Quantifying non-stationarity effects on organization of atmospheric turbulent eddy motion by Benford's law

• Benford's law is first applied to quantify the organization degrees of the multi-scaled structures.
• Multi-scale first digit entropy is proposed to be a quantified measure.
• Multi-scale first digit entropy is sensitive to organization degrees of the multi-scaled structures.

How to quantify the effects of non-stationarity on organization of atmospheric turbulent eddy motions has drawn less attention in recent literatures. Here Benford's law (BL), which states that the first digit (1 through 9) in many datasets follows a monotonically decreasing logarithmic distribution, is used to address this issue for the first time. A quantifier called multi-scale first digit entropy (MFDE) is adopted, which is based on the deviation of BL from the practical first digit distribution of multi-scaled vertical velocity increments, and marked differences are detected in stationary and non-stationary series. The MFDE values of stationary records do not change much with different scales while increase significantly for non-stationary ones as time scales increase. Due to the close relationship between MFDE and the multi-scale Shannon entropy (MSSE), the above results indicate that the non-stationary series are more organized than the stationary ones. Especially, the MFDE can also be used to quantify the different organization degrees of the multi-scaled structures existing in surface vertical velocity records.