Astrophysical jets and pulsar-fed supernova remnants (plerions) are expected to develop highly organized magnetic structures dominated by concentric loops of toroidal held, B-phi. It has been argued that such structures could explain the polarization properties of some jets and contribute to their lateral confinement through magnetic tension forces. A concentric toroidal field geometry is also central to the Rees-Gunn model for the Crab Nebula, the archetypal plerion, and leads to the deduction that the Crab pulsar's wind must have a weak magnetic field. Yet this kind of equilibrium between magnetic and gas pressure forces, the "equilibrium Z-pinch" of the controlled fusion literature, is well known to be susceptible to disruptive localized instabilities, even when the magnetic field is weak and/or boundary conditions (e.g., a dense external medium) slow or suppress global modes. Thus, the magnetic field structures imputed to the interiors of jets and plerions are unlikely to persist for very long To determine the growth rates of Z-pinch instabilities under astrophysical conditions, I derive a dispersion relation that is valid for the relativistic fluids of which jets and plerions may be composed, in the ideal magnetohydrodynamics (MHD) limit. The dominant instabilities are kink (m = 1) and pinch (m = 0) modes. The former generally dominate, destroying the concentric field structure and probably driving the system toward a more chaotic state in which the mean field strength is independent of radius (and in which resistive dissipation of the field may be enhanced). I estimate the timescales over which the field structure is likely to be rearranged and relate these to distances along relativistic jets and radii from the central pulsar in a plerion. I conclude that the central tenet of the Rees-Gunn model for the Crab Nebula, the existence of a concentric toroidal held well outside the pulsar wind's termination shock, is physically unrealistic. With this assumption gone, there is no dynamical reason to conclude that the magnetic energy flux carried by the pulsar wind is much weaker than the kinetic energy flux. Abandoning the principal conclusion of Rees & Gunn would resolve a long-standing puzzle in pulsar wind theory.
