Separation of magnetic field lines

The field lines of magnetic fields that depend on three spatial coordinates are shown to have a fundamentally different behavior from those that depend on two coordinates. Unlike two-coordinate cases, a flux tube in a magnetic field that depends on all three spatial coordinates that has a circular cross section at one location along the tube characteristically has a highly distorted cross section at other locations. In an ideal evolution of a magnetic field, the current densities typically increase. Crudely stated, if the current densities increase by a factor σ, the ratio of the long to the short distance across a cross section of a flux tube characteristically increases by e2σ, and the ratio of the longer distance to the initial radius increases as eσ. Electron inertia prevents a plasma from isolating two magnetic field structures on a distance scale shorter than c/ωpe, which is about 10 cm in the solar corona, and reconnection must be triggered if σ becomes sufficiently large. The radius of the sun, R⊙=7×1010cm is about e23 times larger, so when σ≳23, two lines separated by c/ωpe at one location can be separated by the full scale of any magnetic structures in the corona at another. The conditions for achieving a large exponentiation, σ, are derived, and the importance of exponentiation is discussed.