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Previous numerical studies have shown that in protostellar outflows, themass-velocity distribution $m(v)$ can be well described by a broken power law$\propto v^{- \gamma}$. On the other hand, recent observations of a sample ofoutflows show that the CO intensity-velocity distribution, closely related to$m(v)$, follows an exponential law $\propto \exp(-v/v_0)$. In the present work,we revisit the physical origin of the mass-velocity relationship $m(v)$ injet-driven protostellar outflows. We investigate the respective contributionsof the different regions of the outflow, from the swept-up ambient gas to thejet. We performed 3D numerical simulations of a protostellar jet propagatinginto a molecular cloud using the hydrodynamical code Yguazu-a. The code takesinto account atomic and ionic species and was modified to include the H$_2$gas. We find that by excluding the jet contribution, $m(v)$ is satisfyinglyfitted with a single exponential law, with $v_0$ well in the range ofobservational values. The jet contribution results in additional components inthe mass-velocity relationship. This empirical mass-velocity relationship isfound to be valid locally in the outflow. The exponent $v_0$ is almost constantin time and for a given level of mixing between the ambient medium and the jetmaterial. In general, $v_0$ displays only a weak spatial dependence. A simplemodeling of the L1157 outflow successfully reproduces the various components ofthe observed CO intensity-velocity relationship. Our simulations indicate thatthese components trace the outflow cavity of swept-up gas and the materialentrained along the jet, respectively. The CO intensity-velocity exponentiallaw is naturally explained by the jet-driven outflow model. The entrainedmaterial plays an important role in shaping the mass-velocity profile.

H$_2$ mass-velocity relationship from 3D numerical simulations of jet-driven molecular outflows