Stratocumulus-capped mixed layers derived from a three-dimensional model

Results of a three-dimensional numerical model are analysed in a study of turbulence and entrainment within mixed layers containing stratocumulus with or without parameterized cloud-top radiative cooling. The model eliminates most of the assumptions invoked in theories of cloud-capped mixed layers, but suffers disadvantages which include poor resolution and large truncation errors in and above the capping inversion.For relatively thick mixed layers with relatively thick capping inversions, the cloud-top radiative cooling is found to be lodged mostly within the capping inversion when the cooling is confined locally to the upper 50 m or less of the cloud. It does not then contribute substantially towards increased buoyancy flux and turbulence within the well mixed layer just below.The optimal means of correlating the entrainment rate, or mixed-layer growth rate, for mixed layers of variable amounts of stratocumulus is found to be through functional dependence upon an overall jump Richardson number, utilizing as scaling velocity the standard deviation of vertical velocity existing at the top of the mixed layer (near the center of the capping inversion). This velocity is found to be a fraction of the generalized convective velocity for the mixed layer as a whole which is greater for cloud-capped mixed layers than for clear mixed layers.     

The depth of the daytime mixed layer at two coastal sites: A model and its validation

A mathematical model of mixed-layer depth based on the thermodynamic analysis of Tennekes (1973) is generalized to include advection and subsidence. The effects of advection on mixed-layer depth have been modelled by setting the model equations in a Lagrangian frame, performing an approximate first integral in order to derive the spatial dependence of the model variables, and using these spatial forms to give a set of Eulerian equations. The effects of subsidence have been modelled by imposing a subsidence velocity on the top of the mixed layer as well as allowing subsidence-induced warming above that layer.The model thus derived consists of a system of non-linear differential equations which may be numerically solved to elucidate the temporal behaviour of mixed-layer depth. The boundary conditions necessary for such a solution are drawn from field studies at two coastal sites: one with a relatively simple coastline and essentially flat land under agricultural use, the other with a considerably more complex coastline, rolling relief and mixed land use (agricultural, parkland and urban). In both cases the modelled evolution of mixed-layer depth is in good agreement with the measured depth.The sensitivity of the model to all the input variables is investigated by examining the dependence of the maximum mixed-layer depth on each of these variables in an artificial set.     

On eddy diffusion profiles in oceanic bottom boundary layers associated with cold eddies and filaments

The oceanic bottom boundary-layer model of Weatherly and Martin (1978) is used to study the vertical structure of the eddy diffusivity in a region with initially imposed bottom mixed-layer thickness. Because of near-bottom oceanic features, such as the Cold Filament (Weatherly and Kelley, 1982) and cold eddies (Ebbesmeyer et al., 1988), the bottom mixed-layer thickness is not the sole result of boundary-layer mixing; this is the incentive for this study. For a given geostrophic forcing and imposed mixed-layer depth, a formula for the eddy diffusion coefficient is found. This parameterization of the eddy diffusivity improves previous formulas used in oceanic and atmospheric boundary layers in the upper portion of the boundary layer. A simple model of a Cold Filament-like feature demonstrates the structure of the bottom boundary layer, the bottom mixed layer, and the relation between the two. A lens-like cross section of cold blobs, often used in analytical models, may be inappropriate if bottom friction is important.     

An applied model for the height of the daytime mixed layer and the entrainment zone

A model is presented for the height of the mixed layer and the depth of the entrainment zone under near-neutral and unstable atmospheric conditions. It is based on the zero-order mixed-layer height model of Batchvarova and Gryning (1991) and the parameterization of the entrainment zone depth proposed by Gryning and Batchvarova (1994). However, most zero-order slab type models of mixed-layer height may be applied. The use of the model requires only information on those meteorological parameters that are needed in operational applications of ordinary zero-order slab type models of mixed-layer height: friction velocity, kinematic heat flux near the ground and potential temperature gradient in the free atmosphere above the entrainment zone. When information is available on the horizontal divergence of the large-scale flow field, the model also takes into account the effect of subsidence, although this is usually neglected in operational models of mixed-layer height owing to lack of data. Model performance is tested using data from the CIRCE experiment.     

