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Parameterizing turbulence in the atmospheric boundary layer as a function of space and time is essential for weather and climate models. Here, we explore a model for wavenumber–frequency spectra based on a linear random advection approach to characterize sheared convective atmospheric boundary layer flows. Building on previous works, we obtain the wavenumber–frequency spectrum as a product of the wavenumber spectrum and a Gaussian frequency distribution, whose mean and variance are given by the mean advection and random sweeping velocities, respectively. The applicability of the model is tested with direct numerical simulation data in the mixed layer and the entrainment zone for the streamwise and vertical velocity components and buoyancy. To obtain a fully analytical model, we propose using a von Kármán wavenumber spectrum parameterized by the characteristic variances and integral length scales. These parameters are height dependent and vary considerably with the relative balance of buoyancy and shear forces. The introduced analytical model relies on fitting parameters obtained from numerical data in the relevant range of scales. The comparison of the von Kármán–based spectra for velocity and buoyancy to simulation results shows that the main features of the measured spectra are captured by the model.

Determining the shallow-convective mass flux at cloud base is the principle closure needed in convective parameterizations. Closure methods are usually developed and tested based on large-eddy simulations of a limited number of idealized cases. Here we evaluate how well common closures reproduce the magnitude and variability of the cloud-base mass flux observed during the recent EUREC4A field campaign upstream Barbados. The true observed mass flux is estimated as a residual of the sub-cloud layer mass budget from the circular dropsonde arrays at the 200km scale. To assess the closures, we diagnose all parameters of the chosen closures from the same dropsonde data or from coincident turbulence and cloud measurements of a second aircraft. Preliminary results suggest that (i) both variability in the area fraction and vertical velocity scales should be accounted for to reproduce the observed mass flux variability, (ii) methods using the turbulence kinetic energy to approximate the vertical velocity scale outperform methods based on the subcloud convective velocity scale, and (iii) the closure formulations should be general enough to remain useful when applied to other data and regimes.

The magnitude of the entrainment buoyancy flux, and hence the growth rate of the convective boundary layer, does not increase monotonically with wind shear. Explanations for this have previously been based on wind-shear effects on the turbulence kinetic energy. By distinguishing between turbulent and non-turbulent regions, we provide an alternative explanation based on two competing wind-shear effects: the initial decrease in the correlation between buoyancy and vertical velocity fluctuations, and the increase in the turbulent area fraction. The former is determined by the change in the dominant forcing; without wind shear, buoyancy fluctuations drive vertical velocity fluctuations and the two are thus highly correlated; with wind shear, vertical velocity fluctuations are partly determined by horizontal velocity fluctuations via the transfer of kinetic energy through the pressure–strain correlation, thus reducing their correlation with the buoyancy field. The increasing turbulent area fraction, on the other hand, is determined by the increasing shear production of turbulence kinetic energy inside the entrainment zone. We also show that the dependence of these conditional statistics on the boundary-layer depth and on the magnitude of the wind shear can be captured by a single non-dimensional variable, which can be interpreted as an entrainment-zone Froude number. This work is partly supported by grant PID2019-105162RB-I00 funded by MCIN/AEI/10.13039/501100011033 and by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft (DFG). The authors also acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC). The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

We employed direct numerical simulations to estimate the error on chemical calculation in simulations with regional chemical-transport models induced by neglecting subgrid chemical segregation due to inefficient turbulent mixing in an urban boundary layer with strong and heterogeneously distributed surface emissions. In simulations of initially segregated reactive species with an entrainment-emission configuration with an A–B–C second-order chemical scheme, urban surface emission fluxes of the homogeneously emitted tracer A result in a very large segregation between the tracers and hence a very large overestimation of the effective chemical reaction rate in a complete-mixing model. This large effect can be indicated by a large Damköhler number (Da) of the limiting reactant. With heterogeneous surface emissions of the two reactants, the resultant normalised boundary-layer-averaged effective chemical reaction rate is found to be in a Gaussian function of Da, and it is increasingly overestimated by the imposed rate with an increased horizontal scale of emission heterogeneity. Coarse-grid models with resolutions commensurable to regional models give reduced yet still significant errors for all simulations with homogeneous emissions. Such model improvement is more sensitive to the increased vertical resolution. However, such improvement cannot be seen for simulations with heterogeneous emissions when the horizontal resolution of the model cannot resolve emission heterogeneity. This work highlights particular conditions in which the ability to resolve chemical segregation is especially important when modelling urban environments.

We explore the generalization properties of an echo state network applied as a reduced dynamical model to predict flux-driven two-dimensional turbulent convection. To this end, we consider a convection domain at fixed height with a variable ratio of buoyancy fluxes at the top and bottom boundaries, which break the top-down symmetry in comparison to the standard Rayleigh-B\'enard case thus leading to highly asymmetric mean and fluctuation profiles across the layer. Our direct numerical simulation model describes a convective boundary layer in a simple way. The data are used to train and test a recurrent neural network in the form of an echo state network. The input to the echo state networks is obtained in two different ways, either by a proper orthogonal decomposition or by a convolutional autoencoder. In both cases, the echo state network reproduces the turbulence dynamics and the statistical properties of the buoyancy flux, and is able to model unseen data records with different flux ratios.

Turbulence in the atmospheric boundary layer (ABL) plays an important role in the weather and climate system as it governs the meteorological exchanges of momentum, energy, and moisture between the free atmosphere and the Earth’s surface. Motivated by the need for conceptual physics-based models that parameterize turbulence in the ABL in terms of spectra at all spatio-temporal scales, we explore a linear random advection approach to characterize different scenarios of sheared convective boundary layer flows. As a main result, we obtain the wavenumber–frequency spectrum as a product of the wavenumber spectrum and a Gaussian frequency distribution, whose mean and variance are given by the mean advection and random sweeping velocities, respectively. The applicability of the model is evaluated with direct numerical simulations of the mixed layer and entrainment zone for the streamwise and vertical velocity components as well as buoyancy. To obtain a fully analytical formula for the linear random advection approach, we propose using a von-Kàrmàn wavenumber spectrum parameterized by the characteristic convective velocity and length scales. These scales are height-dependent and vary considerably with the relative balance of buoyancy and shear forces. The comparison of the von-Kàrmàn-based model for velocity and buoyancy to simulation results shows that the main features of the measured spectra are captured by the model.

