Obukhov Length Scale Research Articles

The coupling of winds, ocean turbulence, and High Salinity Shelf Water in the Terra Nova Bay Polynya

The Terra Nova Bay (TNB) Polynya in the Western Ross Sea of Antarctica is a major producer of High Salinity Shelf Water (HSSW), a precursor to Antarctica Bottom Water (AABW). Processes occurring in and around the polynya can therefore effect change in the lower limb of overturning circulation in this region. Here, we use data from a densely-instrumented upper-ocean mooring, deployed for 1 year in a region of active HSSW formation within TNB, to examine the coupling of surface brine rejection and vertical mixing to katabatic wind forcing. We find a high correlation between salinity and winds during the wintertime HSSW production season at the mooring site, with a lag-response of 20 h in near-surface (∼47 m) salinity to winds measured at the nearby Automatic Weather Station (AWS) Manuela. Salinity and temperature measurements show a fully destratified water column by June, with a lag-response of near-seabed (∼360 m) salinity to near-surface salinity of just 5 h. Measurements of turbulent kinetic energy (TKE) dissipation rate (ɛ) from moored pulse-coherent acoustic Doppler current profilers (ADCPs) show general agreement with classic boundary layer scaling (BLS), and calculations of a vertical mixing timescale using the Obukhov length scale average to ∼2.5 h during austral winter, consistent with the 5-h lag time observed in the salinity data. Comparisons to data from concurrent mooring deployments along the southern boundary of TNB, as well as to previously published assessments of model simulations and data from Climatic Long-term Interaction for the Mass-balance in Antarctica (CLIMA) moorings, allow us to explore spatial variability in the coupling of winds and salinity across TNB and to speculate on possible HSSW circulation pathways.

Assessment of Wind over Complex Terrain Considering the Effects of Topography, Atmospheric Stability and Turbine Wakes

This study proposes a microscale flow model to estimate mean wind speed, fluctuating wind speed and wind direction over complex terrain considering the effects of topography, atmospheric stability, and turbine wakes. Firstly, the effect of topography is considered using Computational Fluid Dynamics (CFD). Next, a mesoscale model is presented to account for the effect of atmospheric stability. The effect of turbine wakes on the mean and fluctuating wind speeds are then represented by an advanced wake model. The model is validated using the measurement data of a wind farm located in the North of Japan. The measured wind data by Lidar at a reference height are horizontally extrapolated to a nearby met mast hub height and validated by a cup anemometer. Moreover, a novel averaging method is proposed to calculate a directional equivalent Monin–Obukhov length scale to account for the effect of atmospheric stability. Finally, the measured wind data at the reference height are vertically extrapolated and validated at the lidar location. The predicted mean and fluctuating wind speeds show good agreement with the measurements.

Stratification and temporal evolution of mixing regimes in diurnally heated river flows

Direct numerical simulations of stratified open channel flows subject to a varying surface heat flux are performed. The influence of the diurnal heating time on the spatial and temporal variation of mixing in the flow and the characteristics of the mean flow state are examined. The control parameters are the bulk stability parameter lambda_{B}, defined through the ratio of the channel height delta and a bulk Obukhov length scale mathscr{L}_{B}, and the diurnal time scale hat{t}, defined as the ratio of the heating time to an eddy turnover time. The Prandtl number Pr and Reynolds number Re_{tau} have values of 1 and 400. Simulations are performed over hat{t} = 1 to 24 and lambda_{B} = 0.6 to 26. Two key flow features are used to classify the flow regimes observed, namely the laminar layer depth (LLD) and stratified layer depth (SLD) where the LLD is defined as the depth from the free surface when the buoyancy Reynolds number Re_{B} approx 7 and the SLD is the depth from the free surface when the turbulent Froude number Fr approx 1. This study attempts to characterise how these length scales vary across the diel cycle. The LLD is a viscous length scale and a regime map of a viscous parameter, the bulk Obhukov Reynolds number Re_mathscr{L}, and hat{t} is presented to classify the LLD behaviour. A regime map of lambda _{B} and hat{t} is presented to classify the behaviour of the SLD. Three classifications for each layer depth behaviour within a diel cycle form the basis of the regime maps for this paper: a neutral flow where the LLD or SLD does not exist (denoted by NL and NS), a stratified flow where the LLD or SLD are diurnally varying (denoted as DL and DS) and a persistent layer of the LLD or SLD (denoted as PL and PS). The transition between the NL to DL is hat{t} propto Re_{mathscr{L}}^{4.5}, DL to PL is hat{t} propto Re_{mathscr{L}}^{- 0.5}, NS to DS is hat{t} propto lambda _{B}^{0} and DS to PS is hat{t} propto lambda _{B}^{1}. The regime maps may be used as a predictive tool to determine when suppressed mixing regimes occur in rivers. At each flow depth, the flow sweeps though a range of mixing states across the diel cycle. The local mixing efficiency are briefly assessed and found to scale well with the instantaneous Fr number according to the regimes proposed by Garanaik and Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323-333).

