The most widely known toroidal fusion device is the tokamak; this word is of Russian origin and is a contraction of toroidal, kamera (chamber), magnet and katuschka (coil). The essential features of a tokamak are suggested in Fig. 10.5, and the functions of the various components can be described as follows. The plasma torus is viewed as a single-winding secondary of a transformer. A current flow in the primary transformer winding will therefore induce a current in the plasma toms by transformer action. The resultant toroidal plasma current I provides for Ohmic heating and also generates the poloidal magnetic field B0. Further, the toroidal magnetic field coils generate the toroidal field B0 which is perpendicular to B0. These two magnetic fields combine by vector addition,

В = B^ +Be (10.9)

to provide the basis for the rotational transform in Fig. 10.3. Further, the stabilizing coils, Fig. 10.5, are to restrain the plasma from expanding as is the natural tendency of a current ring. Then, the total field vector В has to incorporate also this stationary field component. The expansion tendency arises from the difference between the poloidal magnetic pressure, B02/ 2|lo (Sec. 9.1), on the outside and the inside of the toms. Since B0 is larger on the inside, the associated higher magnetic pressure will steer the plasma ring towards an increased major radius. It is therefore seen that toroidal and poloidal magnetic fields alone cannot confine the plasma in a tokamak. Of additional need is a vertical magnetic field Bv, produced by the stabilizing coils shown in Fig. 10.5, in order to produce an inward directed j X Bv force which prohibits the outward expansion. If the toroidal plasma current is sufficiently strong, only a small Bv is required to stabilize the plasma. Hence, a tokamak may be characterized as a toroidal device featuring a large plasma current and a strong toroidal magnetic field such that

ВФ>Ве>Ву ■ (10.10)

The toroidal plasma current needed to supply the poloidal magnetic induction

is generated by a toroidal electric field E = (0,Еф,0) which is achieved here by the transformer’s temporally changing magnetic flux 4Vans that penetrates the hole in the toms.

I Stabilizing

The connection between the various physical field quantities involved is readily evident from the following. Since one of Maxwell’s Equations tells us

V x E= ,

dt

we conclude from Stoke’s Law,

that the work done per unit charge by the induced electric field over a closed path along the secondary transformer winding-which essentially is the plasma ring itself-is equal to the rate of change of magnetic flux, 4/(Tans, through the closed winding loop, i. e. through the hole in the toms. Hence, ds = 2к dR еф here and A=7tR2=constant, which has been assumed in Eq.(10.12). The induced toroidal electric field is then found from Eq.(10.12) to be of the magnitude
2nR dt

The flux change is achieved either by an air-core or an iron-core transformer as shown in Fig. 10.5. It follows then that a tokamak cannot operate in steady — state but only as a pulsed device unless non-inductive schemes of plasma current drive can be applied. The density of the plasma current driven by the induced toroidal electric field is subsequently found via Ohm’s Law which can be generalized for a conducting fluid in a B-field to be

T] j = E + VxB (10.14)

where T) is the specific resistivity of the plasma and V is the fluid velocity as defined and used in Eq.(6.21). Additional current terms can occur in Eq.(10.14), which, however, are of second order and therefore not itemized here. The resistivity of a thermal plasma is governed by collisional kinetic effects and can be shown from a plasma physics consideration to vary as

т)~{кТе)Ш • (Ю.15)

As the plasma is heated, the Coulomb cross section as vr’4 (Eq. 3.15) decreases and consequently the resistivity, which is proportional to <asvr>, drops accordingly.

Note that the induction of a plasma current suggests an easy way to heat the plasma; that is through Ohmic dissipation

Рон= i71 j2d’r , (10.16)

Volume

where the symbol OH refers to the common label of this effect. It is evident from the proportionality given in Eq.(10.15) that this heating method becomes less efficient at higher plasma temperatures (>1 keV) and will not suffice up to the required thermonuclear temperatures (~10 keV). For that, other methods must be employed of which some we discussed in Sec. 8.2.

Another effect associated with electric field induction in a plasma is noted here. The frequency of collisions between electrons and ions (compare with 1/Тц of Eq.(9.41)) is

(t«) 7 06 a, vr00 vi3 (10.17)

and shows that high-energy electrons undergo relatively few collisions and therefore predominantly carry the induced current. Consider now an electron from the high energy tail of its velocity distribution which moves in the direction opposite to E. Due to the low collision frequency, it will gain further energy making thus a collision with an ion even less likely. This in turn allows it to be further accelerated by E. This phenomenon is called ‘electron runaway’. If E, and hence the velocity gain of the electrons, is sufficiently large, the Coulomb cross section drops so quickly that these runaway electrons never encounter a collision and thus form a beam of accelerated electrons disengaged from the main part of

Closed Magnetic Systems the distribution.