  Low-order equal-time statistics of a barotropic flow on a rotating sphere are
investigated. The flow is driven by linear relaxation toward an unstable zonal
jet. For relatively short relaxation times, the flow is dominated by
critical-layer waves. For sufficiently long relaxation times, the flow is
turbulent. Statistics obtained from a second-order cumulant expansion are
compared to those accumulated in direct numerical simulations, revealing the
strengths and limitations of the expansion for different relaxation times.