Turbulent Inflow Generation for Large-Eddy Simulation of Winds around Complex Terrain

Accurate turbulent inflow conditions are needed to broaden the application of the large-eddy simulation technique to predict winds around arbitrarily complex terrain. We investigate the concept of buoyancy perturbations with colored noise to trigger turbulence in upstream flows approaching complex terrain regions. Random perturbations are imposed on the source term in the pseudo-temperature transport equation. These perturbations are effective within three-dimensional boxes and scaled using a bulk Richardson number defined for each box. We apply the turbulent inflow generation technique to predict winds around the Askervein and Bolund Hills under neutrally stratified conditions. We find that a common value for the bulk Richardson number works well for a variety of flow problems. Additionally, we show that the height of the perturbation box plays an important role in the accuracy of the predictions around complex terrain. We consistently obtained good results for both simulation cases when the perturbation box height was made a fraction of the Obukhov length scale.

Scaling of Moored Surface Ocean Turbulence Measurements in the Southeast Pacific Ocean

AbstractEstimates of turbulence kinetic energy (TKE) dissipation rate (ε) are key in understanding how heat, gas, and other climate‐relevant properties are transferred across the air‐sea interface and mixed within the ocean. A relatively new method involving moored pulse‐coherent acoustic Doppler current profilers (ADCPs) allows for estimates of ε with concurrent surface flux and wave measurements across an extensive length of time and range of conditions. Here, we present 9 months of moored estimates of ε at a fixed depth of 8.4 m at the Stratus mooring site (20°S, 85°W). We find that turbulence regimes are quantified similarly using the Obukhov length scale and the newer Langmuir stability length scale , suggesting that ocean‐side friction velocity implicitly captures the influence of Langmuir turbulence at this site. This is illustrated by a strong correlation between surface Stokes drift and that is likely facilitated by the steady Southeast trade winds regime. In certain regimes, , where is the von Kármán constant and is instrument depth, and surface buoyancy flux capture our estimates of well, collapsing data points near unity. We find that a newer Langmuir turbulence scaling, based on and , scales ε well at times but is overall less consistent than . Monin‐Obukhov similarity theory (MOST) relationships from prior studies in a variety of aquatic and atmospheric settings largely agree with our data in conditions where convection and wind‐driven current shear are both significant sources of TKE, but diverge in other regimes.

