Flux Richardson Number Research Articles

A case study of the life cycle of a stratus‐lowering coastal‐fog event in Newfoundland, Canada

AbstractWe present a case study of a coastal‐fog stratus‐cloud‐lowering event on September 13–14, 2018, during the C‐FOG field campaign conducted along the east coast of Newfoundland, Canada. The goal of this work is to understand the mechanisms governing the life cycle of a 4‐hr‐long coastal‐fog event that resulted from the complex interplay of dynamic, thermodynamic, and microphysical processes. In addition to standard meteorological measurements, turbulence, irradiance, droplet‐size spectra, tethered‐balloon wind and thermodynamic profiles, visibility, precipitation, and spatial heterogeneity of microphysics measurements are presented to discriminate and interpret the fog formation, development, and dissipation. After sunset, strong radiative cloud‐top cooling induced top‐down convection length scales that can be characterised with the Thorpe scale. Top‐down mixing and turbulence kinetic energy generated due to buoyant/shear mixing are characterised using the flux and bulk Richardson number near the surface. Use of these parameters is unique in the analysis of fog events and helped describe mixing processes. Downward mixing led to fog droplet formation that precipitated from the cloud base, which in turn cooled the sub‐cloud layers via droplet evaporation and moistened the air beneath the cloud. Once fog formed, it was affected by dry‐air entrainment from its top. As a result, the fog thinned, creating patchy fog that was characterised by remarkable oscillations in visibility near the surface. Dissipation of the fog was driven by strong turbulence above the fog layer and horizontal thermal advection demonstrated using the temperature tendency equation. This work provides novel measurements and analysis techniques that have previously not been used to understand the mechanisms governing stratus‐lowering events. These observations and analyses help highlight processes and explain mechanisms related to the fog life cycle that are inherently challenging to predict in mesoscale models.

Modelling the Apparent von Kármán Parameter in Thermally and Sedimentologically Stratified Air Flows

AbstractThrough a field experiment on a sandy beach at Pingtan, China, we propose the first model to characterize the variability of apparent von Kármán parameter in the thermally and sedimentologically stratified air flows. The model shows that is negatively correlated to flux Richardson number as well as sediment transport rate. The atmospheric stability can be either from thermal stratification or sedimentological stratification, neglecting either of them can lead to substantial errors in predicting shear velocity and sediment transport rate.

The butterfly effect and the transition to turbulence in a stratified shear layer

In a stably stratified shear layer, multiple competing instabilities produce sensitivity to small changes in initial conditions, popularly called the butterfly effect (as a flapping wing may alter the weather). Three ensembles of 15 simulated mixing events, identical but for small perturbations to the initial state, are used to explore differences in the route to turbulence, the maximum turbulence level and the total amount and efficiency of mixing accomplished by each event. Comparisons show that a small change in the initial state alters the strength and timing of the primary Kelvin–Helmholtz instability, the subharmonic pairing instability and the various three-dimensional secondary instabilities that lead to turbulence. The effect is greatest in, but not limited to, the parameter regime where pairing and the three-dimensional secondary instabilities are in strong competition. Pairing may be accelerated or prevented; maximum turbulence kinetic energy may vary by up to a factor of 4.6, flux Richardson number by 12 %–15 % and net mixing by a factor of 2.

Energy- and flux-budget theory for surface layers in atmospheric convective turbulence

The energy- and flux-budget (EFB) theory developed previously for atmospheric stably stratified turbulence is extended to the surface layer in atmospheric convective turbulence. This theory is based on budget equations for turbulent energies and fluxes in the Boussinesq approximation. In the lower part of the surface layer in the atmospheric convective boundary layer, the rate of turbulence production of the turbulent kinetic energy (TKE) caused by the surface shear is much larger than that caused by the buoyancy, which results in three-dimensional turbulence of very complex nature. In the upper part of the surface layer, the rate of turbulence production of TKE due to the shear is much smaller than that caused by the buoyancy, which causes unusual strongly anisotropic buoyancy-driven turbulence. Considering the applications of the obtained results to the atmospheric convective boundary-layer turbulence, the theoretical relationships potentially useful in modeling applications have been derived. The developed EFB theory allows us to obtain a smooth transition between a stably stratified turbulence to a convective turbulence. The EFB theory for the surface layer in a convective turbulence provides an analytical expression for the entire surface layer including the transition range between the lower and upper parts of the surface layer, and it allows us to determine the vertical profiles for all turbulent characteristics, including TKE, the intensity of turbulent potential temperature fluctuations, the vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature), the integral turbulence scale, the turbulence anisotropy, the turbulent Prandtl number, and the flux Richardson number.

