A Generalized Richardson Number Diagnostic for Turbulence in the Free Atmosphere

TL;DR

This paper introduces a new Richardson number, Ri_new, that incorporates horizontal shear to better diagnose turbulence in the free atmosphere, outperforming traditional metrics across various regions and conditions.

Contribution

The paper develops and validates a generalized Richardson number that includes horizontal shear, improving turbulence prediction accuracy over existing diagnostics.

Findings

Ri_new outperforms Ri_old and TI1 in turbulence detection.

Peak predictive skill occurs at Kmh/Kmv ratios around 5000.

Ri_new shows consistent improvements across regions and seasons.

Abstract

A new Richardson number formulation, Ri_new, is introduced to improve the diagnosis of turbulence in the stratified free atmosphere, particularly near jet stream regions. 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 (Kmh/Kmv). This extends the classical Richardson number Ri_old, which includes only vertical shear, and provides a physically based measure of the balance between stratification and three-dimensional shear production. 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. Sensitivity analyses show that the predictive skill of Ri_new is maximized for Kmh/Kmv values in the range 10^3-10^4, with peak performance near 5000 and weak dependence on horizontal resolution. 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. Over oceans, performance remains high and Ri_new still provides the best overall discrimination skill. 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.