The influence of far field stratification on shear-induced turbulent mixing

We compare the properties of the turbulence induced by the breakdown of Kelvin–Helmholtz instability (KHI) at high Reynolds number in two classes of stratified shear flows where the background density profile is given by either a linear function or a hyperbolic tangent function, at different values of the minimum initial gradient Richardson number ${{Ri}}_0$ . Considering global and local measures of mixing defined in terms of either the irreversible mixing rate $\mathscr {M}$ associated with the time evolution of the background potential energy, or an appropriately defined density variance dissipation rate $\chi$ , we find that the proliferation of secondary instabilities strongly affects the efficiency of mixing early in the flow evolution, and also that these secondary instabilities are highly sensitive to flow perturbations that are added at the point of maximal (two-dimensional) billow amplitude. Nevertheless, mixing efficiency does not appear to depend strongly on the far field density structure, a feature supported by the evolution of local horizontally averaged values of the buoyancy Reynolds number ${Re}_b$ and gradient Richardson number ${Ri}_g$ . We investigate the applicability of various proposed scaling laws for flux coefficients $\varGamma$ in terms of characteristic length scales, in particular discussing the relevance of the overturning ‘Thorpe scale’ in stratified turbulent flows. Finally, we compare a variety of empirical model parameterizations used to compute diapycnal diffusivity in an oceanographic context, arguing that for transient flows such as KHI-induced turbulence, simple models that relate the ‘age’ of a turbulent event to its mixing efficiency can produce reasonably robust mixing estimates.

Les travaux sur la turbulence : les origines, Toucans, Cost-ES0905 et influence de l'entropie

Le schéma de turbulence Toucans est utilisé dans la configuration opérationnelle Alaro du modèle Aladin depuis début 2015. Son développement a été initié, guidé et en grande partie conçu par Jean-François Geleyn. Ce développement a commencé avec le prédécesseur du schéma Toucans, le schéma « pseudo-pronostique » en énergie cinétique turbulente, lui-même basé sur l'ancien schéma de turbulence de Louis, mais étendu dans Toucans à un schéma pronostique. Le schéma Toucans a pour objectif de traiter de manière cohérente les fonctions qui dépendent de la stabilité verticale de l'atmosphère, de l'influence de l'humidité et des échelles de longueur de la turbulence (de mélange et de dissipation). De plus, de nouvelles caractéristiques ont été ajoutées : une représentation améliorée pour les stratifications très stables (absence de nombre de Richardson critique), une meilleure représentation de l'anisotropie, un paramétrage unifié de la turbulence et des nuages par l'ajout d'une deuxième énergie turbulente pronostique et la paramétrisation des moments du troisième ordre. The Toucans turbulence scheme is a turbulence scheme that is used in the operational Alaro configuration of the Aladin model since early 2015. Its development was initiated, guided and to a large extend authored by Jean-François Geleyn. The development started with the predecessor of the Toucans scheme, the "pseudo-prognostic" turbulent kinetic energy scheme which itself was built on the "Louis" turbulence scheme, but extended to a prognostic scheme. The Toucans scheme aims for a consistent treatment of stability dependency functions, influence of moisture, and turbulence length scales. Additionally, new features were added to the turbulence scheme: improved representation of turbulence in very stable stratification (absence of critical gradient Richardson number), better representation of anisotropy, unified parameterization of turbulence and clouds via addition of second prognostic turbulence energy, and parameterization of third order moments.

COEFFICIENTS OF EDDY DIFFUSION OF MOMENTUM AND HEAT IN ATMOSPHERIC BOUNDARY LAYER: NUMERICAL STUDY

The changes in turbulent eddy mixing in the atmospheric boundary layer (ABL) are investigated with the use of the mesoscale RANS turbulence model with account for effects of internal gravitational waves, which support momentum transfer under condition of very stable stratification. A focus was put on analysis of behavior of the coefficients of vertical eddy diffusion of momentum and heat. The behavior of the turbulent eddy mixing parameters was found to be consistent with measurements in the laboratory and atmosphere. In particular, the flow Richardson number () during the transient flow to a strongly stable state can behave nonmonotonically, growing with increasing gradient Richardson number () to the state of saturation at a certain gradient Richardson number ( Ri@ 1 ), which separates two different turbulent regimes: strong mixing and weak mixing.

