THE DISTRIBUTION OF FLUCTUATION MEASUREMENTS WITHIN THE RANDOM-PHASE MODEL

A variety of diagnostics measure fluctuating quantities at a small number of locations. A common method to estimate the wave-number spectrum S(k) at a given frequency is to make many measurements of the "local" wave number k(l), which is the phase difference between two locations divided by their separation. The random-phase model allows the synthesis of a typical signal from a given S(k). We apply local wave-number analysis to such synthetic signals and show that, in the limit of small separation, the power-weighted mean of k(l) accurately reflects the mean of S(k), and the width of the distribution is exactly square-root 2 times smaller than the width of S(k). We also show that no information on the shape of S(k) exists in the measurements. Finally, we generalize these results to finite separations, finite measurement volumes, and any number of locations.