A frequency-averaged quasilinear model is derived and experimentally validated for the evolution of ion acoustic turbulence (IAT) along the centerline of a 100-A class, LaB_{6} hollow cathode. Probe-based diagnostics and a laser induced fluorescence system are employed to measure the properties of both the turbulence and the background plasma parameters as they vary spatially in the cathode plume. It is shown that for the three discharge currents investigated, 100 A, 130 A, and 160 A, the spatial growth of the total energy density of the IAT in the near field of the cathode plume is exponential and agrees quantitatively with the predicted growth rates from the quasilinear formulation. However, in the downstream region of the cathode plume, the growth of IAT energy saturates at a level that is commensurate with the Sagdeev limit. The experimental validation of the quasilinear model for IAT growth and its limitations are discussed in the context of numerical efforts to describe self-consistently the plasma processes in the hollow cathode plume.