Robust relation of streamwise velocity autocorrelation in atmospheric surface layers based on an autoregressive moving average model

We construct an autoregressive moving average (ARMA) model consisting of the history and random effects for the streamwise velocity fluctuation in boundary-layer turbulence. The distance to the wall and the boundary-layer thickness determine the time step and the order of the ARMA model, respectively. Based on the autocorrelation's analytical expression of the ARMA model, we obtain a global analytical expression for the second-order structure function, which asymptotically captures the inertial, dynamic and large-scale ranges. Specifically, the exponential autocorrelation of the ARMA model arises from the autoregressive coefficients and is modified to logarithmic behaviour by the moving-average coefficients. The asymptotic expressions enable us to determine model coefficients by existing parameters, such as the Kolmogorov and the Townsend–Perry constants. A consequent double-log expression for the characteristic length scale is derived and is justified by direct numerical simulation data with $Re_\tau \approx 5200$ and field-measured neutral atmospheric surface layer data with $Re_\tau \sim O(10^6)$ from the Qingtu Lake Observation Array site. This relation is robust because it applies to $Re_\tau$ from $O(10^4)$ to $O(10^6)$ , and even when the statistics of natural ASL deviate from those of canonical boundary-layer turbulence, e.g. in the case of imbalance in energy production and dissipation, and when the Townsend–Perry constant deviates from traditional values.