Highly collimated protostellar jets are believed to originate in the c
losest environment of a fully convective young stellar object, which p
resumably carries a magnetosphere built up by a strong stellar dynamo
and is surrounded by an accretion disk. The light cylinder of the rota
ting stellar magnetosphere is located within the jet radius, requiring
a relativistic treatment of the magnetohydrodynamics. Here, we presen
t numerical solutions of the relativistic 2D force-balance of the magn
etic field, described by the Grad-Schluter-Shafranov equation, using a
model topology of a star-disk-jet scenario. The resulting magnetic fi
eld structure allows simultaneously for wind outflow towards a cylindr
ical jet and for mass accretion along dipolar field lines. The asympto
tic jet radius is several light cylinder radii. The outflow is initial
ly poorly collimated with an opening angle of 65 degrees and then rapi
dly collimates within a distance of 0.3 jet radii along the jet axis.
We calculate the dynamical parameters of a steady, cold plasma motion
in the calculated collimating jet magnetosphere for different plasma m
agnetisations. For the asymptotic poloidal jet velocity as a function
of the magnetisation we numerically derive a power law. If the magneti
sation is constant on all flux surfaces, we find that the plasma energ
y and poloidal plasma velocity increase towards the outer flux surface
s, while the density decreases. The asymptotic fast magnetosonic Mach-
number of the how turned out to be independent of the magnetisation an
d is generally of the order of 2.5, The asymptotic structure of the ma
gnetosonic surfaces is parallel to the jet axis.