Manifestation of random phase in a finite ensemble of a turbulent fluid

A field can be specified by a set of spectral coefficients, {c(j)(t)}, rela tive to an expansion basis. In fluid turbulence, one often assumes the cond ition that if one were to extract data from an ensemble of realizations, on e would find that the average, [c(i)*(t)c(i')(t)], would vanish unless i = i', as in the case of a homogeneous ensemble in which the i's represent wav e vectors. We analytically treat an extension to the case in which one has a larger number of coefficients than realizations introducing intrinsic lin ear dependences that vitiate the stated condition.