The dynamics and spectra of the quasi-diagonal direct interaction approximation (QDIA) closure for inhomogeneous two-dimensional turbulence over mean (single realization) topography are compared with results from direct numerical simulations (DNS). A more efficient version of the closure, termed the cumulant update QDIA (CUQDIA), has also been formulated and tested. Studies are performed for a range of resolutions, for large scale Reynolds numbers between very low ($R_{L} < 1$) and moderate ($R_{L} \approxeq 300$) and for wide ranges of topographic spectra and initial mean field and transient spectra. The QDIA-type closures are shown to be computationally tractable for general inhomogeneous flows, particularly in cumulant update form, and to perform extremely well when the turbulence is weak. At low ($R_{L} \approxeq 60$) to moderate ($R_{L} \approxeq 300$) Reynolds numbers the presence of significant amplitude small-scale mean fields and topography reduces the under-estimation of small-scale transient kinetic energy that is characteristic of the Eulerian direct interaction approximation (DIA). A regularized version of the CUQDIA closure (RCUQDIA) in which interactions are localized in wavenumber space, depending on specified cut-off ratios, has also been tested at moderate Reynolds number for cases when the small-scale mean fields and topography are weak. Excellent agreement has been found between the RCUQDIA closure and DNS results for turbulent flows with properties broadly similar to atmospheric spectra.