Machine Learning for Improving Surface-Layer-Flux Estimates

Flows in the atmospheric boundary layer are turbulent, characterized by a large Reynolds number, the existence of a roughness sublayer and the absence of a well-defined viscous layer. Exchanges with the surface are therefore dominated by turbulent fluxes. In numerical models for atmospheric flows, turbulent fluxes must be specified at the surface; however, surface fluxes are not known a priori and therefore must be parametrized. Atmospheric flow models, including global circulation, limited area models, and large-eddy simulation, employ Monin–Obukhov similarity theory (MOST) to parametrize surface fluxes. The MOST approach is a semi-empirical formulation that accounts for atmospheric stability effects through universal stability functions. The stability functions are determined based on limited observations using simple regression as a function of the non-dimensional stability parameter representing a ratio of distance from the surface and the Obukhov length scale (Obukhov in Trudy Inst Theor Geofiz AN SSSR 1:95–115, 1946), z/L. However, simple regression cannot capture the relationship between governing parameters and surface-layer structure under the wide range of conditions to which MOST is commonly applied. We therefore develop, train, and test two machine-learning models, an artificial neural network (ANN) and random forest (RF), to estimate surface fluxes of momentum, sensible heat, and moisture based on surface and near-surface observations. To train and test these machine-learning algorithms, we use several years of observations from the Cabauw mast in the Netherlands and from the National Oceanic and Atmospheric Administration’s Field Research Division tower in Idaho. The RF and ANN models outperform MOST. Even when we train the RF and ANN on one set of data and apply them to the second set, they provide more accurate estimates of all of the fluxes compared to MOST. Estimates of sensible heat and moisture fluxes are significantly improved, and model interpretability techniques highlight the logical physical relationships we expect in surface-layer processes.

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.

Study of coherent structures and heat flux transportation under different stratification stability conditions in the atmospheric surface layer

A field experiment was carried out on a dry flat bed of Qingtu Lake in Minqin, China to measure the velocity and temperature in the atmospheric surface layer at high Reynolds number. The facility, the Qingtu Lake Observation Array, permits synchronous multi-point measurements of three-dimensional wind velocity and temperature at different wall-normal and spanwise positions. Data were selected to analyze the characteristics of the large-scale coherent structures under different stratification stability conditions. The results confirm the previous findings that the positive buoyancy can be considered as a wall-normal force to have a “lifting” effect increasing the size of large-scale structures and further illustrate that negative buoyancy in the stable condition has an opposite effect leading to small structures with less wall-normal coherence. The stability dependence of structure inclination angles for all three velocity components is built by a parametric model indicating a log-linear trend under near-neutral and unstable conditions. In addition, the unique measurement array at this facility also permits us to study the three-dimensional form of these features and the associated temperature fluctuations. By computing the space average for momentum and heat flux, we found that the threshold stability parameter z/L=−0.1, where z = 2.5 m and L is the Obukhov length scale, can act as a role to determine the dominant flow by shear-driven and buoyancy-driven, respectively. Last, the streamwise length scales for temperature component are investigated where a smaller length scale corresponds to a stronger buoyancy condition, but with larger values at higher heights.

Lettau\u2019s Contribution to the Obukhov Length Scale: A Scientific Historical Study

It has been repeatedly assumed that Heinz Lettau found the Obukhov length in 1949 independently of Obukhov in 1946. However, it was not the characteristic length scale, the Obukhov length L, but the ratio of height and the Obukhov length (z/L), the Obukhov stability parameter, that he analyzed. Whether Lettau described the parameter z/L independently of Obukhov is investigated herein. Regardless of speculation about this, the significant contributions made by Lettau in the application of z/L merit this term being called the Obukhov–Lettau stability parameter in the future.

Roughness-induced secondary flows in stably stratified turbulent boundary layers

Spanwise heterogeneous roughness, more specifically, spanwise-adjacent strips of relatively high and low roughness, is known to cause large-scale secondary flows in the vertical domain in neutral turbulent boundary layers. In this work, we study the response of secondary vortices to thermal stratification with focus on the stably stratified case. We systematically vary the Monin–Obukhov (MO) length scale from L/h = 0.3, which corresponds to relatively strong stratification, to ∞, which corresponds to the neutral condition. Here, L is the MO length and h is an outer length scale. The results show that stable stratification suppresses the vertical motions and reduces the vertical size and strength of the first off-wall secondary vortex. Moreover, the reduced vertical extent of the first off-wall vortex and the shear near its top together give rise to a second vortex, and the second to a third vortex. This leads to a stack of secondary vortices in a stably stratified boundary layer flow with decreasing strength when moving away from the wall.

