Although several lines of evidence suggest that jets from young stars are driven magnetically from accretion disks, existing observations of field strengths in the bow shocks of these flows imply that magnetic fields play only a minor role in the dynamics at these locations. To investigate this apparent discrepancy we performed numerical simulations of expanding magnetized jets with stochastically variable input velocities with the AstroBEAR MHD code. Because the magnetic field B is proportional to the density n within compression and rarefaction regions, the magnetic signal speed drops in rarefactions and increases in the compressed areas of velocity-variable flows. In contrast, B ~ n^0.5 for a steady-state conical flow with a toroidal field, so the Alfven speed in that case is constant along the entire jet. The simulations show that the combined effects of shocks, rarefactions, and divergent flow cause magnetic fields to scale with density as an intermediate power 1 > p > 0.5. Because p > 0.5, the Alfven speed in rarefactions decreases on average as the jet propagates away from the star. This behavior is extremely important to the flow dynamics because it means that a typical Alfven velocity in the jet close to the star is significantly larger than it is in the rarefactions ahead of bow shocks at larger distances, the one place where the field is a measurable quantity. We find that the observed values of weak fields at large distances are consistent with strong fields required to drive the observed mass loss close to the star. For a typical stellar jet the crossover point inside which velocity perturbations of 30 - 40 km/s no longer produce shocks is ~ 300 AU from the source.