In Chs. 4 and 5, we have introduced some basic considerations of magnetic confinement. To begin this more detailed look, we consider the containment capacity of magnetic fields and subsequently discuss specific reductions.

The macroscopic fluid description of a plasma does not clearly suggest how individual charged particles could be confined by a magnetic field. Even if the external В-field is taken to consist of smooth straight field lines, the plasma could generate internal fields which might cause drift motion leading to particle escape from the confinement region. An effective containment will require the plasma particles to be confined to some region for a sufficient time period. We may therefore assert that, in general, this is associated with a state of the macroscopic fluid in which all identified forces are exactly in balance providing thereby a time-independent solution. We consider therefore Eq.(6.25) for steady state, i. e. 3Vj/5t = 0, and obtain for the isotropic case (Vj-V)V, = 0 in the absence of an electric field, the defining equation

Vpj = pCjjx B. (9.1)

Imposing this equation for plasma ions, j = i, and electrons, j = e, we write for the total plasma pressure gradient

Vp = W Pi + pe)

= (р-У; + pcee)x В (9.2)

= jxB.

Evidently, j is the electric current density which must also satisfy Maxwell’s equation

j = Vx— . (9.3)

V0

We note that Eq.(9.2) asserts that in equilibrium, there exists an equivalence between the pressure gradient force and the Lorentz force. Further, the electric current density j and the magnetic field В are perpendicular to Vp; that is, as suggested in Fig. 9.1, j and В lie on an isobaric surface because everywhere Vp is a normal to the surfaces p = constant.

Introducing Eq.(9.3) into (9.2) leads, upon some differential vector analysis, to the expression

>2

or, equivalently,

(9.5b)

This relationship is an important characterization of a hydrodynamic equilibrium state. It makes evident that the total sum of kinetic pressure and magnetic field energy density

Emag _ BH _ B2 V ~ 2 ~2/л0’ [2]

where H is the local field strength, will be a constant everywhere within the confined plasma. Hence, the pressure gradient Vp induces a current in the direction shown in Fig. 9.1 and which, by induction, causes an according decrease in the magnetic field in the plasma. From a particle perspective, we could as well have used Eq.(5.36) to derive a drift due to the pressure gradient force — Vp with the velocity

_ VP x B тал

VD. Vp——————— T

NqB

for a fluid element containing N particles. In view of the differences in ion and electron drifts, we define the electric current density by

В x Vp

В and label it the diamagnetic current. The above expression is essentially equivalent with the previous current density definition in Eq.(9.2) which, when cross-multiplied by B, results in Eq. (9.8), recognizing that j is perpendicular to B.

Characterizing the thermodynamic state of the plasma by the Ideal Gas Law

pV = N*kT (9.9)

with N* as the total number of particles in volume V, we obtain an expression for the total kinetic pressure for an ensemble of ions and electrons as

р = ЖкТ+^кТе

V V

= NikT + N±T

= (Ni + Ne )kT.

Both ions and electrons are, in this last expression, taken to be characterized by the same temperature Tj = Te = T for their Maxwellian distributions.

The extent of coupling between the magnetic pressure Eq.(9.6) and corresponding kinetic pressure Eq.(9.10) is a characteristic of a particular magnetic system and defines the dimensionless beta-parameter as the ratio of these two pressure terms:

a _ ( Ni + Ne )kT

B2/2p0

In a plasma with a pressure gradient the magnetic field is, as suggested by Eq.(9.5b), low where the particle pressure-or mass density-is high and vice — versa. Some magnetic configurations exist which produce a diamagnetic current such that the internal magnetic field may, in a limited region, be reduced to a minimum value or even to B=0; obviously, the local (З-value would then approach infinity. As a consequence, it is common to define (3 as the ratio of maximum particle pressure-e. g. at the centre of the plasma cylinder, if we refer to Fig. 9.1-to the maximum magnetic pressure-e. g. at the outer surface of the
plasma cylinder-thereby limiting the P-value to P < 1. In most magnetic configurations, fusion plasma confinement requires an imposed magnetic pressure which significantly exceeds the intrinsic particle kinetic pressure. A low beta facility is typically characterized by P < 0.1. In Ch. 4 it was shown that the fusion power density varies as p2 so that a high P facility is desirable recognizing however a limitation due to plasma instabilities.