17. “FORCE-FREE” FIELDS
An equilibrium of some interest is the case of force-free fields. Of course, by
definition all equilibrium situations are “force-free”, but in MHD that description is
usually reserved for the special case where the Lorentz force vanishes, i.e.,
J ! B = 0!!!.
(17.1)
The pressure is constant ( !p = 0 ) and the current is everywhere parallel to the magnetic
field, i.e.,
J = ! (x)B!!!,
(17.2)
or
!"B=
#
B!!!.
µ0
(17.3)
(As an aside, vector fields with the property that they are everywhere parallel to their curl
are called Beltrami fields.) Note that Taking the divergence of Equation (17.3) and using
! " B = 0 yields
B ! "# = 0!!!,
(17.4)
while taking the curl along with ! " B = 0 gives
2
$#'
1
! B"& ) B=
!# * B!!!.
µ0
% µ0 (
2
(17.5)
Equation (17.4) says that ! (x) is constant along a magnetic field line. Together,
Equations (17.4) and (17.5) are two equations that, in principle, can be solved for the
unknowns B and ! . (In practice, however, this presents some very subtle mathematical
issues.)
A case of particular interest occurs when ! = constant and the geometry is
cylindrical. We also introduce the notation ! " # / µ0 , which has units of L!1 . Then
Equation (17.4) is satisfied automatically, and, with Br = 0 , the ! and z components of
Equation (17.5) are
d 2 B!
dB
r
+ r ! " 1 " # 2 r 2 B! = 0!!!,
2
dr
dr
(17.6)
d 2 Bz
dB
r
+ r z + ! 2 r 2 Bz = 0!!!.
2
dr
dr
(17.7)
2
(
)
and
2
These are both forms of Bessel’s equation. The solution of Equation (17.6) is
B! = B0 J1 ( " r )!!!,
(17.8)
and the solution of Equation (17.7) is
1
Bz = B0 J 0 ( ! r )!!!,
(17.9)
where J 0 and J1 are called Bessel functions of the zeroth and first order, respectively.
We remark that Bz (r) changes sign when ! r = j0,1 = 2.4048 , where jn, k denotes the kth
zero of the nth order Bessel function. (These are extensively tabulated.) These solutions
are shown in the figure (where Bz is labeled as B! ).
Let a conducting wall be located at r = a . A large variety of screw pinch equilibria
are represented by varying the varying the parameter ! a . From the preceding remarks,
we know that the axial field changes sign (“reverses”) inside the wall if ! a > 2.4048 .
This parameter regime is called the Reversed-field Pinch (or RFP). If ! a << 1 , then since
J! (z) ~ (z / 2)! / "(! + 1) for z << 1 , we have B! ~ " r / 2 and Bz ~ 1 ; the poloidal ( ! )
component of the magnetic field increases linearly (implying that the axial current
density J z is constant), and the axial (z) component of the magnetic field is constant.
This parameter regime is called the tokamak. The intermediate range of ! a (sometimes
called the paramagnetic pinch, since the axial field is larger in the center than at the
edge) has proven to be less interesting experimentally.
Of course, there is no a priori reason to expect that ! will be constant. In general, it
must be determined from the solution of Equations (17.4) and (17.5), along with
appropriate boundary conditions. (The allowable form the boundary conditions for these
equations is a subtle mathematical problem.) We will return to this point when we
discuss MHD relaxation later in this course.
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