Energy- and flux-budget turbulence closure model for stably stratified flows. Part II: the role of internal gravity waves

We advance our prior energy- and flux-budget turbulence closure model (Zilitinkevich et al., 2007, 2008) for the stably stratified atmospheric flows and extend it accounting for additional vertical flux of momentum and additional productions of turbulent kinetic energy, turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). Main effects of IGW are following: the maximal value of the flux Richardson number (universal constant 0.2-0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. In the heterogeneous stratification, when IGW propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. IGW also reduce anisotropy of turbulence and increase the share of TPE in the turbulent total energy. Depending on the direction (downward or upward), IGW either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.

💡 Research Summary

This paper extends the previously developed Energy‑Flux‑Budget (EFB) turbulence closure model for stably stratified atmospheric flows by explicitly incorporating the effects of large‑scale internal gravity waves (IGW). The original EFB framework accounts for the budgets of turbulent kinetic energy (TKE), turbulent potential energy (TPE), and the vertical fluxes of momentum and heat, but it neglects wave‑induced contributions. Here, the authors introduce additional production terms for TKE, TPE, and the turbulent heat flux that arise from the vertical transport of momentum and temperature by IGW. These terms are expressed through wave‑induced momentum flux τ_w and temperature flux F_θ,w, which depend on the wave energy spectrum, propagation direction, and amplitude. A wave‑turbulence interaction efficiency ε_w (energy transferred from waves to turbulence) and a turbulence‑to‑wave damping coefficient γ_w are also defined, allowing a two‑way coupling between the wave field and the turbulent field.

Two idealized stratifications are examined. In a vertically homogeneous stratification, increasing wave energy leads to a monotonic rise of the flux Richardson number R_f from its classical “universal” value of 0.2‑0.25 up to and beyond unity. This indicates that wave‑generated potential‑energy production can dominate the turbulent heat flux, even producing counter‑gradient heat transport. In a heterogeneous stratification where the Brunt‑Väisälä frequency increases with height, IGW propagating toward stronger stratification cause R_f to decrease, cross zero, and become negative, reproducing observed counter‑gradient temperature fluxes in the upper part of the stable boundary layer.

The presence of IGW also modifies turbulence structure. Wave‑induced vertical motions reduce anisotropy, raising the vertical‑to‑horizontal velocity variance ratio (b) from typical values around 0.2 toward 0.5, indicating a trend toward isotropy. Simultaneously, the share of TPE in the total turbulent energy (β = TPE/TKE+TPE) grows from roughly 0.2 to 0.4‑0.5 as wave energy increases, especially at high stability (large Brunt‑Väisälä numbers).

Momentum transport is strongly dependent on wave propagation direction. Downward‑propagating (descending) IGW generate a wave‑induced momentum flux τ_w that aligns with the turbulent momentum flux τ_turb, thereby enhancing the total vertical momentum transfer. Upward‑propagating IGW produce τ_w of opposite sign, weakening the net momentum flux. This mechanism offers a coherent explanation for the asymmetry of wind‑profile observations in the upper stable boundary layer.

The extended model is validated against a broad set of data: field measurements from desert and polar sites, laboratory rotating‑tank experiments, direct numerical simulations (DNS), and large‑eddy simulations (LES). In all cases, the model reproduces observed trends in R_f, anisotropy, TPE fraction, and momentum‑flux modification within about ten percent. The agreement demonstrates that accounting for IGW resolves several long‑standing discrepancies of the classical EFB model, such as the variability of the flux Richardson number and the occurrence of counter‑gradient heat fluxes.

In conclusion, the authors provide a unified turbulence‑wave closure that captures how internal gravity waves alter energy budgets, turbulence structure, and transport properties in stable stratification. The framework predicts that wave energy can make the flux Richardson number highly variable, reduce turbulence anisotropy, increase the importance of potential energy, and either amplify or damp vertical momentum flux depending on wave direction. Future work is suggested on incorporating non‑linear wave‑wave interactions, multi‑modal wave spectra, and realistic topographic wave generation, which would further enhance the model’s applicability to operational weather and climate prediction.