Turbulent mixed convection in channel flows with heterogeneous surfaces is studied using direct numerical simulations. The relative importance between buoyancy and shear effects, characterized by the bulk Richardson number $Ri_b$, is varied in order to cover the flow regimes of forced, mixed and natural convection, which are associated with different large-scale flow organization. The heterogeneous surface consists of streamwise-aligned ridges, which are known to induce secondary motion in case of forced convection. The large-scale streamwise rolls emerging under smooth-wall mixed convection conditions are significantly affected by the heterogeneous surfaces and their appearance is considerably reduced for dense ridge spacings. It is found that the formation of these rolls requires larger buoyancy forces than over smooth walls due to the additional drag induced by the ridges. Therefore, the transition from forced convection structures to rolls is delayed towards larger $Ri_b$ for spanwise heterogeneous surfaces. The influence of the heterogeneous surface on the flow organization of mixed convection is particularly pronounced in the roll-to-cell transition range, where ridges favor the transition to convective cells at significantly lower $Ri_b$. In addition, the convective cells are observed to align perpendicular to the ridges with decreasing ridge spacing. We attribute this reorganization to the fact that flow parallel to the ridges experience less drag than flow across the ridges, which is energetically more beneficial. Furthermore, we find that streamwise rolls exhibit very slow dynamics for $Ri_b=1$ and $Ri_b=3.2$ when the ridge spacing is in the order of the rolls' width. For these cases the up- and downdrafts of the rolls move slowly across the entire channel instead of being fixed in space as observed for the smooth-wall cases., 34 pages manuscript with 19 figures submitted to Journal of Fluid Mechanics

Two zero-order bulk models (ZOMs) are developed for the velocity, buoyancy, and moisture of a cloud-free barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. The models differ in the entrainment closure assumption: in the first one, termed the ‘‘energetics-based model,’’ the negative and positive areas of the buoyancy flux are assumed to match between the model and the actual CBL; in the second one, termed the ‘‘geometric-based model,’’ the modeled CBL depth is assumed to match different definitions of the actual CBL depth. Parameterizations for these properties derived from direct numerical simulation (DNS) are employed as entrainment closure equations. These parameterizations, and hence the resulting models, are free from the potential singularity at finite wind strength that has been a major limitation in previous bulk models. The proposed ZOMs are verified using the DNS data. Model results show that the CBL depths obtained from the energetics-based model and previous ZOMs correspond to the height that marks the transition from the lower to the upper entrainment-zone sublayer; this reference height is few hundred meters above the height of the minimum buoyancy flux. It is also argued that ZOMs, despite their simplicity compared to higher-order models, can accurately represent CBL bulk properties when the relevant features of the actual entrainment zone are considered in the entrainment closures. The vertical structure of the actual entrainment zone, if required, can be constructed a posteriori using the available relationships between the predicted zero-order CBL depth and various definitions of the actual CBL depth.

Direct numerical simulations resolving meter and submeter scales in the cloud-top region of stratocumulus are used to investigate the interactions between a mean vertical wind shear and in-cloud turbulence driven by evaporative and radiative cooling. There are three major results. First, a critical velocity jump exists, above which shear significantly broadens the entrainment interfacial layer (EIL), enhances cloud-top cooling, and increases the mean entrainment velocity; shear effects are negligible when the velocity jump is below . Second, a depletion velocity jump exists, above which shear-enhanced mixing reduces cloud-top radiative cooling, thereby weakening the large convective motions; shear effects remain localized within the EIL when the velocity jump is below . The critical velocity jump and depletion velocity jump are provided as a function of in-cloud and free-tropospheric conditions, and one finds and for typical subtropical conditions. Third, the individual contributions to the mean entrainment velocity from mixing, radiative cooling, and evaporative cooling strongly depend on the choice of the reference height where the entrainment velocity is calculated. This result implies that the individual contributions to the mean entrainment velocity should be estimated at a comparable height while deriving entrainment-rate parameterizations. A strong shear alters substantially the magnitude and the height where these individual contributions reach their maxima, which further demonstrates the importance of shear on the dynamics of stratocumulus clouds.

In this work, direct numerical simulation (DNS) of the stratocumulus cloud-top mixing layer is used to test various approaches to estimate the turbulence kinetic energy (TKE) dissipation rate ε from one-dimensional (1D) intersections that resemble experimental series. Results of these estimates are compared with “true” (DNS) values of ε in buoyant and inhomogeneous atmospheric flows. We focus on recently proposed methods of the TKE dissipation-rate retrievals based on zero crossings and recovering the missing part of the spectrum. These methods are tested on fully resolved turbulence fields and compared to standard retrievals from power spectra and structure functions. Anisotropy of turbulence due to buoyancy is shown to influence retrievals based on the vertical velocity component. TKE dissipation-rate estimates from the number of crossings correspond well to spectral estimates. The method based on the recovery of the missing part of the spectrum works best for Pope’s model of the dissipation spectrum and is sensitive to external intermittency. This allows for characterization of external intermittency by the Taylor-to-Liepmann scale ratio. Further improvements of this method are possible when the variance of the velocity derivative is used instead of the number of zero crossings per unit length. In conclusion, the new methods of TKE dissipation-rate retrieval from 1D series provide a valuable complement to standard approaches.

We report on the Azores Stratocumulus Measurements of Radiation, Turbulence and Aerosols (ACORES) campaign, which took place around Graciosa and Pico Islands/Azores in July 2017. The main objective was to investigate the vertical distribution of aerosol particles, stratocumulus microphysical and radiative properties, and turbulence parameters in the eastern North Atlantic. The vertical exchange of mass, momentum, and energy between the free troposphere (FT) and the cloudy marine boundary layer (MBL) was explored over a range of scales from submeters to kilometers. To cover these spatial scales with appropriate measurements, helicopter-borne observations with unprecedented high resolution were realized using the Airborne Cloud Turbulence Observation System (ACTOS) and Spectral Modular Airborne Radiation Measurement System–Helicopter-Borne Observations (SMART-HELIOS) instrumental payloads. The helicopter-borne observations were combined with ground-based aerosol measurements collected at two continuously running field stations on Pico Mountain (2,225 m above sea level, in the FT), and at the Atmospheric Radiation Measurement (ARM) station on Graciosa (at sea level). First findings from the ACORES observations we are discussing in the paper are as follows: (i) we have observed a high variability of the turbulent cloud-top structure on horizontal scales below 100 m with local temperature gradients of up to 4 K over less than 1 m vertical distance, (ii) we have collected strictly collocated radiation measurements supporting the relevance of small-scale processes by revealing significant inhomogeneities in cloud-top brightness temperature to scales well below 100 m, and (iii) we have concluded that aerosol properties are completely different in the MBL and FT with often-complex stratification and frequently observed burst-like new particle formation.

