B FRC formation and equilibrium

A FRC is a variety of compact toroids with the following characteristics: no appreciable toroidal field, values of fl of the order of unity, no rotational transform, all the equilibrium currents maintained by diamagnetism, a scrape-off layer exhausting heat and particles outside the coil system. FRCs are reviewed in [Tuszewski, 1988].

FRCs were accidentally discovered in the 1960s in 0-pinches. In order to understand the main features of this configuration it is useful to consider the main formation scheme (the 0-pinch formation) illustrated in Figure B.15:

a. the discharge tube is filled with neutral gas and a bias magnetic field is applied; the gas is pre-ionized, freezing the magnetic field in the plasma at a temperature of a few electronvolts;

Ionized Propulsion System
Figure B.15. FRC formation sequence.

b. the current in the theta pinch coils is reversed on a fast timescale, inducing in this way a plasma current along d (and an axial field opposite to the bias field) that causes the plasma and bias field to implode radially;

c. the oppositely directed magnetic field lines reconnect near the end of the 0-pinch coil, forming a closed magnetic field configuration;

d. the large magnetic tension at the reconnection region causes the FRC to contract in the axial direction until an equilibrium configuration is achieved.

During Phase b, heating occurs due to the implosion shock followed by slow compression; resistive heating also occurs during the annihilation of the bias field and is characterized by a resistive dissipation higher than in the classical case.

The main feature of interest in FRCs is that, in order to achieve an equilibrium configuration, the average fl of the plasma must be high. Using simple analytical models (confirmed by full numerical analysis), it can be shown that

with rs and rc the separatrix radius and the flux conserver radius, respectively. Since rs < rc, this implies beta values larger than 50%. Nevertheless, the plasma maintains remarkable stability properties.

The flux ^ of the axial magnetic field between the null point and the separatrix can be shown to be bound by two values:

with Be being the magnetic field outside the separatrix (determined by the poloidal coil current), and the two boundary values obtained for k = 0 and k = 1, respectively. From Equation (B.35) it is also possible to determine an expression for the parameter S

S = M2^sPi,£e) = 2-3/2(rc/Pi-e)(rs72r2)(2+k)/2 (B.36)

with pie the ion gyration radius (=gyroradius) in the external magnetic field. Therefore, S is always lower than the value obtained for rs = rc (fl = 50%) and k = 0 (i.e.,

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