Boundary Layer Stability Regime at DACCIWA Site Using Gradient Richardson Number

Meteorological data including air temperature and wind speed which were collected from DACCIWA measurement site at a tropical agricultural field site in Ile-Ife (7.55oE, 4.56oE), south-western Nigeria have been used to classify boundary layer stability regimes using gradient Richardson number. Three categories were considered to deduce the pattern of stability conditions namely stable, unstable and neutral conditions for 3-hourly intervals at 0.00, 03.00, 06.00, 09.00, 12.00, 15.00, 18.00 and 21.00 hours from 15th June to 31st July 2016. The data were sampled every 1sec and stored subsequently as 10 minutes averages for all the measured parameters. The data was further reduced to 30 minutes averages for easy analysis and manipulation in the calculation of gradient Richardson number used for boundary layer stability regime characterization. The results showed that the month of June 2016 had prevalence of stable regime from 0:00 – 6:00 am and 6:00 pm; 9:00 am was predominantly neutral and shared similar pattern with 9:00 pm. Unstable regime was slightly observed at 12:00 pm and majorly observed at 3:00 pm. The month of July had a little shift from what was observed in the month of June. Predominance of neutral conditions was observed from 9:00 pm to 9:00 am; Hours of 12:00 – 3:00 pm were dominated by unstable regime while 6:00 pm was dominated by stable regime.

A note on analytic solutions for entraining stratified gravity currents

High-Reynolds-number steady currents of relatively dense fluid propagating along a horizontal boundary become unstable and mix with the overlying fluid if the gradient Richardson number across the interface is less than 1/4. The process of entrainment produces a deepening mixing layer at the interface, which increases the gradient Richardson number of this layer and eventually may suppress further entrainment. The conservation of the vertically averaged buoyancy and momentum flux, as the current advances along the boundary, leads to two integral constraints relating the downstream flow with that upstream of the mixing zone. These constraints are equivalent to imposing a Froude number in the upstream flow. Using the ansatz that the dowstream velocity and buoyancy profiles in the current have a lower well-mixed region overlain by an interfacial layer of constant gradient, we can use these two constraints to quantify the total entrainment of ambient fluid into the flow as a function of the gradient Richardson number of the downstream flow. This leads to recognition that both subcritical and supercritical currents may develop downstream of the mixing zone. However, as the mixing increases and the interfacial layer gradually deepens, there is a critical point at which these two solution branches coincide. For each upstream Froude number, we can also determine the downstream flow with maximal entrainment. This maximal entrainment solution coincides with the convergence point of the supercritical and subcritical branches. We compare this with the entrainment predicted for those solutions with a gradient Richardson number of 1/4, which corresponds to the marginally stable case. As the upstream Froude number $Fr_{u}$ increases, the maximum depth of the interfacial mixing layer gradually increases until eventually, for $Fr_{u}>2.921$, the whole current may become modified through entrainment. We discuss the relevance of these results for mixing in gravity-driven flows.

The effect of stable thermal stratification on turbulent boundary layer statistics

The effects of stable thermal stratification on turbulent boundary layers are experimentally investigated for smooth and rough walls. For weak to moderate stability, the turbulent stresses are seen to scale with the wall shear stress, compensating for changes in fluid density in the same manner as done for compressible flows. This suggests little change in turbulent structure within this regime. At higher levels of stratification turbulence no longer scales with the wall shear stress and turbulent production by mean shear collapses, but without the preferential damping of near-wall motions observed in previous studies. We suggest that the weakly stable and strongly stable (collapsed) regimes are delineated by the point where the turbulence no longer scales with the local wall shear stress, a significant departure from previous definitions. The critical stratification separating these two regimes closely follows the linear stability analysis of Schlichting (Z. Angew. Math. Mech., vol. 15 (6), 1935, pp. 313–338) for both smooth and rough surfaces, indicating that a good predictor of critical stratification is the gradient Richardson number evaluated at the wall. Wall-normal and shear stresses follow atmospheric trends in the local gradient Richardson number scaling of Sorbjan (Q. J. R. Meteorol. Soc., vol. 136, 2010, pp. 1243–1254), suggesting that much can be learned about stratified atmospheric flows from the study of laboratory scale boundary layers at relatively low Reynolds numbers.

