Solar ﬂares, current sheets, and ﬁnite-time singularities ...yuril/sol_sing.pdf · 1. Solar...

Solar flares, current sheets,and finite-time singularitiesin magnetohydrodynamics

Yuri Litvinenko

Department of Mathematics, University of Waikato, New Zealand

Flares and FTS – p.1/44

Outline

Solar flares: observations and theories

Reconnecting current sheets: basic properties

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

Finite-time singularities in Hall MHD


GOES X-ray image of a solar flare (9/9/2005)


TRACE close-up of the same flare


The flaring solar corona (Yohkoh X-ray image)


SDO EUV image of an eruption (31/08/2012)


Why study solar flares?

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

Energetic particles, accelerated in solar flares, are dangerous toastronauts 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 impacthigh-frequency radio communication and GPS navigationsystems.

Power grid vulnerability, pipeline corrosion, ...

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


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 toE ≃ 1032 erg:

B = 100 G, L = 1010 cm =⇒ B2

8πL3 ≥ E

Flare timetf ≃ 103 s:

tf ≤ 100 tA, tA =L

vA, vA =

B√4πρ


Resistive energy release rate

W = I2R = I2L

Sη

I ∼ BL, S ≃ L2 =⇒ W ∼ η

Problem:η ∼ T−3/2 =⇒ W is too small.

Solution: flattenI into a sheet.

l ≪ L, S ≃ lL =⇒ W ∼ η

l

The famous Sweet–Parker (1957) scaling:

l ∼W ∼ η1/2


Reconnecting current sheet


Solar flare geometry

(Sturrock, 1968)


Observational evidence for reconnection

(McKenzie, 2001)


RHESSI X-ray images of a current sheet

(Sui and Holman, 2003)


Flare geometry in three dimensions

(Machado et al., 1983)


Hugh Hudson’s archive of flare cartoons


More flare cartoons


Even more ...


A recent flare cartoon

(Janvier, 2014)


Current sheet formation at a neutral line


Parameters of a solar active region

Typical values:

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

B = 102 G, vA =B√4πρ

≃ 109 cm s−1

Dimensionless resistivity:

η =c2

4πvALσ≃ 10−14.5

Dimensionless ion inertial length:

di =c

Lωpi

≃ 10−6.5 > η1/2


Dimensionless equations of Hall MHD

Momentum equation:

∂tv + (v · ∇)v = −∇p+ J × B

Ohm’s law:E + v × B = di(J × B −∇pe)

Incompressibility:∇ · v = 0

Maxwell’s equations:

∇× E = −∂tB, ∇ · B = 0, J = ∇× B


212D Hall MHD solution

v(x, y, t) = ∇φ× z +W z

B(x, y, t) = ∇ψ × z + Zz

Planar components:

∂t(∇2φ) + [∇2φ, φ] = [∇2ψ,ψ]

∂tψ + [ψ, φ] = di[ψ,Z]

Axial components:

∂tW + [W,φ] = [Z,ψ]

∂tZ + [Z, φ] = [W,ψ] + di[∇2ψ,ψ]


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


v and B at t = 0


Reconnection in a resistive viscous plasma

η 6= 0, ν 6= 0 =⇒

ψ → ψ + 2η

∫

(α− β)dt

W → W + 2ν

∫

(f + g)dt


The similarity reduction (a system of ODEs)

α− 2αγ − 2diαh = 0

β + 2βγ + 2diβh = 0

f − 2γf + 2αh = 0

g + 2γg + 2βh = 0

h+ 4αg + 4βf = 0


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)


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

ω(t) =ω0

1 + ω0t

FTS:ω0 < 0 =⇒ ω(t) → ∞ as t→ 1/|ω0|


Effect of the Hall term on the magnetic field

di > 0

t≪ 1:

α(t) ≈ 1 + (2γ0 + 2dih0)t

β(t) ≈ 1 − (2γ0 + 2dih0)t

The angle between the magnetic separatrices,ψ = 0:

tanθ

2=

(

β

α

)1/2

≈ 1 − (2γ0 + 2dih0)t


Collapse to a current sheet:dih0 = +10−4

0.5 1 1.5 2 2.5 3t

2.55

7.510

12.515

ln Α

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



0.5 1 1.5 2 2.5 3t

-14-12-10-8-6-4-2

ln Β

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



0.5 1 1.5 2 2.5 3t

-5

5

10

15

lnHh diL

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


Collapse to a current sheet:dih0 = −10−4

0.5 1 1.5 2 2.5 3t

-2

-1

1

2

3ln Α

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


Collapse to a current sheet:dih0 = −10−4

0.5 1 1.5 2 2.5 3t

-3-2-1

123ln Β

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


Finite-time singularity: a mechanical analogy

FTS: h(t) → ∞ as t→ ts

h+ U ′(h) = 0

U(h) = −1

2(d2

ih4 + a2h2) +

1

2(d2

ih4

0+ a2h2

0) − 8(α0g0 + β0f0)

2

a2 = −2[4di(α0g0 − β0f0) − 8α0β0 + d2

ih2

0]


Finite-time singularity: a mechanical analogy


Criterion for the FTS absence

α0β0(α0 + dif0)(β0 − dig0) ≥ 0

di(α0g0 − β0f0) − 2α0β0 ≥ 0

(Litvinenko and McMahon, 2014)


Behaviour near the singularity (h > 0)

τ = (ts − t) → 0, Γ =

∫ t

0

γ(t′)dt′ → Γs

α ≈ 1

4(β0 − dig0)exp(2Γs)τ

−2

β ≈ 4α0β0(β0 − dig0) exp(−2Γs)τ2

dif ≈ − 1

4(β0 − dig0)exp(2Γs)τ

−2

dig ≈ −(β0 − dig0) exp(−2Γs)

dih ≈ τ−1


An intermediate asymptotic solution

h(t) ≈(

h0 cosh(at) +h0

asinh(at)

)

×

1 − d2

i

16a2

(

h0 +h0

a

)2

exp(2at)

−1

Singularity time,h(ts) = ∞:

ts =1

2aln

16a2

d2

i

(

h0 +h0

a

)

−2


Homework problem

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


Solar flares as finite-time singularities?


An estimate for the solar flare onset time

L = 109.5 cm, T = 106 K, n = 109 cm−3,

B = 102 G, vA =B

√

4πmpn≃ 109 cm s−1

di =c

Lωpi

≃ 10−6.5

a ≃ h0 ≃ 1 =⇒ ts ≃L

vA

ln1

di

≃ 30 s


Summary

Observational data and theoretical models strongly suggest thatreconnecting current sheets, separating the interacting magneticfluxes in the solar atmosphere, play the key role in thedynamics and energetics of solar flares.

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

Predictions made using the exact self-similar solutions mayhave important implications for magnetic reconnection in thelaboratory and space plasmas.


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