Temporal surrogates of spatial turbulent statistics: The Taylor hypothesis revisited

The Taylor hypothesis, which allows surrogating spatial measurements requiring many experimental probes by time series from one or two probes, is examined on the basis of a simple analytic model of turbulent statistics. The main points are as follows: (i) The Taylor hypothesis introduces systematic errors in the evaluation of scaling exponents. (ii) When the mean wind $V_0$ is not infinitely larger than the root-mean-square longitudinal turbulent fluctuations $v_{\mathrm{T}},$ the effective Taylor advection velocity $V_{\mathrm{ad}}$ should take the latter into account. (iii) When two or more probes are employed the application of the Taylor hypothesis and the optimal choice of the effective advecting wind $V_{\mathrm{ad}}$ need extra care. We present practical considerations for minimizing the errors incurred in experiments using one or two probes. (iv) Analysis of the Taylor hypothesis when different probes experience different mean winds is offered.