<p>Turbulent shear stresses play a pivotal role in momentum transfer within atmospheric flows. In stably stratified atmospheric boundary layers, it is commonly assumed that the total turbulent shear stress follows a power-law distribution, which in turn serves as the basis for deriving various profiles of eddy viscosity and wind speed. Although observational data and numerical simulations offer partial support for this key assumption, a rigorous theoretical verification of the power-law behavior of turbulent shear stress has remained elusive. By applying Lie group analysis to the Ekman equations, this study provides a rigorous mathematical proof of this power-law relationship. This brief report therefore enhances our physical understanding of atmospheric dynamics.</p>