The notion of statistical equilibrium rests upon the idea that the time-scale of somechaotic, small-scale process is appreciably smaller than that of the large-scale processwith which it interacts. Put a slightly different way, statistical equilibrium implies thatthe large-scale process of interest evolves slowly compared with the small-scale processone wishes to regard in an ensemble-integrated sense. It is instructive to examine thenature of convective statistical equilibrium in a few, relatively simple, cases.As a first example we use the classical Prandtl convective boundary layer, consistingof a semi-infinite, Boussinesq fluid overlying an infinite horizontal plate. The fluid iscooled at a constant, infinitesimal rate such that the volume-integrated cooling is constant,and the plate is held at a fixed temperature. The lower plate must be considereddynamically rough, so that molecular diffusivities may be neglected, but we will considerstatistical aspects of the flow at distances from the plate that are large compared withOn physical grounds, we expect the fluid to convect turbulently.
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