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 smooth particle magnetohydrodynamic simulations . Evaluated for a set of parameters which are mainly suitable for protostellar jets ( with density ratios between the jet and the ambient medium \eta \approx 3 - 10 , and ambient Mach number M _ { a } \approx 24 ) , these simulations are also compared with baseline non-magnetic and adiabatic calculations . Two initial magnetic field topologies ( in \sim equipartition with the gas , \beta = p _ { th } / p _ { B } \simeq 1 ) are considered : ( i ) a helical field and ( ii ) a longitudinal field , both of which permeate both the jet and the ambient medium . 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 , but this effect is larger in the presence of the helical field . In both magnetic configurations , low-amplitude , approximately equally spaced ( \lambda \approx 2 - 4 R _ { j } ) internal shocks ( which are absent in the HD systems ) are produced by MHD Kelvin-Helmholtz 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 \sim twice ) in the presence of radiative cooling relative to the adiabatic cases . Besides , a large density ratio , \eta , 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 . Magnetic fields may leave also important signatures on the head morphology of the radiative cooling jets . The amplification of the nonparallel components of the magnetic fields , particularly in the helical field geometry , reduces the postshock compressibility and increases the postshock cooling length . This may lead to stabilization of the cold shell of shocked material that develops at the head against both the Rayleigh-Taylor and global thermal instabilities . As a consequence , the clumps that develop by fragmentation of the shell in the HD jets tend to be depleted in the helical field geometry . The jet immersed in the longitudinal field , on the other hand , still retains the clumps although they have their densities decreased relative to the HD counterparts . As stressed before ( Cerqueira , de Gouveia Dal Pino & Herant 1997 ) , since the fragmented shell structure resembles the knotty pattern commonly observed in HH objects behind the bow shocks of protostellar jets , this result suggests that , as long as ( equipartition ) magnetic fields are present , they should probably be predominantly longitudinal at the head of these jets . Subject headings : ISM : jets and outflows – MHD – stars : pre-main-sequence - stars : mass loss