The National Science Foundation (NSF) provided the funding support for this workshop through grant number AGS-1945681. We are grateful to the following: Susannah Burrows (Pacific Northwest National Laboratory), Hermann Gerber (Gerber Scientific), and Fan Yang (Brookhaven National Laboratory) for looking over the meeting summary for accuracy; Jessica Brassard of Michigan Technological University for invaluable support and advising on the strategy, organization, and logistics of the workshop, as well as for the artistic musings shown in Figure 1; Peter Sullivan and Wojciech Grabowski for facilitating the coordination and co-sponsorship by the Geophysical Turbulence Program; Kris Marwitz of NCAR for administrative and logistical support; And to the more than 60 attendees who actively engaged in brainstorms and discussions, without whom this meeting summary would not have been possible Clouds and aerosols, ubiquitously embedded in turbulent flows, are central to the prediction of weather and climate. The purpose of the workshop described here was to explore scientific questions and set priorities for a large-scale aerosol-cloud-turbulence laboratory facility. (Here, “facility” denotes one or more aerosol/cloud chambers and the associated instrumentation and technical/scientific staff.) Specifically, at the workshop we attempted to gauge community interest and to obtain a sense of priorities for the scientific challenges likely to be amenable to laboratory investigation.

Large-scale circulations around a city are co-modulated by the urban heat islandand by regional wind patterns. Depending on these variables, the circulations fall into different regimes ranging from advection-dominated (plume regime) to convection-driven (bubble regime). Using dimensional analysis and large-eddy simulations, this study investigates how these different circulations scale with urban and rural heat fluxes, as well as up stream wind speed. Two dimensionless parameters are shown to control the dynamics of the flow: (1) the ratio of rural to urban thermal convective velocities that contrasts their respective buoyancy fluxes and (2) the ratio of bulk inflow velocity to the convection velocity in the rural area. Finally, the vertical flow velocities transecting the rural to urban transitions are used to developa criterion for categorizing different large-scale circulations into plume, bubble or transitional regimes. The findings have implications for city ventilation since bubbleregimes are expected to trap pollutants, as well as for scaling analysis in canonical mixed-convection flows.

Conventional analysis has shown that strong wind shear enhances the entrainment buoyancy flux in the convective boundary layer. By conditioning the entrainment zone into regions of turbulent (i.e., strongly vortical) and nonturbulent (i.e., weakly vortical) flow, some unexpected aspects of this process are revealed. It is found that turbulent regions contribute the most to the entrainment buoyancy flux, but that as wind shear increases, the magnitude of the buoyancy flux in turbulent regions remains approximately constant, or even decreases, despite substantially stronger buoyancy fluctuations. The reason is that the correlation between buoyancy and vertical velocity fluctuations decreases with increasing wind shear, to the extent that it compensates the stronger buoyancy fluctuations. In free convection, this correlation is high because the vertical velocity is mainly determined by the buoyancy force acting in the same direction. Under strong shear conditions, buoyancy is no longer the only external source of vertical velocity fluctuations and their correlation consequently decreases. Hence, shear enhancement of the buoyancy flux in the entrainment zone is primarily due to an increase of the turbulent area fraction, rather than a change of flux inside the turbulent regions. © 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses)

We use dimensional analysis and direct numerical simulations to characterize specific humidity statistics in the equilibrium (quasi-steady) entrainment regime of cloud-free convective boundary layers that grow into linearly stratified free atmospheres. The first three moments and the mean vertical flux are studied for arbitrary combinations of free-atmosphere lapse-rates and surface fluxes of buoyancy and specific humidity. First, we find the combination of these parameters that distinguishes between the entrainment-drying regime and the surface-moistening regime. We also provide a zero-order model describing both regimes. Second, we parametrize the variances in the mixed layer and in the entrainment zone separately, based on convective and entrainment-zone scales, respectively. We show that the large variances in the entrainment zone are not only due to large production rates, but also due to low dissipation rates. Third, we provide the skewness for any regime between the pure drying limit and the pure moistening limit. The variation of the skewness indicates that knowing the sign of the skewness near the surface is often insufficient to distinguish between drying and moistening regimes, in contrast to previous conjectures. In a more general context, this paper further supports the applicability of direct numerical simulations to investigate the atmospheric boundary layer, as inferred from the degree of Reynolds number similarity observed in the results and from the consistency of the derived parametrizations with field measurements.

The heat exchange between floating ice and the underlying ocean is determined by the interplay of diffusive fluxes directly at the ice–ocean interface and turbulent fluxes away from it. In this study, we examine this interplay through direct numerical simulations of free convection. Our results show that an estimation of the interface flux ratio based on direct measurements of the turbulent fluxes can be difficult because the flux ratio varies with depth. As an alternative, we present a consistent evaluation of the flux ratio based on the total heat and salt fluxes across the boundary layer. This approach allows us to reconcile previous estimates of the ice–ocean interface conditions. We find that the ratio of heat and salt fluxes directly at the interface is 83–100 rather than 33 as determined by previous turbulence measurements in the outer layer. This can cause errors in the estimated ice-ablation rate from field measurements of up to 40% if they are based on the three-equation formulation.

