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 (R-0/r(0) = 1/eta, where R-0 and r(0) are the major and the minor torus radii, respectively). The total flux IF is represented by the sum of the flux outside the torus envelope (To) and the internal flux within the torus body (Psi(i)). Analogously, the total inductivity is expressed as L = L-o + L-i. The outside self-inductivity is determined directly from the magnetic flux To, utilizing Psi(o) = 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 and L-o(eta) depend only very weakly on the form of j(r). On the other hand, Psi(i) and L-i do not depend on 77, but depend on the form of j(r). In the range 0.02 less than or similar to eta less than or similar to 0.5, the numerical values of L-o can be very well fitted by the function of the form L-o(fitl) (eta) = -A log(eta) - B. Such a relation is analogous to that for a slender torus, although the coefficients are different. For eta less than or similar to 0.01 the slender-torus approximation (L-o(*)) matches the numerical results better than ourfunction L-o(fitl), whereas for thicker tori, L-o(fitl) becomes more appropriate. It is shown that, beyond eta greater than or similar to 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 eta approximate to 0.55. In the range eta greater than or similar to 0.5, the numerical values of L-o can be very well expressed by the function L-o(fit2) (eta) = c(1)(1-eta)(c2). Furthermore, since the internal flux and inductivity become larger than that outside the envelope, Psi(i) and L-i become larger than To and L-o. The total inductivity L-tot(fit) = L-o(fit) + L-i, calculated tot 0 by appropriately employing our functions L-o(fit1) and L-o(fit2), never deviates by more than 1% from the numerically determined values of L-tot.