Stratification and mixing processes associated with hypoxia in a shallow lake (Lake Kasumigaura, Japan)

Physical processes associated with hypoxia events in a shallow lake, Lake Kasumigaura, Japan, are investigated with long-term mooring observations at the middle of the lake basin. Results show that strong stratification during the summer suppresses vertical mixing resulting in hypoxia in the bottom boundary layer. The dissolved oxygen decreases due to the limited oxygen supply under strongly stratified conditions. The intensity of the stratification is controlled by the ratio between surface wind stress and the surface incoming buoyancy flux. The Monin–Obukhov length scale explains the stratification and shows good agreement with the Froude number and Wedderburn number. The stratification is also enhanced by the heat flux towards the sediments, reaching O (100) W m−2, during the summer season. In addition to field observations, this study conducted numerical simulations to further investigate mixing structure. Results from numerical simulations indicate that the bottom sediment heat flux significantly contributes to mixing and stratification in the lake. This study suggests that mixing processes associated with winds and heat flux at surface and bottom sediments are the key to understanding hypoxia events in Lake Kasumigaura.

Stratified Ekman layers evolving under a finite-time stabilizing buoyancy flux

Stratified flow in nocturnal boundary layers is studied using direct numerical simulation (DNS) of the Ekman layer, a model problem that is useful to understand atmospheric boundary-layer (ABL) turbulence. A stabilizing buoyancy flux is applied for a finite time to a neutral Ekman layer. Based on previous studies and the simulations conducted here, the choice of $L_{\mathit{cri}}^{+}=Lu_{\ast }/\unicode[STIX]{x1D708}\approx 700$ ($L$ is the Obukhov length scale and $u_{\ast }$ is the friction velocity) provides a cooling flux that is sufficiently strong to cause the initial collapse of turbulence. The turbulent kinetic energy decays over a time scale of $4.06L/u_{\ast }$ during the collapse. The simulations suggest that imposing $L_{\mathit{cri}}^{+}\approx 700$ on the neutral Ekman layer results in turbulence collapse during the initial transient, independent of Reynolds number, $Re_{\ast }$. However, the long-time state of the flow, i.e. turbulent with spatial intermittency or non-turbulent, is found to depend on the initial value of $Re_{\ast }$ since the cooling flux and resultant stratification increase with $Re_{\ast }$ for a given $L^{+}$. The lower-$Re_{\ast }$ cases have sustained turbulence with shear and stratification profiles that evolve in a manner such that the gradient Richardson number, $Ri_{g}$, in the near-surface layer, including the low-level jet, remains subcritical. The highest $Re_{\ast }$ case has supercritical $Ri_{g}$ in the low-level jet and turbulence does not recover. A theoretical discussion is performed to infer that the bulk Richardson number, $Ri_{b}$, is more suitable than $L^{+}$ to determine the fate of stratified Ekman layers at late time. DNS results support the implications of $Ri_{b}$ for the effect of initial $Re_{\ast }$ and $L^{+}$ on the flow.

Comments on “Langmuir Turbulence and Surface Heating in the Ocean Surface Boundary Layer”

AbstractUsing large-eddy simulations (LES) it is shown that the depth of a diurnal thermocline h should be scaled by the Zilitinkevich scale LZ, not by the Monin–Obukhov length scale LMO, contrary to the proposition by Pearson et al. Their argument to explain the slower increase of h than LMO using the effect of the preexisting thermocline is also invalid.