In this work, the reconstruction of sub-grid scales in large eddy simulation (LES) of turbulent flows in stratocumulus clouds is addressed. The approach is based on the fractality assumption of turbulent velocity field. The fractal model reconstructs sub-grid velocity field from known filtered values on LES grid, by means of fractal interpolation, proposed by Scotti and Meneveau (Physica D 127, 198–232 1999). The characteristics of the reconstructed signal depend on the stretching parameter d, which is related to the fractal dimension of the signal. In many previous studies, the stretching parameter values were assumed to be constant in space and time. To improve the fractal interpolation approach, we account for the stretching parameter variability. The local stretching parameter is calculated from direct numerical simulation (DNS) data with an algorithm proposed by Mazel and Hayes (IEEE Trans. Signal Process 40(7), 1724–1734, 1992), and its probability density function (PDF) is determined. It is found that the PDFs of d have a universal form when the velocity field is filtered to wave-numbers within the inertial range. In order to investigate Reynolds number (Re) dependence, we compare the inertial-range PDFs of d in DNS and large eddy simulation (LES) of stratocumulus cloud-top and experimental airborne data from Physics of Stratocumulus Top (POST) research campaign. Next, fractal reconstruction of the subgrid velocity is performed and energy spectra and statistics of velocity increments are compared with DNS data. It is assumed that the stretching parameter d is a random variable with the prescribed PDF. We show that the agreement with the DNS is in such case better and the error in mass conservation is smaller compared to the use of constant values of d. The motivation of this study is to reproduce effect of sub-grid scales on a motion of Lagrangian particles (e.g. droplets) in clouds.

Direct numerical simulations are used to characterize wind-shear effects on entrainment in a barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. We consider weakly to strongly unstable conditions $-z_{enc}/L_{Ob}\gtrsim 4$, where $z_{enc}$ is the encroachment CBL depth and $L_{Ob}$ is the Obukhov length. Dimensional analysis allows us to characterize such a sheared CBL by a normalized CBL depth, a Froude number and a Reynolds number. The first two non-dimensional quantities embed the dependence of the system on time, on the surface buoyancy flux, and on the buoyancy stratification and wind velocity in the free atmosphere. We show that the dependence of entrainment-zone properties on these two non-dimensional quantities can be expressed in terms of just one independent variable, the ratio between a shear scale $(\unicode[STIX]{x0394}z_{i})_{s}\equiv \sqrt{1/3}\unicode[STIX]{x0394}u/N_{0}$ and a convective scale $(\unicode[STIX]{x0394}z_{i})_{c}\equiv 0.25z_{enc}$, where $\unicode[STIX]{x0394}u$ is the velocity increment across the entrainment zone, and $N_{0}$ is the buoyancy frequency of the free atmosphere. Here $(\unicode[STIX]{x0394}z_{i})_{s}$ and $(\unicode[STIX]{x0394}z_{i})_{c}$ represent the entrainment-zone thickness in the limits of weak convective instability (strong wind) and strong convective instability (weak wind), respectively. We derive scaling laws for the CBL depth, the entrainment-zone thickness, the mean entrainment velocity and the entrainment-flux ratio as functions of $(\unicode[STIX]{x0394}z_{i})_{s}/(\unicode[STIX]{x0394}z_{i})_{c}$. These scaling laws can also be expressed as functions of only a Richardson number $(N_{0}z_{enc}/\unicode[STIX]{x0394}u)^{2}$, but not in terms of only the stability parameter $-z_{enc}/L_{Ob}$.

In this work, Direct Numerical Simulations (DNS) of atmospheric convective boundary layer flow is used to test various approaches to estimate turbulence kinetic energy dissipation rate (EDR) \(\epsilon \) from one-dimensional (1D) velocity signals. Results of these estimates are compared with “true” DNS values of \(\epsilon \). We focus on methods of EDR retrievals proposed recently in Waclawczyk et al. Atmos. Meas. Tech. 10, 2017. We test these methods and show that they provide a valuable complement to standard approaches. Another goal is to investigate how the presence of anisotropy due to buoyancy affect the various retrieval techniques of \(\epsilon \).

We investigate by means of direct numerical simulation how large-scale circulations produce deviations from Monin–Obukhov similarity theory (MOST) in the limit of free convection, disentangling the role of large-scale downdrafts from updrafts using conditional analysis. We compare the convective boundary layer to two other free-convective flows: Rayleigh–Bénard convection with an adiabatic top lid and classical Rayleigh–Bénard convection. This serves a dual purpose: firstly, to ascertain how changes in the upper boundary conditions and thereby in the large-scale circulations modify the near-surface behaviour and secondly, to assess to what extent we can extrapolate results from idealized systems to the unstable atmospheric surface layer. Using a low-pass filter to define the large scales we find that, whilst deviations from MOST occur within large-scale downdraft regions, strong deviations also occur within large-scale updraft regions. The deviations within updrafts are independent of the filter length scale used to define the large-scale circulations, independent of whether updrafts are defined as ascending air, or as air that is both ascending and positively buoyant, and are not due to changes with height of the updraft area fraction. This suggests that even updraft properties are not just determined locally, but also by outer scales. Cold, strong downdrafts in classical Rayleigh–Bénard convection notably modify the near-surface behaviour compared to the other two systems. For the moderate Reynolds numbers considered, Rayleigh–Bénard convection with an adiabatic top lid thus seems more appropriate than classical Rayleigh–Bénard convection for studying the unstable atmospheric surface layer in the limit of free convection.

Existence of non-turbulent flow patches in the vicinity of the wall of a turbulent flow is known as global intermittency. Global intermittency challenges the conventional statistics approach when describing turbulence in the inner layer and calls for the use of conditional statistics. We extend the vorticity-based conditioning of a flow to turbulent and non-turbulent sub-volumes by a high-pass filter operation. This modified method consistently detects non-turbulent flow patches in the outer and inner layers for stratifications ranging from the neutral limit to extreme stability, where the flow is close to a complete laminarization. When applying this conditioning method to direct numerical simulation data of stably stratified Ekman flow, we find the following. First, external intermittency has a strong effect on the logarithmic law for the mean velocity in Ekman flow under neutral stratification. If instead of the full field, only turbulent sub-volumes are considered, the data fit an idealized logarithmic velocity profile much better; in particular, a problematic dip in the von Kármán measure$\unicode[STIX]{x1D705}$in the surface layer is decreased by approximately 50 % and our data only support the reduced range$0.41\lesssim \unicode[STIX]{x1D705}\lesssim 0.43$. Second, order-one changes in turbulent quantities under strong stratification can be explained by a modulation of the turbulent volume fraction rather than by a structural change of individual turbulence events; within the turbulent fraction of the flow, the character of individual turbulence events measured in terms of turbulence dissipation rate or variance of velocity fluctuations is similar to that under neutral stratification.

