Application of statistical dynamical turbulence closures to data assimilation

We describe the development of an accurate yet computationally tractable statistical dynamical closure theory for general inhomogeneous turbulent flows, coined the quasi-diagonal direct interaction approximation closure (QDIA), and its application to problems in data assimilation. The QDIA provides prognostic equations for evolving mean fields, covariances and higher-order non-Gaussian terms, all of which are also required in the formulation of data assimilation schemes for nonlinear geophysical flows. The QDIA is a generalization of the class of direct interaction approximation theories, initially developed by Kraichnan (1959 J. Fluid Mech. 5 497) for isotropic turbulence, to fully inhomogeneous flows and has been further generalized to allow for both inhomogeneous and non-Gaussian initial conditions and long integrations. A regularization procedure or empirical vertex renormalization that ensures correct inertial range spectra is also described. The aim of this paper is to provide a coherent mathematical description of the QDIA turbulence closure and closure-based data assimilation scheme we have labeled the statistical dynamical Kalman filter. The mathematical formalism presented has been synthesized from recent works of the authors with some additional material and is presented in sufficient detail that the paper is of a pedagogical nature.