Warwick PX420 Solar MHD 2015-2016: Reconnection
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Magnetic reconnection
We have already discussed a force-free magnetic X-point (or a “null-point”). In 2D it is
formed by the field B = B0 (y, x, 0). The field lines are given by parabolae, y 2 −x2 = const,
4
y
2
–4
–2
0
2 x
4
–2
–4
We may calculate the j × B force in this configuration, and find that it is zero everywhere
(the magnetic curvature force is balanced by the magnetic pressure gradient force).
Let us deform this equilibrium, B = B0 (y, α02 x, 0), where α0 is constant. The field lines
are (DIY) y 2 − α02 x2 = const.
The electric field density in this configuration is (DIY)
1
jz =
µ0
∂By ∂Bx
−
∂x
∂y
!
=
α02 − 1
.
µ0
α02 − 1 2
Hence, the magnetic force is j × B =
−α0 x, y, 0 6= 0.
µ0
We can calculate components of the force at different points:
j × B|(0,1,0) =
α02 − 1
ey ,
µ0
j × B|(1,0,0) = −
j × B|(0,−1,0) = −
α02 (α02 − 1)
ex ,
µ0
α02 − 1
ey ,
µ0
j × B|(−1,0,0) =
α02 (α02 − 1)
ex .
µ0
Warwick PX420 Solar MHD 2015-2016: Reconnection
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Thus, the configuration is not in equilibrium. If an X-point is perturbed, the perturbation
grows. The forces generate flows of the plasma. The X-point magnetic geometry evolves
to a current sheet.
The magnetic diffusivity η is very small, but in the vicinity of the current sheet its effect
becomes significant, as in the diffusion equation it is multiplied by the Laplasian of the
magnetic field, which is large as the field has there a very steep profile.
If the magnetic diffusivity η is not zero, the frozen-in condition is violated. It leads to
reconnection of the magnetic field lines: the magnetic field lines can “intersect”, and the
magnetic topology rearranges.
Consider a current sheet. In its very vicinity the resistivity can be taken as finite.
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 forces, moving field lines toward the current sheet from the top and bottom.
Warwick PX420 Solar MHD 2015-2016: Reconnection
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In the current sheet the oppositely directed magnetic field lines get reconnected: an Xpoint appears, and the lines are brought by the inflows to the X-point and are moved away
by the outflows. It creates magnetic tension forces in the horizontal direction. These forces
drive the frozen-in plasma — the sling shot effect.
• The plasma is accelerated along the layer (in the sketch in the horizontal direction),
• and eventually expelled from its ends at some relatively large velocity Vo , which can
be shown to be about the Alfvén speed CA .
• The (incompressible) mass conservation condition gives us LVi = lVo , hence Vi Vo .
It is the Sweet–Parker stationary reconnection.
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. It leads to the increase in the current density j.
• As the diffusivity is not negligible in the reconnection region (in the vicinity of the
current sheet), the current is subject to Ohmic dissipation, hence increase in internal
energy of the plasma.
• The slingshot effect generates bulk flows of plasma, hence increase in its kinetic
energy.
Warwick PX420 Solar MHD 2015-2016: Reconnection
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• The electric field E = −V×B accelerates plasma particles: non-thermal high energy
particles.
The rate of magnetic reconnection is defined as M0 = Vi /Vo ∝ S −1/2 ,
where S = µ0 LCA /η is the Lundquist number. Typically, in the corona S = 107−9 .
There is however a problem: for typical parameters of the corona, the characteristic time
of energy release by magnetic reconnection is about a few tens of days. This is too long
to explain dynamical phenomena (e.g. flares and CME) in the solar atmosphere. The
problem of “fast reconnection” is one of the key problems of modern solar and space
plasma physics. Possible solutions: anomalous resistivity, non-MHD processes...