Influence of Stability and Surface Roughness on Turbulence during the Stable Atmospheric Variability and Transport (SAVANT) Field Campaign

Abstract Harvesting of crops in a weakly sloping Midwestern field during the Stable Atmospheric Variability and Transport (SAVANT) observation campaign allowed for a systematic investigation of the influence of surface roughness and static stability magnitude on the applicability of the Monin–Obukhov similarity (MOST) and hockey-stick transition (HOST) theories during stable boundary layer periods. We analyze momentum flux and turbulent velocity scale VTKE in three regimes, defined using the gradient Richardson number Ri and flux Richardson number Rif as regime 1 (0 < Ri ≤ 0.1 and 0 < Rif ≤ 0.1), regime 2 (0.1 < Ri ≤ 0.23 and 0.1 < Rif ≤ 0.23), and regime 3 (both Ri and Rif > 0.23). After harvest, in regime 1, stability varied from near-neutral to weakly stable and both MOST and HOST were applicable to estimate the momentum fluxes and VTKE as a function of mean wind speed. In regime 2, the momentum flux deviated from the MOST linear relationship as stability increased. In regimes 1 and 2, a HOST-defined threshold wind speed Vs was identified beyond which VTKE increased linearly with wind speed at a rate of 0.26 for all observation heights. Below this threshold wind speed, VTKE behaved independent of mean wind and observation heights. Alternatively, for preharvest periods, MOST was applicable in regimes 1 and 2 for all heights and HOST was applicable with reduced Vs for heights above the crop layer. Regime 3 during pre- and postharvest consisted of strongly stable periods and very weak to weak winds, where MOST was found to be invalid and VTKE remained low and independent of wind speed. The results suggest that roughness due to crops enhances the turbulence generation at lower wind speeds.

Scaling the Mixing Efficiency of Sediment‐Stratified Turbulence

AbstractThe flux Richardson number Rf, also called the mixing efficiency of stratified turbulence, is important in determining geophysical flow phenomena such as ocean circulation and air‐sea transports. Measuring Rf in the field is usually difficult, thus parameterization of Rf based on readily observed properties is essential. Here, estimates of Rf in a strongly turbulent, sediment‐stratified estuarine flow are obtained from measurements of covariance‐derived turbulent buoyancy fluxes (B) and spectrally fitted values of the dissipation rate of turbulent kinetic energy (ε). We test scalings for Rf in terms of the buoyancy Reynolds number (Reb), the gradient Richardson number (Ri), and turbulent Froude number (Frt). Neither the Reb‐based nor the Ri‐based scheme is able to describe the observed variations in Rf, but the Frt‐based parameterization works well. These findings support further use of the Frt ‐ based parameterization in turbulent oceanic and estuarine environments.

Turbulence characteristics within the atmospheric surface layer of the coastal region of Qatar

The atmospheric turbulence characteristics in the coastal region of Qatar are analyzed using the measurements conducted on the shoreline (26.08 N, 51.36 E). The micrometeorological data were collected, from August 2015 to September 2016, using sonic anemometers (20 Hz) at three heights and a weather station atop a 9-m tower. The turbulence characteristics are studied within the framework of Monin–Obukhov similarity theory (MOST), in the presence of the coastal inhomogeneities generated by the sea and land surfaces coming together. The results show the wind from the north-west prevails during the entire test period, with the wind speed higher than that from other directions. The non-dimensional standard deviations of velocity components are found to be consistent the results reported around the world and match suggested MOST scaling, with a relatively greater value for the dissipation rate of turbulent kinetic energy. The flux Richardson number shows a larger scatter under the super-stable and super-unstable regimes. Moreover, the non-dimensional standard deviation of temperature does not align with the suggested model under near-neutral and very stable regimes, and the gradient Richardson number shows some negative values under stable regimes. Two different atmospheric daily stability patterns, ‘orderly’ and ‘disheveled,’ are identified based on the wind conditions. The orderly stability pattern shows a daily descending and ascending trend during the sunrise and sunset periods, respectively, while the disheveled days follow a random pattern with no clear order. The two patterns are then related to the wind continuity and direction relative to the shoreline.