Closure Schemes for Stably Stratified Atmospheric Flows without Turbulence Cutoff

Two recently proposed turbulence closure schemes are compared against the conventional Mellor–Yamada (MY) model for stably stratified atmospheric flows. The Energy- and Flux-Budget (EFB) approach solves the budgets of turbulent momentum and heat fluxes and turbulent kinetic and potential energies. The Cospectral Budget (CSB) approach is formulated in wavenumber space and integrated across all turbulent scales to obtain flow variables in physical space. Unlike the MY model, which is subject to a “critical gradient Richardson number,” both EFB and CSB models allow turbulence to exist at any gradient Richardson number and predict a saturation of flux Richardson number () at sufficiently large . The CSB approach further predicts the value of and reveals a unique expression linking the Rotta and von Kármán constants. Hence, all constants in the CSB model are nontunable and stability independent. All models agree that the dimensionless sensible heat flux decays with increasing . However, the decay rate and subsequent cutoff in the MY model appear abrupt. The MY model further exhibits an abrupt cutoff in the turbulent stress normalized by vertical velocity variance, while the CSB and EFB models display increasing trends. The EFB model produces a rapid increase in the ratio of turbulent potential energy and vertical velocity variance as is approached, suggesting a strong self-preservation mechanism. Vertical anisotropy in the turbulent kinetic energy is parameterized in different ways in MY and EFB, but this consideration is not required in CSB. Differences between EFB and CSB model predictions originate from how the vertical anisotropy is specified in the EFB model.

Turbulent mixing due to the Holmboe wave instability at high Reynolds number

We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution $\overline{U}(z,0)\,\boldsymbol{e}_{x}=U_{0}\,\boldsymbol{e}_{x}\tanh (z/d)$ and initial density distribution $\overline{\unicode[STIX]{x1D70C}}(z,0)=\unicode[STIX]{x1D70C}_{0}[1-\tanh (z/\unicode[STIX]{x1D6FF})]$ away from a hydrostatic reference state $\unicode[STIX]{x1D70C}_{r}\gg \unicode[STIX]{x1D70C}_{0}$. When the ratio $R=d/\unicode[STIX]{x1D6FF}$ of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number $Ri_{g}(0)=Ri_{b}=g\unicode[STIX]{x1D70C}_{0}d/\unicode[STIX]{x1D70C}_{r}U_{0}^{2}$, where $Ri_{g}(z)=-([g/\unicode[STIX]{x1D70C}_{r}]\,\text{d}\overline{\unicode[STIX]{x1D70C}}/\text{d}z)/(\text{d}\overline{U}/\text{d}z)^{2}$ and $Ri_{b}$ is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers ($Re$), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when $R\gg 1$, and so the density interface is sharp, with $Ri_{g}(z)$ now being maximum at the density interface $Ri_{g}(0)=RRi_{b}$. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower-$Re$ experimental and numerical studies, when $Re$ is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As $Re$ increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a $k^{-5/3}$ spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high $Re$, vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high $Re$. Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.

Experimental Study of Mixing and Entrainment in a Horizontal Turbulent Stratified Jet

The mixing efficiency, flux Richardson number Rif, is investigated in a horizontally injected turbulent stratified jet with a co-existence of stable and unstable stratifications. The high resolution experimental data from a developed laser-based technique show the statistical relationship between Rif and the gradient Richardson number Rig. In addition, the data are used to study the development of entrainment by two approaches, and compared with theoretical predictions.