PS420 Solar Magnetohydrodynamics: 2013-2014
Magnetic Reconnection and Solar Flares
1. Magnetic reconnection
We have already discussed an magnetic X-point (or a null point). In 2D it is given by
the equation B = yx̂ + xŷ . The field lines of this magnetic configuration are parabolae,
y 2 ! x 2 = const .
You may calculate the j ! B force in this configuration and find that it is zero
everywhere (the magnetic curvature force is balanced by the gradient of the
magnetic pressure force everywhere).
Let us deform this equilibrium, B = yx̂ + ! 2 xŷ , where α < 1 .
The field lines are y 2 − α 2 x 2 = const .
The electric current density in this configuration is jz =
Hence, the magnetic force is j ! B = "
1 ⎛ ∂By ∂Bx ⎞ α 2 − 1
.
−
⎜
⎟=
µ0 ⎝ ∂x ∂y ⎠
µ0
(! 2 "1)! 2 x
(! 2 "1) y
x̂ +
ŷ .
µ0
µ0
PS420 Solar Magnetohydrodynamics: 2013-2014
Let us calculate the values of the force at different points:
! 2 !1
! 2 !1
j ! B (0,1) =
ŷ, j ! B (0,!1) = !
ŷ,
µ0
µ0
(! !1)!
=!
2
j ! B (1,0)
µ0
2
(! !1)!
=
2
x̂, j ! B (!1,0)
µ0
2
x̂.
Thus, the configuration is not in equilibrium. If an X-point is perturbed, the
perturbation grows. The forces generate flows of the plasma.
If η ≠ 0 this leads to reconnection of the magnetic field lines: the lines are brought by
the inflows to the X-point and are moved away by the outflows.
A similar phenomenon can happen in a current sheet:
The Sweet – Parker stationary solution.
The plasma diffuses into the current layer at some relatively small inflow velocity vi .
(More specifically: there is the total pressure balance across the current sheet; in the
vicinity of the current sheet there are large gradients of the field and hence diffusion;
the total pressure outside the current sheet is getting higher, resulting into a
pressure gradient force, moving field lines toward the current sheet).
PS420 Solar Magnetohydrodynamics: 2013-2014
In the current sheet the oppositely directed magnetic field lines get reconnected,
resulting into the magnetic tension forces in the horizontal direction. These forces
drive the frozen-in plasma – sling shot effect. The plasma is accelerated along the
layer, and eventually expelled from its ends at some relatively large velocity v0 .
What does accelerate the plasma? Let us neglect the gas pressure, !p = 0 , and
consider only stationary reconnection, ! / !t = 0 . The Euler equation gives us
! ( V.!) V = j " B. For the x-component of the equation we get
! ( V.!)Vx = ( j " B) x =
(
1 % $ B2
#
+ ( B.!) Bx * .
'
µ & $x 2
)
!Vx By !Bx
"
.
!x
µ !y
Making an order-of-magnitude estimation (see the sketch above for the geometrical
scales) we get
V2 B B
! o ! o i.
L
µ"
The conservation of mass and the assumed incompressibility of the plasma give us
! / L = Vi / Vo , or Bo / Bi = Vi / Vo (here we used that Bi ,Vi = Bx ,Vy and Bo,Vo = By ,Vx , see
the sketch, and that the frozen-in condition is fulfilled everywhere except the small
region of reconnection shown by the dark bar in the sketch). Hence, we get
B BL B BB
Vo2 ! o i = o i i = CAi2
µ!"
µ! Bo
Along centre of diffusion region: Bx = 0, !By / !x " 0. It leaves !Vx
where C Ai is the Alfven speed.
The rate of magnetic reconnection is defined as M 0 = vi / v0 ! S !1/2 , where
S = µ0 LCA / η is the Lundquist number (c.f. with the Reynolds number).
According to the Sweet – Parker model, the magnetic reconnection takes place on
the hybrid time scale τ 1/A 2τ 1/R 2 , where the first factor is the Alfven transit time across
the current sheet and the second factor is the time of the resistive diffusion across
the plasma.
Typically, in the corona S ≈ 107 −9. This gives the characteristic time of reconnection
to be about a few tens of days. This is too long to explain dynamical phenomena in
the solar atmosphere.
