HOW LONG IS LONG ENOUGH WHEN MEASURING FLUXES AND OTHER TURBULENCE STATISTICS

It is determined how long a time series must be to estimate covariance s and moments up to fourth order with a specified statistical signific ance. For a given averaging time T there is a systematic difference be tween the true flux or moment and the ensemble average of the time mea ns of the same quantities. This difference, referred to here as the sy stematic error, is a decreasing function of T tending to zero for T -- > infinity. The variance of the time mean of the flux or moment, the s o-called error variance, represents the random scatter of individual r ealizations, which, when T is much larger than the integral time scale T of the time series, is also a decreasing function of T. This makes it possible to assess the minimum value of T necessary to obtain syste matic and random errors smaller than specified values. Assuming that t he time series are either Gaussian processes with exponential correlat ion functions or a skewed process derived from a Gaussian, we obtain e xpressions for the systematic and random errors. These expressions sho w that the systematic error and the error variance in the limit of lar ge T are both inversely proportional to T, which means that the random error, that is, the square root of the error variance, will in this l imit be larger than the systematic error. It is demonstrated theoretic ally, as well as experimentally with aircraft data from the convective boundary layer over the ocean and over land, that the assumption that the time series are Gaussian leads to underestimation of the random e rrors, while derived processes with a more realistic skewness and kurt osis give better estimates. For fluxes, the systematic and random erro rs are estimated when the time series are sampled instantaneously, but the samples separated in time by an amount DELTA. It is found that th e random error variance and the systematic error increase by less than 8% over continuously sampled data if DELTA is no larger than the inte gral scale obtained from the flux time series and the cospectrum, resp ectively.