The response of homogeneous turbulence to rapid mean-flow compression in idealized internal combustion engines and rapid compression machines is examined using a hierarchy of closure models for the Reynolds stress anisotropy. This hierarchy is based on a Reynolds stress anisotropy transport equation that is modeled from the exact transport equation for the anisotropy. The hierarchy of models includes a recentlydeveloped non-equilibrium model, which is shown to be in good agreement with more computationally-complex fully differential models. Using this hierarchical approach, the flow physics addressed by each closure is identified and closure accuracy is shown to depend on the degree to which non-equilibrium turbulent flow effects are captured. We examine the evolution of the turbulence kinetic energy, Reynolds stresses, and anisotropy as a function of the degree of non-equilibrium in the flow, which is parameterized by the ratio of characteristic turbulence and mean-flow deformation time scales. By comparing model results to results obtained from rapid distortion theory and higher level closures, prescriptions are provided for the applicability of different closure models based on the magnitude of the non-equilibrium parameter. The theoretical analysis is complemented by comparisons of simulation results with previously established direct numerical simulations for a one-dimensional compression. Finally, we connect these prescriptions with experimental measurements of turbulent and mean-flow time scales for internal combustion engines operating at realistic conditions.