The quasi-diagonal direct interaction approximation (QDIA) closure theory is formulated for the interaction of mean fields, Rossby waves and inhomogeneous turbulence over topography on a generalized $\beta$-plane. An additional small term, corresponding to the solid-body rotation vorticity on the sphere, is included in the barotropic equation and it is shown that this results in a one-to-one correspondence between the dynamical equations, Rossby wave dispersion relations, nonlinear stability criteria and canonical equilibrium theory on the generalized $\beta$-plane and on the sphere. The dynamics, kinetic energy spectra, mean field structures and mean streamfunction tendencies contributed by transient eddies are compared with the ensemble-averaged results from direct numerical simulations (DNS) at moderate resolution. A series of numerical experiments is performed to examine the generation of Rossby waves when eastward large-scale flows impinge on a conical mountain in the presence of moderate to strong two-dimensional turbulence. The ensemble predictability of northern hemisphere flows in 10-day forecasts is also examined on a generalized $\beta$-plane. In all cases, the QDIA closure is found to be in very good agreement with the statistics of DNS except in situations of strong turbulence and weak mean fields where ensemble-averaged DNS fails to predict mean field amplitudes correctly owing to sampling problems even with as many as 1800 ensemble members.