Large‐eddy simulation of very stable boundary layers. Part II: Length scales and anisotropy in stratified atmospheric turbulence

AbstractIn very stable boundary layers (SBLs), turbulence is expected to be weak, anisotropic, and non‐stationary. Moreover, viscous effects likely become relevant under very weak‐wind conditions. This study investigates the stratification regimes based on the relative strength of the forces that control the flow (i.e., buoyancy‐ or viscosity‐dominated flow) and the degree of turbulence anisotropy in large‐eddy simulations of the very SBL with weak winds. The simulations explored here correspond to an Ekman‐layer‐type boundary layer with geostrophic winds equal to and 2 ms, and surface cooling rates of 1 and 3 Khr, at high latitude. According to the buoyancy Reynolds number and the horizontal Froude number , these SBLs are in strongly stable conditions (i.e., and ). Moreover, the vertical profiles of indicate that the turbulent flows are in an energetic state near the surface but gradually transition to a viscosity‐affected stratified state in the upper part of the SBL as viscous effects become important. For the most stable simulation, the results show that even the small scales are somewhat affected by buoyancy. The invariant analysis of the Reynolds stress anisotropy tensor shows that the flow is in an anisotropic state and is governed by the streamwise component of the turbulent flux (one‐component turbulence). Additionally, based on the two‐point spatial correlation function, the shape of the large coherent turbulent structures is nearly isotropic in the vertical direction and anisotropic in the horizontal direction, and their size increases with height. The evaluation of the turbulent kinetic energy budget shows that the most stable simulation is in a non‐stationary state, which impacts the turbulence mixing efficiency measured by the flux Richardson number.

Parameterization of mixing in stratified open channel flow

The dynamics and parameterization of mixing in temporally evolving turbulent open-channel flow is investigated through direct numerical simulations as the flow transitions from an initially neutral state to stable stratification. We observe three distinctly different mixing regimes separated by transitional values of turbulent Froude number $Fr$ : a weakly stratified regime for $Fr >1$ ; an intermediate regime for $0.3< Fr<1$ ; and a saturated regime for $Fr<0.3$ . The mixing coefficient $\varGamma =B/\epsilon _K$ , (where $B$ is the buoyancy flux and $\epsilon _K$ is the dissipation rate of kinetic energy), is well predicted by the parameterization schemes of Maffioli et al. (J. Fluid Mech., vol. 794, 2016) and Garanaik & Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323–333) across all three regimes through instantaneous measurements of $Fr$ and the ratio $L_E/L_O$ , where $L_E$ and $L_O$ are the Ellison and Ozmidov length scales, respectively. The flux Richardson number $R_f = B/(B+\epsilon _K)$ shows linear dependence on the gradient Richardson number $Ri_g$ up to a transitional value of $Ri_g =0.25$ , past which it saturates again to a constant value independent of $Fr$ or $Ri_g$ . By examining the flow as a balance of inertial, shear and buoyancy forces, we derive physically based scaling relationships to demonstrate that $Ri_g \sim Fr^{-2}$ and $Ri_g \sim Fr^{-1}$ in the weakly and moderately stratified regimes and that $Ri_g$ becomes independent of $Fr$ in the saturated regime. Our results suggest that the $L_E/L_O \sim Fr^{-1}$ scaling of Garanaik & Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323–333) in the intermediate regime manifests due to the influence of mean shear. The differences in the relationships between $Fr$ and $L_E/L_O$ for non-sheared flows within this regime are discussed.

