H$_2$ mass-velocity relationship from 3D numerical simulations of
  jet-driven molecular outflows
  Previous numerical studies have shown that in protostellar outflows, the
mass-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 of
outflows 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 $m(v)$ in
jet-driven protostellar outflows. We investigate the respective contributions
of the different regions of the outflow, from the swept-up ambient gas to the
jet. We performed 3D numerical simulations of a protostellar jet propagating
into a molecular cloud using the hydrodynamical code Yguazu-a. The code takes
into 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 satisfyingly
fitted with a single exponential law, with $v_0$ well in the range of
observational values. The jet contribution results in additional components in
the mass-velocity relationship. This empirical mass-velocity relationship is
found to be valid locally in the outflow. The exponent $v_0$ is almost constant
in time and for a given level of mixing between the ambient medium and the jet
material. In general, $v_0$ displays only a weak spatial dependence. A simple
modeling of the L1157 outflow successfully reproduces the various components of
the observed CO intensity-velocity relationship. Our simulations indicate that
these components trace the outflow cavity of swept-up gas and the material
entrained along the jet, respectively. The CO intensity-velocity exponential
law is naturally explained by the jet-driven outflow model. The entrained
material plays an important role in shaping the mass-velocity profile.