According to Townsend’s hypothesis, so-called wall-attached eddies are the main contributors to turbulent transport in the atmospheric surface layer (ASL). This is also one of the main assumptions of Monin–Obukhov similarity theory (MOST). However, previous evidence seems to indicate that outer-scale eddies can impact the ASL, resulting in deviations from the classic MOST scaling. We conduct large-eddy simulations and direct numerical simulations of a dry convective boundary layer to investigate the impact of coherent structures on the ASL. A height-dependent passive tracer enables coherent structure detection and conditional analysis based on updrafts and subsidence. The MOST similarity functions computed from the simulation results indicate a larger deviation of the momentum similarity function ϕ m from classical scaling relationships compared to the temperature similarity function ϕ h. The conditional-averaged ϕ m for updrafts and subsidence are similar, indicating strong interactions between the inner and outer layers. However, ϕ h conditioned on subsidence follows the mixed-layer scaling, while its updraft counterpart is well predicted by MOST. Updrafts are the dominant contributors to the transport of momentum and temperature. Subsidence, which comprises eddies that originate from the outer layer, contributes increasingly to the transport of temperature with increasing instability. However, u′ of different signs are distributed symmetrically in subsidence unlike the predominantly negative θ′ as instability increases. Thus, the spatial patterns of u′ w′ differ compared to θ′ w′ in regions of subsidence. These results depict the mechanisms for departure from the MOST scaling, which is related to the stronger role of subsidence.

Direct numerical simulations resolving meter and submeter scales in the cloud-top region of stratocumulus are used to investigate the interactions between a mean vertical wind shear and in-cloud turbulence driven by evaporative and radiative cooling. There are three major results. First, a critical velocity jump (Du)crit exists, above which shear significantly broadens the entrainment interfacial layer (EIL), enhances cloud-top cooling, and increases the mean entrainment velocity; shear effects are negligible when the velocity jump is below (Du)crit. Second, a depletion velocity jump (Du)dep exists, above which shear-enhanced mixing reduces cloud-top radiative cooling, thereby weakening the large convective motions; shear effects remain localized within the EIL when the velocity jump is below (Du)dep. The critical velocity jump and depletion velocity jump are provided as a function of in-cloud and free-tropospheric conditions, and one finds (Du)crit ’ 124ms21 and (Du)dep’ 3210ms21 for typical subtropical conditions. Third, the individual contributions to the mean entrainment velocity from mixing, radiative cooling, and evaporative cooling strongly depend on the choice of the reference height where the entrainment velocity is calculated. This result implies that the individual contributions to the mean entrainment velocity should be estimated at a comparable height while deriving entrainment-rate parameterizations.Astrong shear alters substantially the magnitude and the height where these individual contributions reach their maxima, which further demonstrates the importance of shear on the dynamics of stratocumulus clouds.

We argue that combining direct numerical simulation (DNS) with large-eddy simulation (LES) and field studies could accelerate current lines of stratocumulus research. LES allows for a faster and more holistic study of the parameter space, but LES is sensitive to details of its formulation because the energetics are tied to unresolved processes in the cloud top region. One way to assess this sensitivity is through field studies. Another way is through DNS. In particular, DNS can be used to test the hypothesis that LES, even with an inadequate representation of the physics of cloud top entrainment, properly quantifies the sensitivity of cloud-topped boundary layers to changing environmental conditions. We support this argument by contrasting theoretical aspects of both techniques, by presenting first DNS results of a stratocumulus-topped boundary layer and discussing their convergence toward Reynolds number similarity, and by showing the consistency of DNS results with LES results and field measurements.

A conceptual bulk model for a dry, convective boundary layer with prescribed horizontally homogeneous and heterogeneous low-level radiative cooling rates is developed. For horizontally homogeneous radiative cooling, the response of the system to varying its prescribed parameters is explored and formulated in terms of nondimensional parameters. Large-eddy simulations with prescribed radiative cooling rates match the results of the bulk model well. It is found that, depending on the strength of the surface coupling, the height of the boundary layer (BL) either increases or decreases in response to increasing radiative BL cooling. Another property of the system is that, for increasing surface temperature, the BL temperature decreases if the prescribed radiative BL cooling rates are strong. This counterintuitive behavior is caused by the formulation of the entrainment rate at the inversion. Heterogeneous radiative BL cooling is found to cause a circulation induced by pressure deviations between the area of weak radiative BL cooling and the area of strong radiative BL cooling. Including the feedback of the induced circulation on the BL in a two-column model leads to a modified equilibrium state, in which a weakened horizontal BL flow of about 1 m s−1 is maintained for differences in radiative BL cooling rates larger than 1 K day−1. Such a circulation strength is comparable to a shallow circulation caused by surface temperature differences of a few kelvins. Spatial differences in radiative BL cooling should therefore be considered as a first-order effect for the formation of shallow circulations.

This paper describes MicroHH 1.0, a new and open-source (www.microhh.org) computational fluid dynamics code for the simulation of turbulent flows in the atmosphere. It is primarily made for direct numerical simulation but also supports large-eddy simulation (LES). The paper covers the description of the governing equations, their numerical implementation, and the parameterizations included in the code. Furthermore, the paper presents the validation of the dynamical core in the form of convergence and conservation tests, and comparison of simulations of channel flows and slope flows against well-established test cases. The full numerical model, including the associated parameterizations for LES, has been tested for a set of cases under stable and unstable conditions, under the Boussinesq and anelastic approximations, and with dry and moist convection under stationary and time-varying boundary conditions. The paper presents performance tests showing good scaling from 256 to 32 768 processes. The graphical processing unit (GPU)- enabled version of the code can reach a speedup of more than an order of magnitude for simulations that fit in the memory of a single GPU.