Applied model for the growth of the daytime mixed layer

A slab model is proposed for developing the height of the mixed layer capped by stable air aloft. The model equations are closed by relating the consumption of energy (potential and kinetic) at the top of the mixed layer to the production of convective and mechanical turbulent kinetic energy within the mixed layer. By assuming that the temperature difference at the top of the mixed layer instantaneously adjusts to the actual meteorological conditions without regard to the initial temperature difference that prevailed, the model is reduced to a single differential equation which easily can be solved numerically. When the mixed layer is shallow or the atmosphere nearly neutrally stratified, the growth is controlled mainly by mechanical turbulence. When the layer is deep, its growth is controlled mainly by convective turbulence. The model is applied on a data set of the evolution of the height of the mixed layer in the morning hours, when both mechanical and convective turbulence contribute to the growth process. Realistic mixed-layer developments are obtained.     

On the horizontal density ratio in the upper ocean

The horizontal density ratio in the upper ocean is examined using SeaSoar data collected over the last 15 years in the Pacific, Atlantic, and Indian Oceans. The horizontal density ratio R is defined to be the ratio of the relative effect of temperature and salinity on density. A front with a horizontal density ratio of 1 is said to be compensated since temperature and salinity gradients compensate in their effect on density. The statistics of density ratio are examined through calculation of conditional probability density functions. Case studies from each of the oceans elucidate processes affecting the density ratio. Global distributions of density ratio are calculated as functions of mixed-layer depth, distance below the mixed layer, and magnitude of thermohaline variability. Compensation is found in all oceans, on 3–4 km horizontal scales, when the mixed layer is deep and significant thermohaline variability exists. The tendency for compensation is stronger as mixed-layer depth increases. Conversely, compensation is not typical in shallow mixed layers, or when thermohaline variability is weak. The thermocline density ratio is found to be 2, in agreement with previous observational studies, and consistent with the process of salt fingering. The transition from R=1 in the mixed layer to R=2 in the thermocline is sharp when the mixed layer is deep. The ubiquity of compensation in the mixed layer is consistent with recent theory that suggests horizontal eddy diffusivity is a growing function of density gradient.     

Summer boundary-layer height at the plateau site of Dome’C,antarctica

Measurements of the mean and turbulent structure of the planetary boundary layer using a sodar and a sonic anemometer, and radiative measurements using a radiometer, were carried out in the summer of 1999–2000 at the Antarctic plateau station of Dome C during a two-month period. At Dome C strong ground-based inversions dominate for most of the year. However, in spite of the low surface temperatures (between −50 and −20 °C), and the surface always covered by snow and ice, a regular daytime boundary-layer evolution, similar to that observed at mid-latitudes, was observed during summertime. The mixed-layer height generally reaches 200–300 m at 1300–1400 LST in high summer (late December, early January); late in the summer (end of January to February), as the solar elevation decreases, it reduces to 100–200 m. A comparison between the mixed-layer height estimated from sodar measurements and that calculated using a mixed-layer growth model shows a rather satisfactory agreement if we assign a value of 0.01–0.02 m s⁻¹ to the subsidence velocity at the top of the mixed layer, and a value of 0.003–0.004 K m⁻¹ to the potential temperature gradient above the mixed layer.     