Self-similar mixing in stratified plane Couette flow for varying Prandtl number

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebioet al.(J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number$Pr$examined by two orders of magnitude from 0.7 up to 70. Significant effects of$Pr$on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number$Ri_{g}$is determined by the length scale ratio$h/L$, where$h$is the half-channel gap depth and$L$is the Obukhov length scale. As$h/L$approaches very large values,$Ri_{g}$asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number$Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$, where$\unicode[STIX]{x1D700}$is the dissipation,$\unicode[STIX]{x1D708}$is the kinematic viscosity and$N$is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio$L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$is the near-wall viscous scale. The flux Richardson number$Ri_{f}\equiv -B/P$, where$B$is the buoyancy flux and$P$is the shear production, is found to be proportional to$Ri_{g}$. This then leads to a turbulent Prandtl number$Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$of order unity, where$\unicode[STIX]{x1D708}_{t}$and$\unicode[STIX]{x1D705}_{t}$are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number$Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$, where$U^{\prime }$is a turbulent horizontal velocity scale, is found to vary like$Ri_{g}^{-1/2}$. All these scalings are consistent with our numerical data and appear to be independent of$Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn,J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of$\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$. With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

The influence of wind speed on surface layer stability and turbulent fluxes over southern Indian peninsula station

Surface to atmosphere exchange has received much attention in numerical weather prediction models. This exchange is defined by turbulent parameters such as frictional velocity, drag coefficient and heat fluxes, which have to be derived experimentally from high-frequency observations. High-frequency measurements of wind speed, air temperature and water vapour mixing ratio (eddy covariance measurements), were made during the Integrated Ground Observation Campaign (IGOC) of Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX) at Mahabubnagar, India (16∘44′N, 77∘59′E) in the south-west monsoon season. Using these observations, an attempt was made to investigate the behaviour of the turbulent parameters, mentioned above, with respect to wind speed. We found that the surface layer stability derived from the Monin–Obukhov length scale, is well depicted by the magnitude of wind speed, i.e., the atmospheric boundary layer was under unstable regime for wind speeds >4 m s−1; under stable regime for wind speeds <2 m s−1 and under neutral regime for wind speeds in the range of 2–3 m s−1. All the three stability regimes were mixed for wind speeds 3–4 m s−1. The drag coefficient shows scatter variation with wind speed in stable as well as unstable conditions.

Convectively unstable turbulent open channel flow with stable surface stratification

We use direct numerical simulations to examine fully developed turbulent open channel flow where the near wall region is unstably stratified and the outer boundary layer is stably stratified. The simulations are a model for flow in shallow turbid river channels with incident solar radiation. The aim is to determine under what conditions and by what mechanism the stably stratified layer is overturned. The flow is attained by applying a radiative heat flux at the free surface of the open channel. The absorption and transmission of the radiation follows the Beer--Lambert law with a constant absorption coefficient. We examine conditions where approximately 20% of the incident radiative heat flux penetrates through to the wall, releasing heat at the wall as a heat flux. The problem is specified by our buoyancy parameter which is analogous to the bulk Obukhov length scale. In the stable outer boundary layer we observe that the flux Richardson number reaches the limiting value, as was observed in the atmospheric boundary layer under sheared convective conditions. References S. M. Mitrovic, L. Hardwick, and F. Dorani. Use of flow management to mitigate cyanobacterial blooms in the Lower Darling River, Australia. J. Plankton Res. , 33(2):229–241, 2011. doi:10.1093/plankt/fbq094 J. R. Taylor, S. Sarkar, and V. Armenio. Large eddy simulation of stably stratified open channel flow. Phys. Fluids , 17(11):116602, 2005. doi:10.1063/1.2130747 M. Garcia-Villalba and J. C. del Alamo. Turbulence modification by stable stratification in channel flow. Phys. Fluids , 23(4):045104, 2011. doi:10.1063/1.3560359 N. Williamson, S. W Armfield, M. P. Kirkpatrick, and S Norris. A canonical model for stratified flow in estuaries and rivers. ANZIAM J. , 54:C88–C101, 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6429 S. W. Armfield, S. E. Norris, P. Morgan, and R. Street. A parallel non-staggered Navier-Stokes solver implemented on a workstation cluster. In S. Armfield, P. Morgan, and K. Srinvas (Eds.) Computational Fluid Dynamics 2002: Proceedings of the Second International Conference on Computational Fluid Dynamics, Sydney, Australia, 15–19 July 2002 , pages 30–45. Springer, 2003. doi:10.1007/978-3-642-59334-5 R. J. Conzemius and E. Fedorovich. Dynamics of Sheared Convective Boundary Layer Entrainment. Part I: Methodological Background and Large-Eddy Simulations. J. Atmos. Sci. , 63(4):1151–1178, 2006. doi:10.1175/JAS3691.1