Towards robust and accurate Reynolds-averaged closures for natural convection via multi-objective CFD-driven machine learning

Robust and accurate Reynolds-averaged stresses and scalar fluxes closure models for natural convection developed by machine learning are presented in this work. In comparison to passive scalar flows, the complexity of turbulence modelling for natural convection problems is increased as the velocity and scalar fields are strongly coupled by the buoyancy force. Therefore, the appropriate Reynolds-averaged closure models for natural convection ought to capture this interaction within the second-moment terms. Previous data-driven turbulence modelling approaches have treated the unclosed terms of the velocity and thermal fields separately, which has lead to inaccurate predictions when handling natural convection problems. In this study, we therefore develop fully coupled buoyancy-extended models by using a novel multi-objective and multi-expression machine-learning framework that is based on CFD-driven training (Zhao et al., J. Comput. Phys., 411, 109413, (2020)). The model candidates obtained from a Gene-Expression Programming approach, and thus available in symbolic form, are evaluated by running RANS solvers for different Rayleigh number cases during the model training process. This novel framework is applied to vertical natural convection, with the emphasis of this paper on the importance of coupling the explicit closure model formulations, the choice of cost functions, and the appropriate input flow features (i.e. a generalised flux Richardson number) for developing accurate models. It is shown that the resulting machine-learnt models improve the predictions of quantities of interests, e.g. mean velocity and temperature profiles, across a wide range of Rayleigh numbers.

Impact of Stratification Mechanisms on Turbulent Characteristics of Stable Open-Channel Flows

Abstract Flow over a surface can be stratified by imposing a fixed mean vertical temperature (density) gradient profile throughout or via cooling at the surface. These distinct mechanisms can act simultaneously to establish a stable stratification in a flow. Here, we perform a series of direct numerical simulations of open-channel flows to study adaptation of a neutrally stratified turbulent flow under the combined or independent action of the aforementioned mechanisms. We force the fully developed flow with a constant mass flow rate. This flow forcing technique enables us to keep the bulk Reynolds number constant throughout our investigation and avoid complications arising from the acceleration of the bulk flow if a constant pressure gradient approach were to be adopted to force the flow instead. When both stratification mechanisms are active, the dimensionless stratification perturbation number emerges as an external flow control parameter, in addition to the Reynolds, Froude, and Prandtl numbers. We demonstrate that significant deviations from the Monin–Obukhov similarity formulation are possible when both types of stratification mechanisms are active within an otherwise weakly stable flow, even when the flux Richardson number is well below 0.2. An extended version of the similarity theory due to Zilitinkevich and Calanca shows promise in predicting the dimensionless shear for cases where both types of stratification mechanisms are active, but the extended theory is less accurate for gradients of scalar. The degree of deviation from neutral dimensionless shear as a function of the vertical coordinate emerges as a qualitative measure of the strength of stable stratification for all the cases investigated in this study.

Enhanced Turbulence in the Upper Mixed Layer Under Light Winds and Heating: Implications for Gas Fluxes

AbstractMeasurements of turbulence, as rate of dissipation of turbulent kinetic energy (ε), adjacent to the air‐water interface are rare but essential for understanding of gas transfer velocities (k) used to compute fluxes of greenhouse gases. Variability in ε is expected over diel cycles of stratification and mixing. Monin‐Obukhov similarity theory (MOST) predicts an enhancement in ε during heating (buoyancy flux, β+) relative to that for shear (u*w3/κz where u*w is water friction velocity, κ is von Karman constant, z is depth). To verify and expand predictions, we quantified ε in the upper 0.25 m and below from profiles of temperature‐gradient microstructure in combination with time series meteorology and temperature in a tropical reservoir for winds <4 m s−1. Maximum likelihood estimates of near‐surface ε during heating were independent of wind speed and high, ∼5 × 10−6 m2 s−3, up to three orders of magnitude higher than predictions from u*w3/κz, increased with heating, and were ∼10 times higher than during cooling. k, estimated using near‐surface ε, was ∼10 cm hr−1, validated with k obtained from chamber measurements, and 2–5 times higher than computed from wind‐based models. The flux Richardson number (Rf) varied from ∼0.4 to ∼0.001 with a median value of 0.04 in the upper 0.25 m, less than the critical value of 0.2. We extend MOST by incorporating the variability in Rf when scaling the influence of β+ relative to u*w3/κz in estimates of ε, and by extension, k, obtained from time series meteorological and temperature data.