A Mixed-Layer Model Of The Well-Mixed Stratocumulus-Topped Boundary Layer

Recently a range of sophisticated large-eddy simulations of thecloud-topped boundary layer have been intercompared and furthercompared with observations and single column models. Here we comparethese results with perhaps the simplest model of the cloud-toppedboundary layer, namely a mixed-layer model. Results from the model aredescribed with two aims in mind. Firstly, the good results act as areminder of the success of simple models, and, secondly, we suggestthat a simple mixed-layer model could be used as a baseline for futuremodel intercomparisons.The mixed-layer model is based on two assumptions that follow previousstudies. Firstly, the liquid-water potential temperature and the total waterspecific humidity are assumed to be constant with height in the boundarylayer. Secondly, turbulence entrains air across the inversion into the boundarylayer at a rate that is assumed to be proportional to the jump in radiative flux at the cloud top and inversely proportional to the jump in buoyancy at the inversion. The constant of proportionality is called the entrainment efficiency.Results from the model for the entrainment rate and height evolutionof the boundary layer are compared with the observations and modelsconsidered in a EUCREM intercomparison study. Thepresent mixed-layer model accurately predicts the observed heightevolution of the boundary layer, but over-estimates the entrainmentrate to a similar degree as the large-eddy simulations. We show that,if the subsidence rate is reduced to the value given by observationsrather than the value used in the EUCREM intercomparison study,then the model agrees well with observed value of the entrainment rateif the entrainment efficiency is taken to be 0.6. With this value, themodel also agrees well with a further case study byBechtold et al. An entrainment efficiency of 0.6 is a little higherthan suggested by large eddy simulations, but such simulations do notcurrently resolve the entrainment events explicitly. Hence this pointdeserves further study.     

Mixed-layer heat advection and entrainment during the sea breeze

A model is developed to simulate the potential temperature and the height of the mixed layer under advection conditions. It includes analytic expressions for the effects of mixed-layer conditions upwind of the interface between two different surfaces on the development of the mixed layer downwind from the interface. Model performance is evaluated against tethersonde data obtained on two summer days during sea breeze flow in Vancouver, Canada. It is found that the mixed-layer height and temperature over the ocean has a small but noticeable effect on the development of the mixed layer observed 10 km inland from the coast. For these two clear days, the subsidence velocity at the inversion base capping the mixed layer is estimated to be about 30 mm s^–1 from late morning to late afternoon. When the effects of subsidence are included in the model, the mixed-layer height is considerably underpredicted, while the prediction for the mean potential temperature in the mixed layer is considerably improved. Good predictions for both height and temperature can be obtained when values for the heat entrainment ratio,c, 0.44 and 0.68 for these two days respectively for the period from 1000 to 1300 LAT, were used. These values are estimated using an equation including the additional effects on heat entrainment due to the mechanical mixing caused by wind shear at the top of the mixed layer and surface friction. The contribution of wind shear to entrainment was equal to, or greater than, that from buoyant convection resulting from the surface heat flux. Strong wind shear occurred near the top of the mixed layer between the lower level inland flow and the return flow aloft in the sea breeze circulation.Symbols c entrainment parameter for sensible heat - c _p specific heat of air at constant pressure, 1010 J kg^–1 K^–1 - d ₁ the thickness of velocity shear at the mixed-layer top, m - Q _H surface sensible heat flux, W m^–2 - u _m mean mixed-layer wind speed, m s^–1 - u _* friction velocity at the surface, m s^–1 - w subsidence velocity, m s^–1 - W subsidence warming,^oC s^–1 - w _e entrainment velocity, m s^–1 - w _* convection velocity in the mixed layer, m s^–1 - x downwind horizontal distance from the water-land interface, m - y dummy variable forx, m - Z height above the surface, m - Z _i height of capping inversion, m - Z _m mixed-layer depth, i.e.,Z _i–Z_s, m - Z _s height of the surface layer, m - lapse rate of potential temperature aboveZ _i, K m^–1 - potential temperature step atZ _i, K - u _h velocity step change at the mixed-layer top - _m mean mixed-layer potential temperature, K     

大气混合层的数值研究

本文从干混合层模式数值试验所得各种平均要素,厚度以及其它物理量的变化分析讨论了混合层的一些基本物理特性。并利用Wangara资料进行一些比较和试验。本文目的试图建立一个合理的混合层预报模式,并探索一种适合于数值天气预报模式的边界层参数化方案。     

