  The Quasi-diagonal Direct Interaction Approximation (QDIA) closure equations
are formulated for inhomogeneous classical and quantum fields interacting
through dynamical equations with quadratic nonlinearity and with first or
second order time derivatives. Associated more complex inhomogeneous DIA and
Self-Energy closure equations are expounded as part of the derivation. The QDIA
employs a bare vertex approximation and is only a few times more
computationally intensive than the homogeneous DIA. Examples of applications to
turbulent classical geophysical and Navier Stokes fluids, including
non-Gaussian noise, to classical and quantum Klein Gordon equations with g
phi^3 Lagrangian interaction, and to coupled field-auxiliary field equations
associated lambda phi^4 Lagrangian interaction, are presented.
