Equilibrium and stability of plasma in axially symmetric toroidal systems

Conclusions

In the case when the magnetic surfaces have circular normal cross sections in the neighborhood of a magnetic axis (ε=1), the stability criterion reduces to a limitation on the ratio of the current density to the magnetic field on the magnetic axis j₀R/B₀<2. On the assumption of a uniform axial current, this criterion is equivalent to the Shafranov-Kruskal stability condition RB_θ (a)/aB_φ(a)<1/m for the first mode of oscillation m=1. If the current density falls off on going away from the magnetic axis, the stability criterion when written in the form of the Shafranov-Kruskal condition will correspond to m>1. For example, in the case of the parabolic current density distribution j_ϕ=j₀ (1−ρ₂/a²), it turns out that m=2.

To investigate the effect of the ellipticity of the magnetic surface cross sections, a study was made of the two cases when the external magnetic surface Σ had an elliptical or semielliptical cross section. As can be seen from Figs. 2 and 4, the limiting value of j₀R/B₀ decreases rapidly when the semi-axis ratio is decreased below unity and decreases more smoothly when ε is increased above unity: (ε=l_z/l_r).

In the case when toroidal aspects of the problem are very strongly expressed, when the external boundary of the plasma has a semi-elliptical cross section with a semi-axis ratio ε_∑=1/2 (which corresponds to ε=1), the limiting ratio of the axial current in the plasma to the current in the windings of the solenoid producing the axial magnetic field amounts to J/I≈1/3, which corresponds to a ratio of plasma pressure to magnetic pressure β≈1/2.

To investigate the effect of the toroidal geometry on the stability, we also investigated the case of a plasma torus in which the external magnetic surface was of rectangular cross section. As can be seen from Fig. 6, in the case when the axial magnetic field decreases on moving away from the magnetic axis (b/a>0) the limiting value of the ratio N=j₀R/B₀ is reduced somewhat with increasing toroidality, this reduction in N being associated with a corresponding increase in ε.

In summary, as far as the conditions of hydromagnetic stability for a prescribed magnetic field are concerned, an increase in toroidality should lead to an increase in the limiting plasma current and pressure.