An efficient PBL model for global circulation models-design and validation

An efficient, pianetary boundary layer (PBL) model is developed and validated with empirical data for applications in general circulation models (GCMs). The purpose of this PBL model is to establish the turbulent surface fluxes as a function of the principal external PBL parameters in a numerically efficient way. It consists of a surface layer and a mixed layer matched together with the conditions of constant momentum and heat flux at the interface. An algebraic solution to the mean momentum equations describes the mixed-layer velocity profile and thus determines the surface wind vector. The velocity profile is globally valid by incorporating the effect of variable Coriolis force without becoming singular at the equator. Turbulent diffusion depends on atmospheric stability and is modeled in the surface layer by a drag law and with first-order closure in the mixed layer. Radiative cooling in the stably stratified PBL is considered in a simple manner. The coupled system is solved by an iterative method. In order to preserve the computational efficiency of the large-scale model, the PBL model is implemented into the GISS GCM by means of look-up tables with the bulk PBL Richardson number, PBL depth, neutral drag coefficient, and latitude as independent variables.A validation of the PBL model with observed data in the form of Rossby number similarity theory shows that the internal feedback mechanisms are represented correctly. The model, however, underpredicted the sensible heat-flux. A subsequent correction in the turbulence parameterization yields better agreement with the empirical data. The behavior of the principal internal PBL quantities is presented for a range of thermal stabilities and latitudes.     

Mixing Height Over Water And Its Role On The Correlation Between Temperature And Humidity Fluctuations In The Unstable Surface Layer

Results from an experimental investigation of themixing height over inner Danish waters carriedout from September 1990 to October 1992, are discussed.The statistical analysis of the mixed-layer height (z_i)over the sea does not exhibit the dailyvariation that is characteristic of the mixed layerover land, but it is nearly constant over a24-hour cycle. During summer, the mixed layer ishigher than during winter. A second inversionwas often observed.A case study of the development of the mixed layerover the sea under near-neutral and unstableatmospheric conditions during six consecutivedays is presented. A zero-order mixed-layer heightmodel is applied. In addition to momentum and heatfluxes the effect of subsidence was found to be importantfor the evolution of the mixed layer over the sea. Themodelled evolution of z_i compared successfullywith measurements.We have investigated the influence of themixed-layer height on the correlation coefficient R_qTbetween temperature and humidity fluctuations usingthe values obtained with the model.We found that the evolution of R_qT follows theevolution of the mixing height. An empirical modellinking the surface values of R_qT to z_i and the Obukhov scaling length L has been suggested. The modelreproduces the experimental features.     

A Simple Parametrization for the Concentration Variance Dissipation in a Lagrangian Single-Particle Model

A new quasi-analytical mixed-layer model is formulated describing the evolution of the convective atmospheric boundary layer (ABL) during cold-air outbreaks (CAO) over polar oceans downstream of the marginal sea-ice zones. The new model is superior to previous ones since it predicts not only temperature and mixed-layer height but also the height-averaged horizontal wind components. Results of the mixed-layer model are compared with dropsonde and aircraft observations carried out during several CAOs over the Fram Strait and also with results of a 3D non-hydrostatic (NH3D) model. It is shown that the mixed-layer model reproduces well the observed ABL height, temperature, low-level baroclinicity and its influence on the ABL wind speed. The mixed-layer model underestimates the observed ABL temperature only by about 10 %, most likely due to the neglect of condensation and subsidence. The comparison of the mixed-layer and NH3D model results shows good agreement with respect to wind speed including the formation of wind-speed maxima close to the ice edge. It is concluded that baroclinicity within the ABL governs the structure of the wind field while the baroclinicity above the ABL is important in reproducing the wind speed. It is shown that the baroclinicity in the ABL is strongest close to the ice edge and slowly decays further downwind. Analytical solutions demonstrate that the \(\mathrm{e}\)-folding distance of this decay is the same as for the decay of the difference between the surface temperature of open water and of the mixed-layer temperature. This distance characterizing cold-air mass transformation ranges from 450 to 850 km for high-latitude CAOs.     

