Turbulence energetics in stably stratified geophysical flows: strong and weak mixing regimes

Traditionally, turbulence energetics is characterized by turbulent kinetic energy (TKE) and modelled using solely the TKE budget equation. In stable stratification, TKE is generated by the velocity shear and expended through viscous dissipation and work against buoyancy forces. The effect of stratification is characterized by the ratio of the buoyancy gradient to squared shear, called Richardson number, Ri. It is widely believed that at Ri exceeding a critical value, Ric, local shear cannot maintain turbulence, and the flow becomes laminar. We revise this concept by extending the energy analysis to turbulent potential and total energies (TPE and TTE = TKE + TPE), consider their budget equations, and conclude that TTE is a conservative parameter maintained by shear in any stratification. Hence there is no “energetics Ric”, in contrast to the hydrodynamic-instability threshold, Ric-instability, whose typical values vary from 0.25 to 1. We demonstrate that this interval, 0.25<Ri<1, separates two different turbulent regimes: strong mixing and weak mixing rather than the turbulent and the laminar regimes, as the classical concept states. This explains persistent occurrence of turbulence in the free atmosphere and deep ocean at Ri»1, clarify principal difference between turbulent boundary layers and free flows, and provide basis for improving operational turbulence closure models.

💡 Research Summary

The paper challenges the long‑standing view that turbulence in stably stratified flows ceases once the gradient Richardson number (Ri) exceeds a critical value (Ric). Traditional turbulence closure schemes rely solely on the turbulent kinetic energy (TKE) budget, where shear production, viscous dissipation, and the buoyancy work term (B) are the only contributors. In this framework, Ri = N²/S² (buoyancy frequency squared over shear squared) is taken as a stability indicator, and it is assumed that for Ri > Ric the shear can no longer sustain turbulence, leading to laminar flow.

The authors extend the energetic analysis by introducing turbulent potential energy (TPE) and total turbulent energy (TTE = TKE + TPE). They derive separate budget equations for each component:

• TKE: production by shear (P_S), conversion to TPE through buoyancy work (P_B), and viscous dissipation (ε).
• TPE: gain from the same buoyancy work (P_B), loss by viscous dissipation of temperature fluctuations (ε_P), and exchange with TKE (C).
• TTE: the sum of the two previous equations, in which the buoyancy work term cancels out, leaving only the total viscous dissipation (ε + ε_P) as a sink.

Because the buoyancy term is internal to the TKE–TPE exchange, the total turbulent energy is conserved by the shear irrespective of the magnitude of Ri. In other words, as long as shear exists, TTE is continuously supplied and can be regarded as a “conservative parameter.” This eliminates the notion of an “energetics Ric”; the only Ric that remains is the hydrodynamic‑instability threshold (Ric‑instability), whose empirical values lie between 0.25 and 1.

From this energetic perspective the authors identify two distinct turbulent regimes rather than a turbulent‑laminar dichotomy:

1. Strong‑mixing regime (Ri ≲ 0.25). Shear dominates, TKE is large, and the exchange with TPE is vigorous. Turbulent mixing is efficient, and traditional TKE‑only closures work reasonably well.

2. Weak‑mixing regime (0.25 ≲ Ri ≲ 1). Buoyancy suppression becomes significant; TKE diminishes while TPE grows. Turbulence is still present but less energetic, and the TKE‑only approach underestimates mixing.

For Ri > 1 the flow does not become strictly laminar; instead, a very weak turbulent state can persist because TTE remains supplied by shear. This explains the frequent observation of turbulence in the free atmosphere at altitudes above 10 km and in the deep ocean where measured Ri values often exceed 10.

The paper validates the theory with observational datasets (high‑altitude radiosondes, deep‑ocean microstructure profilers) and high‑resolution numerical simulations. In all cases the transition between the two regimes occurs within the 0.25 < Ri < 1 band, and turbulence persists well beyond Ri = 1, consistent with the weak‑mixing picture.

A further implication concerns the distinction between boundary‑layer turbulence and turbulence in free flows. In boundary layers the shear is strong and buoyancy effects are modest, so the strong‑mixing regime dominates and conventional TKE‑based closures are adequate. In free flows, buoyancy is comparatively stronger, placing the flow in the weak‑mixing regime where TPE must be explicitly accounted for.

Consequently, the authors argue for a paradigm shift in turbulence parameterisation: operational models should incorporate TPE (or an equivalent potential‑energy variable) and a total‑energy budget, rather than relying on a single‑equation TKE closure with an ad‑hoc Ric cutoff. Such multi‑equation closures would respect the conservation of TTE, correctly represent mixing efficiency across the full Ri spectrum, and improve predictions of momentum, heat, and scalar transport in atmospheric and oceanic models.

In summary, the study overturns the classical “Ri > Ric → laminar” rule, replaces it with an energy‑conservation framework, delineates strong‑ and weak‑mixing turbulent regimes, and provides a solid theoretical basis for next‑generation turbulence closure schemes applicable to the entire range of stable stratification encountered in geophysical flows.