A Generalized Richardson Number Diagnostic for Turbulence in the Free Atmosphere

Abstract

A new Richardson number formulation, Ri_new, is introduced to improve the diagnosis of turbulence in the stratified free atmosphere. The formulation is derived from the turbulent kinetic energy budget and accounts for both vertical wind shear and horizontal shear (deformation and divergence), weighted by the ratio of horizontal to vertical eddy viscosities (K_mh/K_mv). This extends the classical Richardson number Ri_old, which accounts only for vertical shear. The diagnostics Ri_new , Ri_old ,and the widely used Turbulence Index 1 (TI1), computed from ERA5 reanalysis, are evaluated using more than 247 million automated turbulence reports from commercial aircraft (2017--2024). Across various turbulence intensity thresholds, Ri_new consistently outperforms the other diagnostics, resulting in higher AUC values and improved probability of detection at operationally relevant false-alarm rates. The highest skill is obtained for K_mh/K_mv approximately 5000. Seasonal and regional evaluations indicate that the added value of Ri_new is largest where turbulence generation involves both vertical and horizontal shear, such as over the contiguous United States and during summer. Ri_new remains the best-performing diagnostic in all regions and seasons. Spatial case studies show that Ri_new identifies 83--98% of observed moderate-or-greater turbulence events compared with 54--85% for Ri_old. This substantial improvement in detection comes with a much smaller increase in false alarms, confirming that Ri_new provides a more physically realistic representation of turbulence-prone regions. These results demonstrate that incorporating horizontal wind shear into the Richardson number yields a physically consistent and statistically robust improvement in turbulence diagnostics, with relevance for research and operational applications.