We describe the development of a computationally tractable statistical dynamical turbulence closure for inhomogeneous two-dimensional turbulence and its application to problems in atmospheric dynamics. Based on a generalization of Kraichnan’s direct interaction approximation and a quasi-diagonal approximation to the covariances the quasi-diagonal direct interaction approximation (QDIA) is formulated for the interaction of mean fields, Rossby waves and inhomogeneous turbulence over topography on a generalized β-plane has a one-to-one correspondence between the dynamical equations, Rossby wave dispersion relations, nonlinear stability criteria and canonical equilibrium theory on the sphere. We consider not only the underlying theoretical basis but also the numerical methodology required to integrate the resulting non-Markovian integro-differential closure equations over the long time periods typical of the growth and decay of coherent structures in the atmosphere. We discuss the problem of vertex renormalization and consider a regularization methodology in which eddy-eddy, eddy-topographic and eddy-mean field interactions are localized in wavenumber space. We examine application of the closure to a range of problems in numerical weather prediction such as the role of non-Gaussian initial perturbations and small-scale noise in determining error growth; the parameterization of subgrid-scale energy and enstrophy transfers; and the development of generalized Kalman filter methods for data assimilation in strongly nonlinear flows. We also consider the relationship to minimum enstrophy, maximum entropy and entropy production arguments. Throughout the dynamics, kinetic energy spectra, mean field structures and mean streamfunction tendencies contributed by transient eddies are compared with ensemble-averaged results from direct numerical simulations (DNS).