The effect of sedimentation on stratocumulus entrainment is investigated using direct numerical simulations of a cloud-top mixing layer driven by radiative and evaporative cooling. The simulations focus on the meter and submeter scales that are expected to be relevant for entrainment, and the finest grid spacing is Δx = 26 cm. The entrainment velocity is investigated from the analysis of the integrated-buoyancy evolution equation, which is exactly derived from the flow evolution equations. The analysis shows that sedimentation interacts with entrainment through two different mechanisms. As previously reported, sedimentation prevents droplets from evaporating in the entrainment zone, which in turn reduces the entrainment velocity. Here it is shown that sedimentation also promotes a positive buoyancy flux that directly opposes entrainment. The strengths of both mechanisms are characterized by two different settling numbers, which allow for predicting which meteorological conditions favor the reduction of entrainment by sedimentation. These new insights allow for including sedimentation in a parameterization of the entrainment velocity. The reduction of the entrainment velocity by sedimentation predicted by the parameterization and observed in the simulations is 3 times larger than previously reported in large-eddy simulations, which implies that meter- and submeter-scale turbulence plays an important role in the interaction of entrainment with sedimentation. On the whole, analysis and simulations indicate that stratocumulus entrainment is more sensitive to the cloud droplet number density due to sedimentation than previously thought.

Mixing processes at the boundary of clouds often include typical length-scales of several metres. Such length-scales are too large for current Lagrangian models but they are also poorly resolved by typical Eulerian-based large-eddy simulations. Here, a bulk formulation is introduced for direct numerical simulations. Two main assumptions sustain this approach: the continuum approximation and the liquid-phase diffusion approximation. The formulation includes the small-scale features that originate from microscopic droplet dynamics: sedimentation, finite-time condensation/evaporation, inertial effects and the low diffusion of liquid droplets with respect to vapour. The methodology is applied to the study of the buoyancy reversal instability that occurs at the top of stratocumulus clouds as a consequence of evaporative cooling. The inclusion of sedimentation, low liquid-phase diffusion and finite-time evaporation have a negative impact on instability when compared with the equilibrium formulation. The combined effect of all these small-scale features reduces mixing at the cloud top by at least 90%. This strong reduction is explained by a condensate-free middle layer, which emerges when droplets leave the cloud interface due to sedimentation.

The growth and decay of a convective boundary layer (CBL) over a surface with a constant surface temperature that develops into a linear stratification is studied, and a mathematical model for this system is derived. The study is based on direct numerical simulations with four different Reynolds numbers; the two simulations with the largest Reynolds numbers display Reynolds number similarity, suggesting that the results can be extrapolated to the atmosphere. Because of the interplay of the growing CBL and the gradually decreasing surface buoyancy flux, the system has a complex time evolution in which integrated kinetic energy, buoyancy flux, and dissipation peak and subsequently decay. The derived model provides characteristic scales for bulk properties of the CBL. Even though the system is unsteady, self-similar vertical profiles of buoyancy, buoyancy flux, and velocity variances are recovered. There are two important implications for atmospheric modeling. First, the magnitude of the surface buoyancy flux sets the time scale of the system; thus, over a rough surface the roughness length is a key variable. Therefore, the performance of the surface model is crucial in large-eddy simulations of convection over water surfaces. Second, during the phase in which kinetic energy decays, the integrated kinetic energy never follows a power law, because the buoyancy flux and dissipation balance until the kinetic energy has almost vanished. Therefore, the applicability of power-law decay models to the afternoon transition in the atmospheric boundary layer is questionable; the presented model provides a physically sound alternative

Direct numerical simulation and laboratory experiments are used to investigate turbulent convection beneath a horizontal ice–water interface. Scaling laws are derived that quantify the dependence of the melt rate of the ice on the far-field temperature of the water under purely thermally driven conditions. The scaling laws, the simulations, and the laboratory experiments consistently yield that the melt rate increases by two orders of magnitude, from ⋍101 to ⋍103 mm day−1, as the far-field temperature increases from 4° to 8°C. The strong temperature dependence of the melt rate is explained by analyzing the vertical structure of the flow: For far-field temperatures below 8°C, the flow features a stably stratified, diffusive layer next to the ice that shields it from the warmer, turbulent outer layer. The stratification in the diffusive layer diminishes as the far-field temperature increases and vanishes for far-field temperatures far above 8°C. Possible implications of these results for ice–ocean interfaces are discussed. The drastic melt-rate increase implies that turbulence needs to be considered in the analysis of ice–water interfaces even in shear-free conditions.

In the atmospheric boundary layer, some turbulence-like structures are maintained up to a stable density stratification several times higher than the above-mentioned linear stability analysis predicts. Nonetheless, the cessation of turbulence in such flows exposed to stable density stratification is a well-recognized problem. For non-rotating configurations, namely stably stratified channel- and free-shear flows, it has been shown that this cessation does not occur as an on–off process but is rather a complex transition from a turbulent to a laminar state. When stratification increases gradually, this transition begins with the localized absence of turbulent eddies in an otherwise turbulent flow, and has recently been shown to also occur in stably stratified Ekman flow. This localized absence of turbulence bears a striking resemblance to the absence of turbulence on some or all scales even close to the surface which is sometimes observed in the atmosphere and has been termed Global Intermittency. We propose here a method based on the intermittency factor together with high-pass-filtered flow fields that successfully distinguishes between turbulent and non-turbulent patches in Ekman flow.

Direct numerical simulations of the turbulent temporally evolving cloud-top mixing layer are used to investigate the role of evaporative cooling by isobaric mixing locally at the stratocumulus top. It is shown that the system develops a horizontal layered structure whose evolution is determined by molecular transport. A relatively thin inversion with a constant thickness h = κ/we is formed on top and travels upwards at a mean velocity we ≃ 0.1(κ |bs|χc2)1/3, where κ is the mixture-fraction diffusivity, bs < 0 is the buoyancy anomaly at saturation conditions χs and χc is the cross-over mixture fraction defining the interval of buoyancy reversing mixtures. A turbulent convection layer develops below and continuously broadens into the cloud (the lower saturated fluid). This turbulent layer approaches a self-preserving state that is characterized by the convection scales constructed from a constant reference buoyancy flux Bs = |bs|we/χs. Right underneath the inversion base, a transition or buffer zone is defined based on a strong local conversion of vertical to horizontal motion that leads to a cellular pattern and sheet-like plumes, as observed in cloud measurements and reported in other free-convection problems. The fluctuating saturation surface (instantaneous cloud top) is contained inside this intermediate region. Results show that the inversion is not broken due to the turbulent convection generated by the evaporative cooling, and the upward mean entrainment velocity we is negligibly small compared to the convection velocity scale w* of the turbulent layer and the corresponding growth rate into the cloud.