PS420 Solar Magnetohydrodynamics: 2013-2014
There have been several methods suggested to speed this process up: the Petchek
reconnection based upon the assumption that the inflows form shock waves,
anomalous resistivity, etc. The problem of “fast reconnection” is one of the key
problems of modern solar and space plasma physics.
Energy conversion in magnetic reconnection:
The input energy is the energy stored in the magnetic field, .
Change of B because of reconnection generates steep gradients of B, hence
increase in ! " B . This leads to the increase in the current density. As the diffusivity
is not negligible in the reconnection region, it is subject to Ohmic dissipation, hence
increase in internal energy of the plasma.
Also, the slingshot effect generates bulk flows of plasma, hence increase in its
kinetic energy.
The electric field E = !V " B accelerates plasma particles: non-thermal high energy
particles.
PS420 Solar Magnetohydrodynamics: 2013-2014
2. Solar Flares
Classification of solar flares:
Hα classification
Radio flux
at 5000
Importance Area
Area
MHz in
Class
(Sq.
10-6 solar s.f.u.
Deg.)
disk
S
2.0
200
5
1
2.0–5.1 200–500 30
2
5.2–
500–1200 300
12.4
3
12.5– 1200–
3000
24.7
2400
4
>24.7 >2400
3000
Soft X-ray class
Importanc Peak flux
e class
in 1-8 Å w/m2
A
B
C
10-8 to 10-7
10-7 to 10-6
10-6 to 10-5
M
10-5 to 10-4
X
>10-4
PS420 Solar Magnetohydrodynamics: 2013-2014
Composite energy spectrum from a large flare:
And its interpretation:
PS420 Solar Magnetohydrodynamics: 2013-2014
There is a relation between the soft X-ray flux (thermal emission) and hard X-ray and
microwave fluxes (non-thermal emission): the Neupert effect. The Neupert effect
simply notes that the hard X-rays occur during the rise phase of the soft X-rays.
The time derivative of the soft X-ray light curve resembles the hard X-ray light curve.
PS420 Solar Magnetohydrodynamics: 2013-2014
The standard model of solar flare:
1. Magnetic free energy is stored in the corona, due to either motions of the
photospheric footpoints of loops or to the emergence of current-carrying field
from below the photosphere.
2. A cool, dense filament forms, suspended by the magnetic field, over the
neutral line.
3. The field evolves slowly through equilibrium states, finally reaching a
nonequilibrium which causes the closed field to rise and erupts outward.
4. The reconnection of the field below the rising filament provides plasma
heating and particle acceleration that we call the flare.
5. The accelerated particles follow the field lines and interact with the
chromosphere, heating it and causing the evaporation upflows of the plasma.
6. “Post-flare loops” are formed over the neutral line; they gradually cool down
by radiation.
7. More field lines are involved in reconnection; the reconnection site is going up,
forming new post-flare loops situated above the previously created ones.
8. A multi-temperature arcade is formed with “older” cooler loops being below
“new” hotter loops.
PS420 Solar Magnetohydrodynamics: 2013-2014
An observational “illustration” of this structure in soft X-ray:
PS420 Solar Magnetohydrodynamics: 2013-2014
Red: UV continuum
Blue: 171 Å pass band,
~1 MK
Green: 195 Å pass
band,
>1.5 MK
Possible scenario of a solar flare:
(From Takasaki et al., The Astrophysical Journal, 2004)
PS420 Solar Magnetohydrodynamics: 2013-2014
Statistics of solar flares (from Aschwanden 2004):
The power law distribution: The flare frequency f against
energy E: f(E) = E0 E-a
The total heating rate released by all flares:
where
PS420 Solar Magnetohydrodynamics: 2013-2014
Evaluating the integral, obtain
Typically:
If a<2 – the bulk of heating is delivered by large events.
If a>2 – the bulk of heating is delivered by small events.
Coronal heating by flares:
Active regions:
Quiet Sun:
Predicted frequency distribution that fulfil the heating
requirement
PS420 Solar Magnetohydrodynamics: 2013-2014
The detected radiation of the EUV and SXT nanoflares roughly
corresponds to a third of the total coronal heating requirement.
PS420 Solar Magnetohydrodynamics: 2013-2014
Variation of the emitted X-ray flux with solar cycle:
Correlation of emitted X-ray flux with sunspot numbers: