How Stochastic Structural Stability Theory Relates to Traditional Statistical Closures

The stochastic structural stability theory (SSST) is a technique\footnote{B. F. Farrell and P. J. Ioannou, J.\ Atmos.\ Sci.\ \textbf{60}, 2101 (2003).} that can be used for understanding the statistical behavior of drift-wave--zonal-flow systems.\footnote{B. F. Farrell and P. J. Ioannou, Phys.\ Plasmas \textbf{16}, 112903 (2009).} The method involves parameterizing the nonlinear DW--DW interactions as white noise while keeping the correct behavior of the DW--ZF interactions. The SSST can be interpreted as an intermediate step between the fundamental amplitude equations and conventional statistical closures. Unlike typical closures which describe only the mean-square ZF, the SSST retains a ZF amplitude. We discuss the relationship between the SSST and more traditional closures of the DW--ZF problem.\footnote{J.\ A.\ Krommes and C.-B.\ Kim, Phys.\ Rev.\ E.\ \textbf{62}, 8508 (2000).} In particular, we examine the physical content of a closure of the SSST equations, illustrating with the Generalized Hasegawa--Mima equation. Studies are also made of the Hasegawa--Wakatani system, extending and clarifying the work of Ref.~3. The ideas are relevant for the ultimate control of microturbulence.