We present new adaptive optics echelle spectra of the near-infrared [Fe II] lines in the redshifted and blueshifted jets from the T Tauri star RW Aur. The spectra have an unprecedented combination of high spatial and spectral resolution that makes it possible to trace the dynamics of the flow to a projected distance of only 10 AU from the source. As noted by previous studies, the redshifted flow is much slower than its fainter blueshifted counterpart. Our observations clearly show that both the radial velocities and the emission line widths are larger closer to the source on both sides of the jet. The line widths are 20% - 30% of the jet velocity on both sides of the flow, significantly larger than would be produced by a divergent constant velocity flow. The observed line widths could arise from a layered velocity structure in the jet or from magnetic waves. A bright knot in the redshifted jet has no concomitant increase in line width, implying that it is not heated by a bow shock. Alternate heating mechanisms include planar shocks, ambipolar diffusion and magnetic reconnection.