We investigate the effect of magnetic fields on the propagation dynamics and morphology of overdense, radiatively cooling, supermagnetosonic jets, with the help of fully three-dimensional SPMHD simulations. Evaluated for a set of parameters which are mainly suitable for protostellar jets (with density ratios between the jet and the ambient medium 3-10, and ambient Mach number ~ 24), these simulations are also compared with baseline non-magnetic and adiabatic calculations. We find that, after amplification by compression and re-orientation in nonparallel shocks at the working surface, the magnetic field that is carried backward with the shocked gas into the cocoon improves the jet collimation relative to the purely hydrodynamic (HD) systems. Low-amplitude, approximately equally spaced internal shocks (which are absent in the HD systems) are produced by MHD K-H reflection pinch modes. The longitudinal field geometry also excites non-axisymmetric helical modes which cause some beam wiggling. The strength and amount of these modes are, however, reduced (by ~ twice) in the presence of radiative cooling relative to the adiabatic cases. Besides, a large density ratio between the jet and the ambient medium also reduces, in general, the number of the internal shocks. As a consequence, the weakness of the induced internal shocks makes it doubtful that the magnetic pinches could produce by themselves the bright knots observed in the overdense, radiatively cooling protostellar jets.
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