Flux–Profile Relationships in the Stable Boundary Layer—A Critical Discussion

Flux–profile relationships are crucial for parametrizing surface fluxes of momentum and heat, that are of central relevance for applications such as climate modelling and weather forecast. Nevertheless, their functional forms are still under discussion, and a generally accepted formulation does not exist yet. We reviewed the four main formulations proposed in the literature so far and assessed how they affect the theoretical behaviour of the kinematic heat flux (H0) and the temperature scale (T*) in the stable boundary layer, as well as their consequences on the existence of critical values for both the gradient and the flux Richardson numbers. None of them turned out to be fully consistent with the literature published so far, with two of them leading to very unreliable expressions for both H0 and T*. All considered, a convincing description of flux–profile relationships still needs to be found and seems to represents a considerable challenge.

Energy and flux budget closure theory for passive scalar in stably stratified turbulence

The energy and flux budget (EFB) closure theory for a passive scalar (non-buoyant and non-inertial particles or gaseous admixtures) is developed for stably stratified turbulence. The physical background of the EFB turbulence closures is based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy as well as the turbulent flux of particles. The EFB turbulence closure is designed for stratified geophysical flows from neutral to very stable stratification, and it implies that turbulence is maintained by the velocity shear at any stratification. In a steady-state, expressions for the turbulent flux of the passive scalar and the anisotropic non-symmetric turbulent diffusion tensor are derived, and universal flux Richardson number dependencies of the components of this tensor are obtained. The diagonal component in the vertical direction of the turbulent diffusion tensor is suppressed by strong stratification, while the diagonal components in the horizontal directions are not suppressed, but they are dominant in comparison with the other components of the turbulent diffusion tensor. This implies that any initially created strongly inhomogeneous particle cloud is evolved into a thin pancake in a horizontal plane with very slow increase in its thickness in the vertical direction. The turbulent Schmidt number (the ratio of the eddy viscosity and the vertical turbulent diffusivity of the passive scalar) linearly increases with the gradient Richardson number. The physics of such a behavior is related to the buoyancy force that causes a correlation between fluctuations of the potential temperature and the particle number density. This correlation that is proportional to the product of the vertical turbulent particle flux and the vertical gradient of the mean potential temperature reduces the vertical turbulent particle flux. Considering the applications of these results to the atmospheric boundary-layer turbulence, the theoretical relationships are derived, which allows us to determine the turbulent diffusion tensor as a function of the vertical coordinate measured in the units of the local Obukhov length scale. The obtained relations are potentially useful in modeling applications of particle dispersion in the atmospheric boundary-layer turbulence and free atmosphere turbulence.

How is the Two-Regime Stable Boundary Layer Reproduced by the Different Turbulence Parametrizations in the Weather Research and Forecasting Model?

Five planetary-boundary-layer parametrizations of the Weather Research and Forecasting model are compared with respect to their ability to simulate the very stable and the weakly stable regimes of the stable boundary layer. This is performed for single column models where the large-scale mechanical forcing is represented by geostrophic wind speeds ranging from 0.5 to 12 m $$\hbox {s}^{-1}$$ . The performance of the models is assessed by contrasting the relationships they produce between the turbulence velocity scale and the mean wind speed, between potential temperature gradient and the mean wind speed, and between the flux and gradient Richardson numbers. The level-2.5 Mellor–Yamada–Nakanishi–Niino parametrization simulates the very stable regime the best, mainly because its heat eddy diffusivity decreases with respect to the momentum eddy diffusivity as the stability increases, while the same is not true for the other parametrizations considered.

Statistical characterisation of turbulence for an unsteady gravity current

Abstract

Buoyancy Effects in the Turbulence Kinetic Energy Budget and Reynolds Stress Budget for a Katabatic Jet over a Steep Alpine Slope

Katabatic winds are very frequent but poorly understood or simulated over steep slopes. This study focuses on a katabatic jet above a steep alpine slope. We assess the buoyancy terms in both the turbulence kinetic energy (TKE) and the Reynolds shear-stress budget equations. We specifically focus on the contribution of the slope-normal and along-slope turbulent sensible heat fluxes to these terms. Four levels of measurements below and above the maximum wind-speed height enable analysis of the buoyancy effect along the vertical profile as follow: (i) buoyancy tends to destroy TKE, as expected in stable conditions, and the turbulent momentum flux in the inner-layer region of the jet below the maximum wind-speed height $$z_j$$ ; (ii) results also suggest buoyancy contributes to the production of TKE in the outer-layer shear region of the jet (well above $$z_j$$ ) while consumption of the turbulent momentum flux is observed in the same region; (iii) In the region around the maximum wind speed where mechanical shear production is marginal, buoyancy tends to destroy TKE and our results suggest it tends to increase the momentum flux. The present study also provides an analytical condition for the limit between production and consumption of the turbulent momentum flux due to buoyancy as a function of the slope angle, similar to the condition already proposed for TKE. We reintroduce the stress Richardson number, which is the equivalent of the flux Richardson number for the Reynolds shear-stress budget. We point out that the flux Richardson number and the stress Richardson number are complementary stability parameters for characterizing the katabatic flow apart from the region around the maximum wind-speed height.

