Fundamentals of Plasma Physics

292 Chapter 9. MHD equilibria

stagnation

region

I
r,z
const.



r,z
const.

z z

I
r,z
const.



r,z
const.

high

pressure

low

pressure

collimation

flow flow

J
B

J
B J
B

J
B

(a) (b)

J
B J
B

J
B J
B

I
r,z
const.



r,z
const.

I
r,z
const.



r,z
const.

Figure 9.13: (a) Current channelI(r,z)which has same contours as poloidalflux surfaces
ψ(r,z)so that∇Iis parallel to∇ψin order to have no toroidal acceleration. TheJ×B
force shown by thick arrows is larger at the bottom and canted giving both a higher on-axis
pressure at the bottom and an axial upwardsflow. (b) If theflow stagnates (slows down) for
some reason, then mass accumulates at the top. The frozen-in convected toroidalflux also
accumulates which thus increasesBφand so forcesrto decrease since Ampere’s law gives
2 πBφr=I(r,z)which is constant on a surface of constantI(r,z).The upward-moving,
stagnating plasmaflow with embedded toroidalflux acts somewhat like a zipper for the
surfaces of constantI(r,z)andψ(r,z).

Consider the initial situation shown in Fig.9.13(a) in which a plasma is immersed in

an axisymmetric vacuum poloidal fieldψ(r,z)with a poloidal currentI(r,z).Becauseψ

is assumed to correspond to a vacuum field it is produced by external coils with toroidal

currents. There are thus no toroidal currents in the volume under consideration (i.e., in

the region described by Fig.9.13(a)) and so using Eq.(9.42) it is seen thatψ(r,z)satisfies

∇·

(

r−^2 ∇ψ

)

=0.The poloidal currentI(r,z)has an associated poloidal current density

Jpol=

1

2 π

∇I×∇φ (9.93)

and an associated toroidal magnetic field

Bφ=

μ 0 I(r,z)

2 π

. (9.94)

The poloidalfluxψ(r,z)has an associated poloidal magnetic field

Bpol=

1

2 π

∇ψ×∇φ. (9.95)

IfJpolis parallel toBpolthen∇Iwould be parallel to∇ψso that constantI(r,z)

would be coincident with constantψ(r,z)surfaces as indicated in the figure. On the other