Thickness of the dry convection and large-scale subsidence above deserts

Thickness of dry convection above various deserts of the world is obtained from aerological data, and assimilated data from ECMWF. A mixed layer develops up to a height of about 1 km above the central Sahara, where strong subsidence occurs. However, above many other deserts in Africa and Asia, a deep mixed layer develops up to 4–6 km. These mixed layers develop to a high altitude because the daytime mixed layer links with an existing weakly stratified, near-neutral layer above. Large-scale subsidence does not reach the surface throughout the day, except in the winter season. Mixed-layer height is shallower in the Southern Hemisphere than in the Northern Hemisphere in the summer season.     

Interactions between coherent eddies in the lower convective boundary layer

The processes of interaction between the atmospheric surface and mixed layers in daytime convective conditions over land are studied using a data set obtained during flights by an instrumented aircraft. Profiles of partitioned run-averaged statistics and examples of time series plots are discussed in the light of results from a recently published study by the authors, in which the average structure and flow within coherent eddies was reconstruced using a compositing technique. This evidence is used to support a conceptual model of the mechanisms of interaction between surface-layer plumes and mixed-layer thermal columns. The divergent flow created near the surface by the downdraft arms of the large-scale mixed-layer circulation patterns, forces the development of lines of convergence in the surface layer (the so-called thermal walls), which channel air into the bases of the mixed-layer thermals. Plumes progressively group and merge together with height in the surface and free convection layers, and move along these convergence lines toward large collector plumes at the intersection points, or hubs. Above the hubs are the thermals, and air parcels originating from plumes and their environment are strongly mixed as they rise, leading to an increased difficulty of the conditional sampling method to distinguish between them. The observed influence of mixed-layer convective processes far down into the surface layer, and the form of the averaged profiles, supports recent refinements of the theory of surface-layer structure suggested in Kader and Yaglom (1990).Notation CBL convective boundary layer - SL surface layer - FCL free convection layer - ML mixed layer     

Molecular dissipation of turbulent fluctuations in the convective mixed layer part I: Height variations of dissipation rates

During the Limagne and Beauce experiments, the INAG-IGN Aerocommander FL 280 aircraft made extensive ‘in situ’ measurements of turbulent fluctuations in diurnally evolving convective boundary layers. In this paper, these measurements were used to investigate characteristics of the molecular dissipation of turbulent fluctuations through the mixed layer and well into the overlying stable layer. The dimensionless dissipation rates of turbulent kinetic energy, temperature and humidity variances, and temperature-humidity covariance (ψ, ψ_θ, ψ _qand ψ _θq) were computed and their height variations analysed. The behaviour of the dissipation rate ψ was found to differ significantly from those observed for the other rates. In the lowest region of the mixed layer, ψ does not obey the local free convection prediction. Instead, it follows practically a relationship similar to the one established in the surface layer by Wyngaard et al. (1971). The dissipation rate ψ remains fairly constant in the bulk of the mixed layer (0.3 ≤ z/Z _i≤ 0.8) and shows a very rapid decrease above the inversion. These results confirm those reported previously from the Minnesota and Ashchurch data by Kaimal et al. (1976), Caughey and Palmer (1979), etc. The height variations for the other dissipation rates were found to obey, as expected, the (z/Z _i)^-4/3 decrease predicted under the local free convection similarity hypothesis in the lowest region of the mixed layer. This region extends to the height z/Z _i- 0.4, 0.1, and 0.3, respectively, for ψ_θ, ψ_q, and ψ_θq. Above these levels, the dissipation rates ψ_θ and ψ_q show, on average, a slight increase to reach peak-values near the mixed-layer top, while the ‘dissipation’ rate ψ _θqchanges sign from positive to negative around the height z/Z _i, - 0.7. These characteristics confirm the fact that the structures of temperature and humidity fluctuations are considerably affected by their entrainment-induced fluctuations. Therefore, an attempt has been made to non-dimensionalize the dissipation rates near the mixed-layer top with the interfacial scaling factors.     