Transition to stably stratified states in open channel flow with radiative surface heating

AbstractDirect numerical simulations (DNS) of turbulent stratified flow in an open channel with an internal heat source following the Beer–Lambert law from the surface are used to investigate the transition from neutral to strongly stable flow. Our buoyancy bulk parameter is defined through the ratio of the domain height ${\it\delta}$ to $\mathscr{L}$, a bulk Obukhov length scale for the flow. We cover the range ${\it\lambda}={\it\delta}/\mathscr{L}=0{-}2.0$, from neutral conditions to the onset of the stable regime, with the Reynolds number range $Re_{{\it\tau}}=200{-}800$, at a Prandtl number of 0.71. The result is a boundary layer flow where the effects of stratification are weak in the wall region but progressively stronger in the outer layer up to the free surface. At ${\it\lambda}\simeq 1$ the turbulent kinetic energy (TKE) budget is in local equilibrium over a region extending from the near-wall region to a free-surface affected region a distance $l_{{\it\nu}}$ from the surface, with $l_{{\it\nu}}/{\it\delta}\sim Re^{-1/2}$. In this equilibrium region the flow can be characterised by the flux Richardson number $R_{f}$ and the local Obukhov length scale ${\it\Lambda}$. At higher ${\it\lambda}$ local mixing limit conditions are observed over an extended region. At ${\it\lambda}=2$ the flux Richardson number approaches critical limit values of $R_{f,c}\simeq 0.18$ and gradient Richardson number $Ri_{c}\simeq 0.2$. At high ${\it\lambda}$, we obtain a flow field where buoyancy interacts with the smallest scales of motion and the turbulent shear stress and buoyancy flux are suppressed to molecular levels. We find that this regime can be identified in terms of the parameter $Re_{\mathscr{L},c}=\mathscr{L}u_{{\it\tau}}/{\it\nu}\lesssim 200{-}400$ (where $u_{{\it\tau}}$ is the friction velocity and ${\it\nu}$ the kinematic viscosity), which is related to the $L_{\ast }$ parameter of Flores and Riley (Boundary-Layer Meteorol., vol. 139 (2), 2011, pp. 241–259) and buoyancy Reynolds number $\mathscr{R}$. With energetic equilibrium attained, the local buoyancy Reynolds number, $Re_{{\it\Lambda}}={\it\Lambda}\langle u^{\prime }w^{\prime }\rangle ^{1/2}/{\it\nu}$, is directly related to the separation of the Ozmidov ($l_{O}$) and Kolmogorov (${\it\eta}$) length scales in the outer boundary layer by $Re_{{\it\Lambda}}\simeq \mathscr{R}\equiv (l_{O}/{\it\eta})^{4/3}$. The inner wall region has the behaviour $\mathscr{R}\sim Re_{\mathscr{L}}Re_{{\it\tau}}$, in contrast to stratified boundary layer flows where the buoyancy flux is non-zero at the wall and $\mathscr{R}\sim Re_{\mathscr{L}}$.

