Fundamentals of Plasma Physics

9.8 Static equilibria 275

9.8 Static equilibria

9.8.1 Static equilibria in two dimensions: the Bennett pinch

The simplest static equilibrium was first investigated by Bennett (1934) and is called the
Bennett pinch orz-pinch (herezrefers to the direction of the current). This configuration,
sketched in Fig.9.7, consists of an infinitely long axisymmetric cylindrical plasma with
axial current densityJz=Jz(r)and no other currents.
The axial currentflowing within a circle of radiusris

I(r)=

∫r

0

2 πr′Jz(r′)dr′ (9.24)

and the axial current density is related to this integrated current by

Jz(r)=

1

2 πr

∂I

∂r

. (9.25)

Since Ampere’s law gives

Bθ(r)=

μ 0 I(r)

2 πr

, (9.26)

Eq.(9.22) can be written

r^2

∂P

∂r

=−

μ 0

8 π^2

∂I^2

∂r

. (9.27)

Integrating Eq. (9.27) fromr=0tor=a,whereais the outer radius of the cylindrical
plasma, gives the relation

∫a

0

r^2

∂P

∂r

dr=

[

r^2 P(r)

]a

0 −^2

∫a

0

rP(r)dr=−

μ 0 I^2 (a)

8 π^2

. (9.28)

The integrated term vanishes at bothr=0andr=asinceP(a)=0by definition. If the
temperature is uniform, the pressure can be expressed asP(r)=n(r)κTand so Eq.(9.28)
can be expressed as

I^2 =

8 πNκT

μ 0

(9.29)

whereN=

∫a
0 n(r)2πrdris the number of particles per axial length. Equation (9.29),
called the Bennett relation, shows that the current required to confine a givenNandTis
independent of the details of the internal density profile. This relation describes the sim-
plest non-trivial MHD equilibrium and suggests that quite modest currents couldcontain
substantial plasma pressures. This relation motivated the design of early magnetic fusion
confinement devices but, as will be seen, it is overly optimistic because simplez-pinch
equilibria turn out to be highly unstable.
Confinement using currents thatflow in the azimuthal direction is also possible, but
this configuration, known as aθ-pinch, is fundamentally transient. In aθ-pinch a rapidly
changing azimuthal current in a coil going around a cylindrical plasma creates a transient
Bzfield as shown in Fig. 9.8. Because the conducting plasma conserves magneticflux,
the transientBzfield cannot penetrate the plasma and so is confined to the vacuum region
between the plasma and the coil. This exclusion of theBzfield by the plasma requires