We study the collimation of relativistic magnetohydrodynamic jets by the pressure of an ambient medium , in the limit where the jet interior loses causal contact with its surroundings .
This follows up a hydrodynamic study in a previous paper , adding the effects of a toroidal magnetic field threading the jet .
As the ultrarelativistic jet encounters an ambient medium with a pressure profile with a radial scaling of p~ { } \propto~ { } r ^ { - \eta } where 2 < \eta < 4 , it loses causal contact with its surroundings and forms a boundary layer with a large pressure gradient .
By constructing self-similar solutions to the fluid equations within this boundary layer , we examine the structure of this layer as a function of the external pressure profile .
We show that the boundary layer always becomes magnetically dominated far from the source , and that in the magnetic limit , physical self-similar solutions are admitted in which the total pressure within the layer decreases linearly with distance from the contact discontinuity inward .
These solutions suggest a ‘ hollow cone ’ behavior of the jet , with the boundary layer thickness prescribed by the value of \eta .
In contrast to the hydrodynamical case , however , the boundary layer contains an asymptotically vanishing fraction of the jet energy flux .