A priori and a posteriori studies for large eddy simulation of the compressible turbulent infinitely fast reacting shear layer are presented. The filtered heat release appearing in the energy equation is unclosed and the accuracy of different models for the filtered scalar dissipation rate and the conditional filtered scalar dissipation rate of the mixture fraction in closing this term is analyzed. The effect of different closures of the subgrid transport of momentum, energy and scalars on the modeling of the filtered heat release via the resolved fields is also considered. Three explicit models of these subgrid fluxes are explored, each with an increasing level of reconstruction and all of them regularized by a Smagorinsky-type term. It is observed that a major part of the error in the prediction of the conditional filtered scalar dissipation comes from the unsatisfactory modeling of the filtered dissipation itself. The error can be substantial in the turbulent fluctuation (rms) of the dissipation fields. It is encouraging that all models give good predictions of the mean and rms density in a posteriori LES of this flow with realistic heat release corresponding to large density change. Although a posteriori results show a small sensitivity to subgrid modeling errors in the current problem, extinction–reignition phenomena involving finite-rate chemistry would demand more accurate modeling of the dissipation rates. A posteriori results also show that the resolved fields obtained with the approximate reconstruction using moments (ARM) agree better with the filtered direct numerical simulation since the level of reconstruction in the modeled subfilter fluxes is increased.

Evaporative cooling monotonically increases as the thermodynamical properties of the inversion allow for more evaporation in shear-free radiatively driven stratocumulus. However, the entrainment velocity can deviate from the evaporative cooling trend and even become insensitive to variations in the inversion properties. Here the efficiency of evaporative cooling at amplifying the entrainment velocity is quantified by means of direct numerical simulations of a cloud top mixing layer. We demonstrate that variations in the efficiency modulate the effect of evaporative cooling on entrainment, explaining the different trends. These variations are associated with the evaporation of droplets in cloud holes below the inversion point. The parametrization of the efficiency provides the evaporative amplification of the entrainment velocity as a function of a single parameter that characterizes the inversion. The resulting entrainment velocities match our experiments and previous measurements to within ±25%. The parametrization also predicts the transition to a broken-cloud field consistently with observations.

The stratocumulus-top mixing process is investigated using direct numerical simulations of a shear-free cloud-top mixing layer driven by evaporative and radiative cooling. An extension of previous linear formulations allows for quantifying radiative cooling, evaporative cooling, and the diffusive effects that artificially enhance mixing and evaporative cooling in high-viscosity direct numerical simulations (DNS) and many atmospheric simulations. The diffusive cooling accounts for 20% of the total evaporative cooling for the highest resolution (grid spacing ;14 cm), but this can be much larger (;100%) for lower resolutions that are commonly used in large-eddy simulations (grid spacing ;5 m). This result implies that the k scaling for cloud cover might be strongly influenced by diffusive effects. Furthermore, the definition of the inversion point as the point of neutral buoyancy hbi(zi) 5 0 allows the derivation of two scaling laws. The in-cloud scaling law relates the velocity and buoyancy integral scales to a buoyancy flux defined by the inversion point. The entrainment-zone scaling law provides a relationship between the entrainment velocity and the liquid evaporation rate. By using this inversion point, it is shown that the radiative-cooling contribution to the entrainment velocity decouples from the evaporative-cooling contribution and behaves very similarly as in the smoke cloud. Finally, evaporative and radiative cooling have similar strengths, when this strength is measured by the integrated buoyancy source. This result partially explains why current entrainment parameterizations are not accurate enough, given that most of them implicitly assume that only one of the two mechanisms rules the entrainment.

The marine planetary boundary layer topped by stratocumulus clouds (STBL) is key for the planetary radiation balance [5]. In its simplest configuration the STBL consists of a lower moist boundary layer which is topped by a dry and warm free atmosphere. The top of the STBL is populated by stratocumulus clouds that emit long wave radiation, cooling the moist boundary layer. Radiative cooling is thought to be the main source of turbulent energy for the STBL.

The heterogeneously heated free convective boundary layer (CBL) is investigated by means of dimensional analysis and results from large-eddy simulations (LES) and direct numerical simulations (DNS). The investigated physical model is a CBL that forms in a linearly stratified atmosphere heated from the surface by square patches with a high surface buoyancy flux. Each simulation has been run long enough to show the formation of a peak in kinetic energy, corresponding to the “optimal” heterogeneity size with strong secondary circulations, and the subsequent transition into a horizontally homogeneous CBL. Scaling laws for the time of the optimal state and transition and for the vertically integrated kinetic energy (KE) have been developed. The laws show that the optimal state and transition do not occur at a fixed ratio of the heterogeneity size to the CBL height. Instead, these occur at a higher ratio for simulations with increasing heterogeneity sizes because of the development of structures in the downward-moving air that grow faster than the CBL thickness. The moment of occurrence of the optimal state and transition are strongly related to the heterogeneity amplitude: stronger amplitudes result in an earlier optimal state and a later transition. Furthermore, a decrease in patch size combined with a compensating increase in patch surface buoyancy flux to maintain the energy input results in decreasing KE and a later transition. The simulations suggest that a CBL with a heterogeneity size smaller than the initial CBL height has less entrainment than a horizontally homogeneous CBL, whereas one with a larger heterogeneity size has more.

Direct numerical simulation of the turbulent Ekman layer over a smooth wall is used to investigate bulk properties of a planetary boundary layer under stable stratification. Our simplified configuration depends on two non-dimensional parameters: a Richardson number characterizing the stratification and a Reynolds number characterizing the turbulence scale separation. This simplified configuration is sufficient to reproduce global intermittency, a turbulence collapse, and the decoupling of the surface from the outer region of the boundary layer. Global intermittency appears even in the absence of local perturbations at the surface; the only requirement is that large-scale structures several times wider than the boundary-layer height have enough space to develop. Analysis of the mean velocity, turbulence kinetic energy, and external intermittency is used to investigate the large-scale structures and corresponding differences between stably stratified Ekman flow and channel flow. Both configurations show a similar transition to the turbulence collapse, overshoot of turbulence kinetic energy, and spectral properties. Differences in the outer region resulting from the rotation of the system lead, however, to the generation of enstrophy in the non-turbulent patches of the Ekman flow. The coefficient of the stability correction function from Monin–Obukhov similarity theory is estimated as $$\beta \approx 5.7$$ in agreement with atmospheric observations, theoretical considerations, and results from stably stratified channel flows. Our results demonstrate the applicability of this set-up to atmospheric problems despite the intermediate Reynolds number achieved in our simulations.

