Calculation of the current loop lines for the magnetic knot

Ranada et al. (1998) proposed a magnetic-knot model for ball lightning. Their current-density vector will be represented in terms of field lines, whose equations are solved in a closed form under a very accurate approximation. This allows easy visualization of the current loops and, more important, their classification according to how many turns a given line requires to become closed. Relatively few lines of the infinity of possibilities will close after one or a few turns; for them, the available magnetic energy can be sufficient to maintain the current high enough to keep the loop conductive. Most loops never close and cannot be excited by the finite magnetic field. Each line is labeled by a number (of a continuum of real numbers), which is related in a simple way to the number of turns to close it. It is also noted that the concept of aerogel, proposed in earlier models, may offer a structural basis for the formation of the current lines. An aerogel filament, if it is formed of carbon or metal, may provide good enough conductivity at much lower temperatures than ionized air.