[Paper Review] A Generalized Richardson Number Diagnostic for Turbulence in the Free Atmosphere

Mohamed Foudad, Miguel A. C. Teixeira+2 more|arXiv (Cornell University)|Feb 25, 2026

Meteorological Phenomena and Simulations0 citations

TL;DR

The paper introduces Ri_new, a generalized Richardson number that includes horizontal wind shear in addition to vertical shear, and demonstrates its superior skill over Ri_old and TI1 in predicting aviation turbulence using ERA5 data and 247M aircraft reports.

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.

Motivation & Objective

• Motivate improved turbulence diagnostics for the stratified free atmosphere, especially near jet streams.
• Derive Ri_new from the full turbulent kinetic energy budget including horizontal deformation and divergence.
• Evaluate Ri_new against Ri_old and TI1 using ERA5 reanalysis and large MADIS ACARS turbulence data.
• Identify the impact of the horizontal-to-vertical eddy viscosity ratio (Kh/Kv) on predictive skill.
• Assess applicability across seasons, regions, and turbulence intensities.

Proposed method

• Derive Ri_new from the full TKE budget, retaining horizontal gradients and introducing a closure with first-order turbulence closure.
• Express Ri_new as Ri_g divided by Pr_t, with Ri_g incorporating N^2 and augmented horizontal deformation terms.
• Define Ri_g = N^2 / [Sv^2 + (Kh/ Kv)(Div^2 + D_ST^2 + D_SH^2)], where Sv is vertical shear and Deformation terms capture horizontal shear.
• Relate Ri_new to Ri_old by setting Kh/ Kv = 0, recovering the classic Ri_old.
• Compare Ri_new, Ri_old, and TI1 using ERA5 fields interpolated to match ACARS observations.
• Utilize 247 million MADIS turbulence reports (2017–2024) and ROC/AUC as the primary verification metrics.

Figure 1: Spatial distribution of ACARS turbulence reports from the NOAA MADIS archive for the period 1 January 2017 to 31 December 2024. Shading indicates the number of reports on a logarithmic scale, binned onto a 1° × 1° latitude–longitude grid. Only reports above 8 km flight altitude are include

Experimental results

Research questions

• RQ1 Does Ri_new improve turbulence discrimination relative to the classical Ri_old and TI1 across turbulence intensities?
• RQ2 What is the optimal Kh/Kv ratio for maximizing predictive skill, and how sensitive is Ri_new to horizontal resolution?
• RQ3 How does Ri_new perform regionally and seasonally, especially where three-dimensional (vertical + horizontal) shear dominates turbulence production?
• RQ4 Is Ri_new robust over oceans and land for upper-troposphere/lower-stratosphere turbulence?
• RQ5 How well do Ri_new-based diagnostics align with observed ACARS turbulence reports across a large global dataset?

Key findings

• Ri_new consistently outperforms Ri_old and TI1 across multiple turbulence intensity thresholds, as shown by higher AUC values.
• An optimal Kh/Kv ratio around 5×10^3 yields the best ROC performance for Ri_new, with values in the 10^3–10^4 range also improving skill.
• Ri_new demonstrates robustness across resolutions and shows largest added value where turbulence is driven by both vertical and horizontal shear (e.g., over the contiguous U.S. and during summer).
• Performance remains high over oceans, with Ri_new providing the best overall discrimination in those regions as well.
• The approach provides a physically consistent and statistically robust improvement in turbulence diagnostics with potential benefits for aviation forecasting and climate impact assessments.

Figure 2: ROC curves for TI1 index (blue), the classical Richardson number (Ri old , black), and the optimal Ri new formulation (red) for turbulence intensity thresholds of (a) EDR $=$ 0.10, (b) EDR $=$ 0.20, and (c) EDR $=$ 0.30 $\mathrm{m}^{2/3}\,\mathrm{s}^{-1}$ . Gray curves show Ri new for diff