Scalewise Return to Isotropy in Stratified Boundary Layer Flows

AbstractAnisotropic turbulence is ubiquitous in atmospheric and oceanic boundary layers due to differences in energy injection mechanisms. Unlike mechanical production that injects energy in the streamwise velocity component, buoyancy affects only the vertical velocity component. This anisotropy in energy sources, quantified by the flux Richardson number Rif, is compensated by a “return to isotropy” (RTI) tendency of turbulent flows. Describing RTI in Reynolds‐averaged models and across scales continues to be a challenge in stratified turbulent flows. Using phenomenological models for spectral energy transfers, the necessary conditions for which the widely‐used Rotta model captures RTI across various Rif and eddy sizes are discussed for the first time. This work unravels adjustments to the Rotta constant, with Rif and scale, necessary to obtain consistency between RTI models and the measured properties of the atmospheric surface layer for planar‐homogeneous and stationary flows in the absence of subsidence. A range of Rif and eddy sizes where the usage of a conventional Rotta model is prohibited is also found. Those adjustments lay the groundwork for new closure schemes.

Flow structure modification and drag reduction induced by sediment stratification in coastal tidal bottom boundary layers

The Karman-Prandtl logarithm formula of velocity distribution has been widely used to depict flow structures and estimate parameters including bed roughness length and friction velocity in coastal tidal bottom boundary layers. However, the basic assumptions are of homogeneous flow, but the effects of stratification should be taken into account. Here, field observations of hydrodynamics and sediment dynamics were carried out in the well-mixed area outside of the Changjiang estuary, China, during May 31-June 2, 2011. Velocity profiles within the ~2 m bottom layer suggest that the flow structure is prominently influenced when the gradient of suspended sediment concentration reaches O (0.1 kg m−4). The direct application of the log-fit method resulted in a pronounced deviation from the actual boundary parameters. The extended log-fit method was used to represent the tidally averaged vertical flow structure, indicating that the modification of the flow structure diminishes down to the bed, and the calculated Z0 based on the lowest layer close to the bed (0.22–0.38 m above the bed in the observation) approaches the real value. The observed data were also used to examine the modified logarithmic velocity distribution model, in which the flux Richardson number is modified with an empirical constant A to parameterize the stratification effects. It was found that the constant A is approximately 1, which is significantly different from previous studies. Results of a vertical one-dimensional model supported the above findings. The sediment stratification effects should be taken into account, and the logarithmic velocity distribution model tends to be applicable in the water layers very close to the bed.

Similarity functions and a new [formula omitted] closure for predicting stratified atmospheric surface layer flows in complex terrain

Most of the k−ε closures for modeling stratified surface layer are derived from the classical similarity functions and might fail in complex terrain due to limitations of the classical similarity functions. Despite the classical similarity functions to limit the flux Richardson number Rf, we present new similarity functions estimated from field measurement in full range of Rf and propose a k−ε model using the new similarity functions to improve predictions of stratified surface layer flows in complex terrain. Measurements show that the classical similarity functions are partially valid in complex terrain and the wind shear under strongly stable conditions is constrained at large Rf. According to numerical studies in two areas of complex terrain, models using the classical similarity functions and the new similarity functions both present good predictions of convective airflows in complex terrain. The new similarity functions are shown to significantly improve the k−ε model in predicting stably stratified airflows in complex terrain by constraining the wind shear at large Rf, while the classical similarity functions without limiting the wind shear lead to significantly misestimating the wind speedup factor under stable conditions. Using the proposed model to predict flows in wind farm could benefit wind resource estimation and wind power forecasting.