A Simplified Bead-on-Wire Model With Relativistic Revision for Magnetized Outflow

Outflows are ubiquitous in various astrophysical objects, yet the mechanism responsible for generating outflows remains unclear. One possibility is that outflows are driven by large-scale magnetic fields. When dealing with magnetically driven outflows, the one-dimensional model proposed by Weber and Davis in 1967 is often employed. However, as it was originally a non-relativistic theory, this limits its application in high-energy astrophysical scenarios. This work attempts to modify the non-relativistic Bernoulli equation into its relativistic form through an analogy method, thereby applying it to relativistic outflows and comparing the results with those of classical theory. We find that when the initial temperature of the outflow is sufficiently high or when the angular velocity of rotation of the magnetic field lines is sufficiently fast, the terminal velocity of the outflow can be relativistic. The relativistically modified Bernoulli equation established in this paper could be conveniently applied in the context of extremely high-velocity outflows from different sources.