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, 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.

Figures (16)

• 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 included. The total number of observations is 247 730 014.
• Figure 2: ROC curves for TI1 index (blue), the classical Richardson number (Ri$_{\mathrm{old}}$, black), and the optimal Ri$_{\mathrm{new}}$ 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$_{\mathrm{new}}$ for different tested $K_{mh}/K_{mv}$ ratios. The diagonal dashed line denotes no skill. Values in parentheses give the area under the ROC curve (AUC). The total number of observations used was 247 730 014, with 1 770 543 ($\approx$ 0.71%), 192 447 ($\approx$ 0.08%), and 19 705 ($\approx$ 0.01%) turbulence events for (a) EDR $\ge$ 0.10 (LOG), (b) EDR $\ge$ 0.20 (MOG), and (c) EDR $\ge$ 0.30 $\mathrm{m}^{2/3}\,\mathrm{s}^{-1}$ (SOG), respectively.
• Figure 3: Probability of Detection (POD) as a function of the ratio $K_{mh}/K_{mv}$ for the new Richardson number formulation Ri$_{\mathrm{new}}$ (solid grey) evaluated at four fixed values of the Probability of False Detection (POFD): (a) 20%, (b) 30%, (c) 40%, and (d) 50%. Horizontal dashed lines show the skill of TI1 (blue) and Ri$_{\mathrm{old}}$ (black), while the vertical red dashed line shows the optimal ratio $K_{mh}/K_{mv}$ that maximizes POD for MOG turbulence (EDR $\ge$ 0.20 m$^{2/3}$ s$^{-1}$).
• Figure 4: ROC curves and 95% confidence intervals for TI1, $Ri_{\mathrm{old}}$, and $Ri_{\mathrm{new}}$ (optimal $K_{mh}/K_{mv}=5000$), computed from 10 000 000 randomly sampled reports with 1000 bootstrap replications (EDR $\ge 0.20$ m$^{2/3}$ s$^{-1}$). The uncertainty bands are smaller than the line thickness (approximately $\pm 1\%$) and are therefore almost imperceptible.
• Figure 5: As in Figure \ref{['fig:fig2']}, but for (a) winter DJF; (b) spring MAM; (c) summer JJA; (d) autumn SON, for MOG turbulence (EDR $\ge$ 0.20 m$^{2/3}$ s$^{-1}$)