  The concept of statistical stability is central to Malkus's 1956 attempt to
predict the mean profile in shear flow turbulence. Here we discuss how his
original attempt to assess this - an Orr-Sommerfeld analysis on the mean
profile - can be improved by considering a cumulant expansion of the
Navier-Stokes equations. Focusing on the simplest non-trivial closure (commonly
referred to as CE2) which corresponds to the quasilinearized Navier-Stokes
equations, we develop an extended Orr-Sommerfeld analysis (EOS) which also
incorporates information about the fluctuation field. A more practical version
of this - minimally extended Orr-Sommerfeld analysis (mEOS) - is identified and
tested on a number of statistically-steady and therefore statistically stable
turbulent channel flows. Beyond the concept of statistical stability, this
extended stability analysis should also improve the popular approach of
mean-flow linear analysis in time-dependent shear flows by including more
information about the underlying flow in its predictions as well as for other
flows with additional physics such as convection.
