 0, J = B ∇× − t ∇ · ∇×

Flares and FTS – p.22/44 21 2D Hall MHD solution

v(x,y,t) = φ zˆ + W zˆ ∇ × B(x,y,t) = ψ zˆ + Zzˆ ∇ ×

Planar components:

∂ ( 2φ) + [ 2φ,φ] = [ 2ψ,ψ] t ∇ ∇ ∇

∂tψ + [ψ,φ] = di[ψ, Z] Axial components:

∂tW + [W, φ] = [Z,ψ]

∂ Z + [Z,φ] = [W, ψ] + d [ 2ψ,ψ] t i ∇

Flares and FTS – p.23/44 A self-similar solution

Stream function: φ = γ(t)xy −

Flux function: ψ = α(t)x2 β(t)y2 −

Axial speed: W = f(t)x2 + g(t)y2

Axial magnetic field: Z = h(t)xy

Flares and FTS – p.24/44 v and B at t = 0

Flares and FTS – p.25/44 Reconnection in a resistive viscous plasma

η = 0, ν = 0 = 6 6 ⇒ ψ ψ + 2η (α β)dt → − Z

W W + 2ν (f + g)dt → Z

Flares and FTS – p.26/44 The similarity reduction (a system of ODEs)

α˙ 2αγ 2d αh = 0 − − i

β˙ + 2βγ + 2diβh = 0

f˙ 2γf + 2αh = 0 −

g˙ + 2γg + 2βh = 0

h˙ + 4αg + 4βf = 0

Flares and FTS – p.27/44 Current sheet formation in 2D MHD

di = 0

f = g = h = 0, γ = γ0

α˙ 2αγ0 = 0 − β˙ + 2βγ0 = 0

Exponential growth, no finite-time singularities (FTS):

α(t) = exp(2γ0t)

β(t) = exp( 2γ0t) − (Chapman and Kendall, 1963; Sulem et al., 1985; Grauer and Marliani, 1998)

Flares and FTS – p.28/44 The simplest model of a finite-time singularity

1D fluid flow, v = v(x, t):

∂tv + v∂xv = 0

v = ω(t)x

ω˙ + ω2 = 0

ω0 ω(t) = 1 + ω0t

FTS: ω0 < 0 = ω(t) as t 1/ ω0 ⇒ →∞ → | |

Flares and FTS – p.29/44 Effect of the Hall term on the magnetic field

di > 0

t 1: ≪ α(t) 1+(2γ0 + 2d h0)t ≈ i β(t) 1 (2γ0 + 2d h0)t ≈ − i

The angle between the magnetic separatrices, ψ = 0:

1 2 θ β / tan = 1 (2γ0 + 2d h0)t 2 α ≈ − i  

Flares and FTS – p.30/44 4 Collapse to a current sheet: dih0 = +10−

ln Α 15 12.5 10 7.5 5 2.5 t 0.5 1 1.5 2 2.5 3

α(0) = β(0) = 1, γ(0) = γ0, f(0) = g(0) = 0

Flares and FTS – p.31/44 4 Collapse to a current sheet: dih0 = +10−

ln Β t -2 0.5 1 1.5 2 2.5 3 -4 -6 -8 -10 -12 -14

α(0) = β(0) = 1, γ(0) = γ0, f(0) = g(0) = 0

Flares and FTS – p.32/44 4 Collapse to a current sheet: dih0 = +10−

lnHh diL 15 10 5 t 0.5 1 1.5 2 2.5 3 -5

α(0) = β(0) = 1, γ(0) = γ0, f(0) = g(0) = 0

Flares and FTS – p.33/44 4 Collapse to a current sheet: d h = 10− i 0 −

ln Α 3 2 1 t 0.5 1 1.5 2 2.5 3 -1 -2

α(0) = β(0) = 1, γ(0) = γ0, f(0) = g(0) = 0

Flares and FTS – p.34/44 4 Collapse to a current sheet: d h = 10− i 0 −

ln Β 3 2 1 t 0.5 1 1.5 2 2.5 3 -1 -2 -3

α(0) = β(0) = 1, γ(0) = γ0, f(0) = g(0) = 0

Flares and FTS – p.35/44 Finite-time singularity: a mechanical analogy

FTS: h(t) as t t →∞ → s

h¨ + U ′(h) = 0

1 2 4 2 2 1 2 4 2 2 2 U(h) = (d h + a h ) + (d h0 + a h0) 8(α0g0 + β0f0) −2 i 2 i −

2 2 2 a = 2[4d (α0g0 β0f0) 8α0β0 + d h0] − i − − i

Flares and FTS – p.36/44 Finite-time singularity: a mechanical analogy

Flares and FTS – p.37/44 Criterion for the FTS absence

α0β0(α0 + d f0)(β0 d g0) 0 i − i ≥

d (α0g0 β0f0) 2α0β0 0 i − − ≥

(Litvinenko and McMahon, 2014)

Flares and FTS – p.38/44 Behaviour near the singularity (h> 0)

t ′ ′ τ =(ts t) 0, Γ = γ(t )dt Γs − → 0 → Z

1 −2 α exp(2Γs)τ ≈ 4(β0 d g0) − i 2 β 4α0β0(β0 d g0)exp( 2Γ )τ ≈ − i − s 1 −2 dif exp(2Γs)τ ≈−4(β0 d g0) − i d g (β0 d g0)exp( 2Γ ) i ≈− − i − s d h τ −1 i ≈

Flares and FTS – p.39/44 An intermediate asymptotic solution

h˙ 0 h(t) h0 cosh(at) + sinh(at) ≈ a ! ×

2 −1 2 ˙ di h0 1 2 h0 + exp(2at)  − 16a a !    Singularity time, h(t ) = : s ∞ −2 2 1 16a h˙ 0 ts = ln 2 h0 + 2a  di a !   

Flares and FTS – p.40/44 Homework problem

Find an exact solution for h(t) in terms of Jacobi elliptic functions.

Flares and FTS – p.41/44 Solar flares as finite-time singularities?

Flares and FTS – p.42/44 An estimate for the solar flare onset time

L = 109.5 cm,T = 106 K, n = 109 cm−3,

2 B 9 −1 B = 10 G, vA = 10 cm s 4πmpn ≃ p c −6.5 di = 10 Lωpi ≃

L 1 a h0 1 = ts ln 30 s ≃ ≃ ⇒ ≃ vA di ≃

Flares and FTS – p.43/44 Summary

Observational data and theoretical models strongly suggest that reconnecting current sheets, separating the interacting magnetic fluxes in the solar atmosphere, play the key role in the dynamics and energetics of solar flares.

The current sheet formation can be modelled as the development of a singularity in the solution for the electric current density at a magnetic neutral line. A finite-time collapse to the current sheet can occur in a weakly collisional plasma, described by Hall magnetohydrodynamics.

Predictions made using the exact self-similar solutions may have important implications for magnetic reconnection in the laboratory and space plasmas.

Flares and FTS – p.44/44