The magnetic flux and self-inductivity of a thick toroidal current

We investigate numerically the magnetic flux and self-inductivity of a toroidal current I of arbitrary aspect ratio (R0/r0 = 1/η, where R0 and r0 are the major and the minor torus radii, respectively). The total flux Ψ is represented by the sum of the flux outside the torus envelope (Ψo) and the internal flux within the torus body (Ψi). Analogously, the total inductivity is expressed as L = Lo + Li. The outside self-inductivity is determined directly from the magnetic flux Ψo, utilizing Ψo = LoI. On the other hand, the internal inductivity is evaluated as the magnetic energy contained in the poloidal field. The calculations are performed for three different radial profiles of the current density, j(r).It is found that Ψo(η) and Lo (η) depend only very weakly on the form of j(r). On the other hand, Ψi and Li do not depend on η, but depend on the form of j(r). In the range 0.02 ≲ η ≲ 0.5, the numerical values of Lo can be very well fitted by the function of the form Lofit1(η) = −A log(η) − B. Such a relation is analogous to that for a slender torus, although the coefficients are different. For η ≲ 0.01 the slender-torus approximation (Lo*) matches the numerical results better than our function Lofit1, whereas for thicker tori, Lofit1 becomes more appropriate. It is shown that, beyond η ≳ 0.1, the departure of the slender-torus analytical expression from the numerical values becomes greater than 10%, and the difference becomes larger than 100% at η 0.55. In the range η 0.5, the numerical values of Lo can be very well expressed by the function Lofit2(η)=c1 (1 − η)c2. Furthermore, since the internal flux and inductivity become larger than that outside the envelope, Ψi and Li become larger than Ψo and Lo. The total inductivity Ltotfit = Lofit + Li, calculated by appropriately employing our functions Lofit1 and Lofit2, never deviates by more than 1% from the numerically determined values of Ltot.