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), relative to an expansion basis. In fluid turbulence, one often assumes the condition that if one were to extract data from an ensemble of realizations, one would find that the average, , would vanish unless i = i('), as in the case of a homogeneous ensemble in which the i's represent wave vectors. We analytically treat an extension to the case in which one has a larger number of coefficients than realizations, introducing intrinsic linear dependences that vitiate the stated condition.