Download Section_32_Magnetic_..

32. MAGNETIC RECONNECTION
We now begin discussions of the dynamics of the magneto-fluid system when
resistivity is included in the model. This is called resistive MHD. There are several
important differences from ideal MHD.
First, the ideal Ohm’s law is no longer valid, so all the things we have been talking
about regarding MHD stability, etc, are no longer valid. In resistive MHD, the force
operator is no longer self-adjoint. We will have to resort to solving differential equations.
Second, in ideal MHD the topology, or “connectedness”, of the magnetic field is
fixed for all time, and the magnetic field lines are co-moving with the fluid. This is not
the case in resisitive MHD. Now the fluid can slip through the field, and the field lines
no longer have integrity in time. Their topology can change. This process is called
magnetic reconnection (although why its called reconnection instead of just connection, I
have no idea), and it plays a fundamental role in the behavior of real plasmas in both
laboratory and astrophysical settings, even when the resistivity, when measured by the
magnetic Reynolds’ number or the Lundquist number, is extremely small.
Consider the configuration shown in the figure.
The magnetic field is B0  B(x)ê y  Bz0 ê z , with Bz0  constant , so the current is
J  J z (x)êz , and the equilibrium condition is dp / dx  J z B . There are conducting walls
at x  L . The system extends to infinity at  y . This equilibrium is called a sheet
pinch. We choose B(x) such that B(0)  0 . Then if k  kê y , F(x)  k  B0  0 at
x  0 , so x  0 is a singular surface.
We now consider the dynamics of the sheet pinch in ideal MHD. The transverse ( x )
magnetic field evolves according to Faraday’s law. We assume instability and write the
time dependence as e t . Then  Bx1  ikEz1 , or
Ez1 
i Bx1
.
k
(32.1)
From the ideal MHD Ohm’s law, Ez1  Vx1B(x) so that
1
Vx1 x  
i Bx1
.
F x 
(32.2)
If   0 , Vx1   at x  0 (where F  0 ). Well behaved solutions require   0 , so
that the sheet pinch is stable in ideal MHD.
Now include resistivity. If the resistivity is constant, the perturbed field evolves
according to
B1

   V1  B0   2 B1 .
t
0
(32.3)
Again assuming instability, the x-component is
 Bx1  iFVx1 

  d 2 Bx1
 k 2 Bx1  ,
2

0  dx

(32.4)
so that

i 
  d 2 Bx1
2
Vx1    Bx1  

k
B
x1
  .
F
0  dx 2

(32.5)
Well behaved solutions are now possible if
 Bx1 

  d 2 Bx1
 k 2 Bx1 
2

0  dx

(32.6)
near F  0 , so that instability is now possible. This is called resistive instability. Note,
however, that  ~  , so that the growth is on a resistive time scale relative to the scale
lengths on the right hand side of Equation (32.6). This means that any unstable growth
will be on a time scale  1 ~  R much slower than the Alfvén time  A , especially if the
Lundquist number S  LVA / ( / 0 )   R /  A  1 ; we expect  A  1 . Recall that
Alfvén waves reflect the effects of inertia. This implies that for time scales much longer
than the Alfvén time, we can neglect inertia in the region away from F  0 . Further,
since the resistivity only affects the solutions near F  0 , we can ignore resistivity as
well in this “outer region”; this region is therefore always in a state of MHD equilibium.
A possible configuration of the sheet pinch including the effects of resistivity is
shown in the figure.
2
The resistivity is important only in a small layer about the singular surface, where F  0 .
Equation (32.6) allows finite Bx1 at x  0 . Field lines that were originally straight can
now break, change their topology, and “reconnect” within this small layer. This is called
magnetic reconnection. The region outside the small layer is governed by ideal MHD
and inertia is ignored.
If there is periodicity in the y-direction, as implied by k  kê y , then magnetic
reconnection results in the formation of magnetic islands, as shown in the figure.
The magnetic island has a separatrix, which separates field lines of different topologies
(open outside the island, and closed inside the island). The magnetic reconnection occurs
at the X-points. The center of the island is called the O-point. Typical flows associated
with magnetic islands are also shown.
The width of magnetic island can be defined by means of flux conservation within
the separatrix. See the figure.
We require
W
 /2
0
0
 By dx 
B
x1
dy .
Near x  0 , By  By0
 x  B0 x / L , so that the left hand side is B0W 2 / 2L .
Bx1  B1 sin ky , the right hand side is just 2B1 / k . Solving for the width, we find
 4B L 
W  1 
 Bk 
(32.7)
With
1/2
,
(32.8)
0
3
where B1 is the amplitude of the perturbed magnetic field.
Magnetic reconnection can occur as a steady state process in which two oppositely
directed magnetic fields are pushed together by external means. The reconnection then
occurs at a constant rate  .
Magnetic reconnection can also occur spontaneously as a resistive instability. The
magnetic island then grows at a rate e t .
We will discuss both possibilities in the next Sections.
Magnetic reconnection is an important phemonenon because ideal MHD constraints
trap energy in the magnetic field. Resistive MHD releases those constraints and allows a
new source of free energy to drive instabilities. Magnetic reconnection is thought by
some to be responsible for “energizing the universe”, by means of solar and stellar flares,
heating of diffuse plasmas, the formation and evolution of astrophysical jets, etc.
Unfortunately, for most cases of interest S  1 (for a tokamak S ~ 10 710 , and even
larger in astrophysical settings). Since   0 when   0 , we expect (and will find)
that  ~ S  , 0    1 . It is difficult to account for the observed rate of energy release
with these slow growth rates. The quest for a cause of “fast” magnetic reconnection has
been alive for five decades, and it still goes on. Undoubtedly it will continue for many
more.
4