Similarity theory based on the Dougherty–Ozmidov length scale

This article describes a local similarity theory for developed turbulence in the stably stratified boundary layer that is based on the Brunt–Väisälä frequency and the dissipation rate of turbulent kinetic energy instead of the turbulent fluxes used in the traditional Monin–Obukhov similarity theory. Based on dimensional analysis (Pi theorem), it is shown that any properly scaled statistics of the small‐scale turbulence are universal functions of a stability parameter defined as the ratio of a reference height z and the Dougherty–Ozmidov length scale, which in the limit of z‐less stratification is linearly proportional to the Obukhov length scale. Measurements of atmospheric turbulence made at five levels on a 20 m tower over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to examine the behaviour of different similarity functions in the stable boundary layer. In the framework of this approach the non‐dimensional turbulent viscosity is equal to the gradient Richardson number, whereas the non‐dimensional turbulent thermal diffusivity is equal to the flux Richardson number. These results are a consequence of the approximate local balance between production of turbulence by shear in the mean flow and viscous dissipation. The turbulence framework based on the Brunt–Väisälä frequency and the dissipation rate of turbulent kinetic energy may have practical advantages for estimating turbulence when the fluxes are not directly available.

A numerical study of the unstratified and stratified Ekman layer

AbstractWe study the turbulent Ekman layer at moderately high Reynolds number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1600 &lt; \mathit{Re} = \delta _{E}G/\nu &lt; 3000$, using direct numerical simulations (DNS). Here, $\delta _{E} = \sqrt{2\nu /f}$ is the laminar Ekman layer thickness, $G$ the geostrophic wind, $\nu $ the kinematic viscosity and $f$ is the Coriolis parameter. We present results for both neutrally, moderately and strongly stably stratified conditions. For unstratified cases, large-scale roll-like structures extending from the outer region down to the wall are observed. These structures have a clear dominant frequency and could be related to periodic oscillations or instabilities developing near the low-level jet. We discuss the effect of stratification and $\mathit{Re}$ on one-point and two-point statistics. In the strongly stratified Ekman layer we observe stable co-existing large-scale laminar and turbulent patches appearing in the form of inclined bands, similar to other wall-bounded flows. For weaker stratification, continuously sustained turbulence strongly affected by buoyancy is produced. We discuss the scaling of turbulent length scales, height of the Ekman layer, friction velocity, veering angle at the wall and heat flux. The boundary-layer thickness, the friction velocity and the veering angle depend on $Lf/u_\tau $, where $u_\tau $ is the friction velocity and $L$ the Obukhov length scale, whereas the heat fluxes appear to scale with $L^+=L u_\tau /\nu $.

Exploring a geophysical process‐based attribution technique for the determination of the atmospheric boundary layer depth using aerosol lidar and near‐surface meteorological measurements

A new objective method for the determination of the atmospheric boundary layer (ABL) depth using routine vertically pointing aerosol lidar measurements is presented. A geophysical process‐based analysis is introduced to improve the attribution of the lidar‐derived aerosol gradients, which is so far the most challenging part in any gradient‐based technique. Using micrometeorological measurements of Obukhov length scale, both early morning and evening transition periods are determined which help separate the turbulence regimes during well‐mixed convective ABL and nocturnal/stable ABL. The lidar‐derived aerosol backscatter signal intensity is used to determine the hourly‐averaged vertical profiles of variance of the fluctuations of particle backscatter signal providing the location of maximum turbulent mixing within the ABL; thus, obtained mean ABL depth guides the attribution by searching for the appropriate minimum of the gradients. An empirical classification of the ABL stratification patterns into three different types is proposed by determining the changes in the near‐surface stability scenarios. First results using the lidar observations obtained between March and July in 2011 at SIRTA atmospheric observatory near Palaiseau (Paris suburb) in France demonstrate that the new attribution technique makes the lidar estimations of ABL depth more physically reliable under a wide spectrum of meteorological conditions. While comparing lidar and nearby radiosonde measurements of ABL depths, an excellent concordance was found with a correlation coefficient of 0.968 and 0.927 for daytime and nighttime measurements, respectively. A brief climatology of the characteristics of the ABL depth, its diurnal cycle, a detailed discussion of the morning and evening transitions are presented.