The entrainment zone (EZ) of a dry, shear-free convective boundary layer growing into a linearly stratified fluid is studied by means of direct numerical simulation. The scale separation between the boundary layer thickness and the Kolmogorov length scale is shown to be sufficient to observe Reynolds number similarity in the statistics of interest during the equilibrium entrainment regime. Contrary to previous considerations, the vertical structure of the entrainment zone is found to be better described by the superposition of two sublayers: 1) an upper EZ sublayer that is dominated by overshooting thermals and is characterized by a penetration depth that scales with the ratio of the convective velocity and the buoyancy frequency of the free troposphere and 2) a lower EZ sublayer that is dominated by troughs of mixed fluid and is characterized by the integral length scale of the mixed layer. Correspondingly, different buoyancy scales are identified. The consequences of this multiplicity of scales on the entrainment rate parameters are evaluated directly, without resorting to any bulk model, through an exact relation among the mean entrainment rate, the local buoyancy increment, and both the turbulent and the finite-thickness contributions to the entrainment ratio A measured at the height of minimum buoyancy flux. The smaller turbulent contribution to A that is usually observed for relatively thick EZs is found to be compensated by the smaller local buoyancy increment instead of by the finite-thickness contribution. The two-layer structure of the entrainment zone is found to affect the exponent of the power-law relation between the normalized mean entrainment rate and the convective Richardson number such that the exponent deviates from 21 for typical atmospheric conditions, although it asymptotically approaches 21 for higher Richardson numbers

A numerical experiment is designed to study the interaction at the stratocumulus top between a mean vertical shear and the buoyancy reversal due to evaporative cooling, without radiative cooling. Direct numerical simulation is used to eliminate the uncertainty introduced by turbulence models. It is found that the enhancement by shear-induced mixing of the turbulence caused by buoyancy reversal can render buoyancy reversal comparable to other forcing mechanisms. However, it is also found that (i) the velocity jump across the capping inversion Du needs to be relatively large and values of about 1 m s21 that are typically associated with the convective motions inside the boundary layer are generally too small and (ii) there is no indication of cloud-top entrainment instability. To obtain these results, parameterizations of the mean entrainment velocity and the relevant time scales are derived from the study of the cloud-top vertical structure. Two overlapping layers can be identified: a background shear layer with a thickness (1/3)(Du) 2 /Db, where Db is the buoyancy increment across the capping inversion and a turbulence layer dominated by free convection inside the cloud and by shear production inside the relatively thin overlap region. As turbulence intensifies, the turbulence layer encroaches into the background shear layer and defines thereby the entrainment velocity. Particularized to the first research flight of the Second Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS II) field campaign, the analysis predicts an entrainment velocity of about 3 mm s21 after 5–10 min—a velocity comparable to the measurements and thus indicative of the relevance of mean shear in that case.

Turbulent entrainment – the process by which turbulence inside of the atmospheric boundary layer entrains air from the free troposphere above it – has been investigated using direct numerical simulations in two configurations, one without a cloud and one with a cloud. With the first configuration, we have learned that the entrainment zone in a convective boundary layer growing into a linearly stratified troposphere is better described in terms of a two-layer structure, with different characteristic scales associated with each of the two sub-layers. With the second configuration, we have explained how wind shear across the entrainment zone capping a stratocumulus cloud can render evaporative cooling as important as radiative cooling in driving convective motions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

A radiatively driven cloud-top mixing layer is investigated using direct numerical simulations. This configuration mimics the mixing process across the inversion that bounds the stratocumulus-topped boundary layer. The main focus of this paper is on small-scale turbulence. The finest resolution (7.4 cm) is about two orders of magnitude finer than that in cloud large-eddy simulations (LES). A one-dimensional horizontally averaged model is employed for the radiation. The results show that the definition of the inversion point with the mean buoyancy of hbi(zi) 5 0 leads to convective turbulent scalings in the cloud bulk consistent with the Deardorff theory. Three mechanisms contribute to the entrainment by cooling the inversion layer: a molecular flux, a turbulent flux, and the direct radiative cooling by the smoke inside the inversion layer. In the simulations the molecular flux is negligible, but the direct cooling reaches values comparable to the turbulent flux as the inversion layer thickens. The results suggest that the direct cooling might be overestimated in lessresolved models like LES, resulting in an excessive entrainment. The scaled turbulent flux is independent of the stratification for the range of Richardson numbers studied here. As suggested by earlier studies, the turbulent entrainment only occurs at the small scales and eddies larger than approximately four optical lengths (60 m in a typical stratocumulus cloud) perform little or no entrainment. Based on those results, a parameterization is proposed that accounts for a large part (50%–100%) of the entrainment velocities measured in the Second Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS II) campaign.

Large-eddy simulation (LES) has been extensively used as a tool to understand how various processes contribute to the dynamics of the stratocumulus layer. These studies are complicated by the fact that many processes are tied to the dynamics of the stably stratified interface that caps the stratocumulus layer, and which is inadequately resolved by LES. Recent direct numerical simulations (DNS) of isobaric mixing due to buoyancy reversal in a cloud-top mixing layer show that molecular effects are in some instances important in setting the cloud-top entrainment rate, which in turn influences the global development of the layer. This suggests that traditional LES are fundamentally incapable of representing cloud-top processes that depend on buoyancy reversal and that numerical artefacts can affect significantly the results. In this study, we investigate a central aspect of this issue by developing a test case that embodies important features of the buoyancy-reversing cloud-top layer. So doing facilitates a one-to-one comparison of the numerical algorithms typical of LES and DNS codes in a well-established case. We focus on the numerical effects only by switching off the subgridscale model in the LES code and using instead a molecular viscosity. We systematically refine the numerical grid and quantify numerical errors, validate convergence and assess computational efficiency of the low-order LES code compared to the high-order DNS. We show that the high-order scheme solves the cloud-top problem computationally more efficiently. On that basis, we suggest that the use of higher-order schemes might be more attractive than further increasing resolution to improve the representation of stratocumulus in LES.