Derivation of the Direct-Interaction Approximation Using Novikov's Theorem

The direct-interaction approximation (DIA)\footnote{R. H. Kraichnan, J. Fluid Mech. {\bf5}, 497 (1959).} is a crucially important statistical closure for both neutral fluids and plasmas. Kraichnan's original derivation proceeded in $k$~space and assumed a large number~$N$ of interacting Fourier modes. That is problematic; the DIA can be formulated even for $N = 3$. In the present work an alternate $x$-space procedure based on Novikov's theorem is described. That theorem is a statement about the correlations of certain Gaussian functionals. Turbulence cannot be Gaussian due to nonlinearity, but Novikov's theorem can be used to formulate self-consistent equations for a Gaussian component of the turbulence. The DIA emerges under the assumption that certain higher-order correlations are small. In essence, this procedure is merely a restatement of Kraichnan's arguments, but it adds additional perspective because the assumption of large~$N$ is not required. Details can be found in a lengthy set of tutorial Lecture Notes.\footnote{J. A. Krommes, A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev's \textsl{Plasma Turbulence} and the resonance-broadening theory of Dupree and Weinstock, J. Plasma Phys., in press.}