Variability of the Mixed-Layer Height Over Mexico City

The diurnal and seasonal variability of the mixed-layer height in urban areas has implications for ground-level air pollution and the meteorological conditions. Measurements of the backscatter of light pulses with a commercial lidar system were performed for a continuous period of almost six years between 2011 and 2016 in the southern part of Mexico City. The profiles were temporally and vertically smoothed, clouds were filtered out, and the mixed-layer height was determined with an ad hoc treatment of both the filtered and unfiltered profiles. The results are in agreement when compared with values of mixed-layer height reconstructed from, (i) radiosonde data, and (ii) surface and vertical column densities of a trace gas. The daily maxima of the mean mixed-layer height reach values \(> 3\hbox { km}\) above ground level in the months of March–April, and are clearly lower (\(< 2.7\hbox { km}\)) during the colder months from September–December. Mean daily minima are typically observed at 0700 local time (UTC ? 6h), and are lowest during the winter months with values on average below 500 m. The data presented here show an anti-correlation between high-pollution episodes and the height of the mixed layer. The growth rate of the convective mixed-layer height has a seasonal behaviour, which is characterized together with the mixed-layer-height anomalies. A clear residual layer is evident from the backscattered signals recorded in days with specific atmospheric conditions, but also from the cloud-filtered mean diurnal profiles. The occasional presence of a residual layer results in an overestimation of the reported mixed-layer height during the night and early morning hours.     

Application of a mixed-layer model to bare soil surfaces

A model that couples the surface energy balance equation, a surface hydraulic resistance equation, and the force-restore soil temperature model to a mixed-layer model of the planetary boundary layer is described. The mixed layer is separated from the soil by a relatively thin surface layer and is overlain by a stable free atmosphere with prescribed profiles of potential temperature and water vapour density. The model is in reasonably good agreement with daytime micrometeorological measurements made at a wet bare site at Agassiz, British Columbia, and a desert site at Pampa de La Joya, Peru. The sensitivity of the mixed-layer model to conditions in the free atmosphere, to the parameters describing the growth of the mixed layer, and to surface roughness lengths, surface hydraulic resistance, and windspeed is examined.     

Air mass modification over Lake Michigan

The modification of a relatively cold air mass over the warm water of Lake Michigan is studied by using a two-dimensional nonlinear mesoscale model. Considerable amounts of heat and water vapor are supplied from the water surface to the lower atmosphere by turbulent eddies. A convective mixed layer develops and grows toward the downwind region with stratocumulus clouds over the lake.The model simulates the warming and moistening of the mixed layer, the development of a boundary layer, the divergence and convergence of wind near the coastlines, and the turbulent fluxes.The model warming of the mixed layer across the lake was about 2.2 °K and the moistening of the mixed layer was about 0.8 g kg^–1, which are comparable to 2.7 °K and 0.8 g kg^–1 observed by Lenschow (1973). The convective boundary layer, which includes the cloud layer, subcloud layer, and superadiabatic layer near the water surface, is well simulated. The tilt of the inversion which coincides with the cloud top is also well reproduced. When a prescribed cooling rate is applied at the cloud top, stronger turbulence and a deeper cloud layer are generated. Without the cooling, the cloud is shallow and the shape of the cloud base is determined by surface conditions. The rise of the inversion is due to upward vertical motion, and deepening of the convective layer in the downwind region.     

Amendment to “Analytical Solution for the Convectively-Mixed Atmospheric Boundary Layer”: Inclusion of Subsidence

In Ouwersloot and Vilà-Guerau de Arellano (Boundary-Layer Meteorol. doi:10.1007/s10546-013-9816-z, 2013, this issue), the analytical solutions for the boundary-layer height and scalar evolutions are derived for the convective boundary layer, based on the prognostic equations of mixed-layer slab models without taking subsidence into account. Here, we include and quantify the added effect of subsidence if the subsidence velocity scales linearly with height throughout the atmosphere. This enables analytical analyses for a wider range of observational cases. As a demonstration, the sensitivity of the boundary-layer height and the potential temperature jump to subsidence and the free tropospheric stability is graphically presented. The new relations show the importance of the temporal distribution of the surface buoyancy flux in determining the evolution if there is subsidence.