A mixed-layer model for regional evaporation

A simple mixed-layer model is developed to describe evaporation into a convective planetary boundary layer (PBL). The model comprises volume budget equations for temperature and humidity, equations to describe transport through the surface layer which is treated as part of the lower boundary, and equations to describe entrainment at the top of the PBL. The ground surface is modelled as a canopy resistance. The model was integrated with canopy resistance, surface-layer resistance and available energy, (R _n – G), input as given functions of time, and the simulated PBL was allowed to grow into an atmosphere with known temperature and humidity profiles.Two variants of the mixed-layer model were tested using data from the KNMI tower site at Cabauw in the Netherlands. These variants differed only in the formulation of entrainment: one used a formulation developed by Driedonks (1982) while the other was a simpler formulation. Simulated evaporation agreed very well with observations irrespective of which entrainment formulation was used, despite discrepancies between simulated and observed PBL height growth which were sometimes quite large for the simpler formulation. Sensitivity analysis of the model confirms that good PBL height-growth predictions are not always a prerequisite for good evaporation predictions.     

Subsidence fronts

Approximate analytical solution of the coupled momentum and heat transfer equations are obtained, using an integral method, for natural convection boundary layers which are produced by a step change of surface temperature. The accuracy of the laminar solution is shown to be good, especially at high Prandtl Number (Pr). The corresponding solutions for the turbulent fluid yield a model of a boundary layer in which the horizontal flow strengthens downstream and is fed by downward motion from above. We call this system, which could possibly simulate the initiation of a sea breeze circulation, a subsidence front. Interesting properties of the solution include the control of the boundary-layer development by the product (PrS), whereS is the slope of the boundary layer, and also the generation of log-linear velocity and temperature profiles by heat flux divergence.     

Numerical simulation of cold easterly circulations over the Canadian Western Plains using a mesoscale boundary-layer model

Arctic outbreaks over the Canadian Western Plains during the late spring period frequently take the form of a cold east-northeasterly flow over a warmer, sloping surface. A mesoscale numerical model is developed in an attempt to simulate such circulations. Following Lavoie (1972) the atmospheric structure of the cold air mass is represented by three layers: a constant flux layer in contact with the earth's surface, a well-mixed planetary boundary layer capped by an inversion, and a deep stratum of overlying stable air. Averaging the set of governing primitive equations through the depth of the mixed layer yields predictive equations for the horizontal wind components, potential temperature, specific humidity, and the height of the inversion. Time-dependent calculations are limited to this layer by parameterizing the interactions between the mixed layer and both the underlying and overlying layers. Precipitation from limited convective clouds, and latent heat within the layer are included in terms of mesoscale variables.A 47.6-km by 47.6-km grid mesh of 1369 points covering the Canadian Prairie Provinces is used to represent the variables. The governing equations are solved numerically with terrain influences, surface roughness, temperature variations, and moisture fluxes allowed to perturb the mixed layer from its initial conditions until resultant mesoscale boundary-layer weather patterns evolve.The mean spring topographic precipitation pattern is successfully reproduced by the simulated late spring upslope flow with limited convective precipitation. Mesoscale planetary boundary-layer weather patterns appear to exert a dominant control over the location and intensity of perturbations in the spring precipitation pattern. The elimination of surface heating significantly reduces the area and intensity of precipitation. A case study based on observed initial conditions showed that the model could reproduce a persistent limited convective precipitation pattern maintained by upslope flow and that a low-level trough exerts a marked influence on the location and the intensity of the precipitation.     

Numerical simulation of the development of the convective boundary layer during a cold air outbreak

During cold air outbreaks, cold and stably stratified air masses are advected from land or ice surfaces over a warmer sea surface. Due to the heating from below, a convective boundary layer develops. For small fetches, convection is organized in the form of horizontal roll vortices, which at greater distances join in a zone with open or closed cells. The formation of the convective boundary layer, and the associated roll vortices, is simulated with a numerical model and results are compared with observations obtained during the MASEX experiment off the east coast of the United States. To validate the model, a comparison with a one-dimensional mixed-layer model is also made, with special attention given to the exact representation of the observed initial and boundary conditions. Comparisons between model results and observations show good qualitative and quantitative correspondence in mean temperature and heat flux profiles respectively at different distances from the coast. Maximum values of vertical velocity are well reproduced. Turbulent kinetic energy is found to be concentrated in the small updraft regions of the rolls, which is in accordance with observations from the MASEX-experiment.     

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.     

Analysis of aircraft measurements of momentum flux in the subcloud layer over the tropical atlantic ocean during gate

The momentum flux data obtained by the gust probe aboard the NOAA DC-6 aircraft during GATE are analyzed. Vertical profiles are obtained for Phases I and III and correlated with vertical wind velocity profiles using the geostrophic departure method. Reasonable agreement is obtained using the horizontal equations of motion with negligible advective acceleration. The vertical profiles of momentum flux and wind speed variance compare well with the numerical model results of Deardorff (1972) and Wyngaard et al. (1974). Vertical distributions of power spectra for vertical eddy motion and cospectra corresponding to the momentum flux components are obtained along with the height variation of the dominant length scales of vertical eddy motion and the dissipation rate of turbulence kinetic energy. When normalized by mixed-layer similarity, these results agree well with previous determinations in the boundary layer over tropical oceans and over land.     

An ice breeze mechanism for boundary-layer jets

The existence of a low-level (z=~1000 m) jet adjacent to a sea-ice boundary is investigated with a two-dimensional numerical model. A thermally-direct ice breeze circulation is induced by specifying an ice-sea surface temperature gradient, with the mean geostrophic wind parallel to the ice edge. Pressure changes associated with over-water mixed-layer development create an increase in geostrophic velocity that accounts for most of the increase in wind speed. A change in initial geostrophic wind direction has significant effects on location and intensity of the low-level jet; geostrophic winds parallel to the ice edge result in stronger jets than occur with cross-ice geostrophic winds. An inertial oscillation simulated by the model in 1-D makes a negligible contribution to the low-level jet.     

人为热源和城市绿化对冬季山谷城市边界层结构影响的数值模拟

为了探讨人为热源和城市绿化对城市边界层的影响,本文在RAMS模式中引入周期性日变化的人为热源和人工改变下垫面状况,初步模拟分析了人为热源和城市绿化对城市边界层结构的影响.结果表明:人为热源具有明显的增温效应,增强了城市的湍流交换,白天增加了大气不稳定度,促进了混合层的发展,夜间降低了大气稳定度,减弱了城市夜间逆温.城市绿化减小了地表反照率,增加了到达地面的净辐射,模拟期间土壤冻结,增加的净辐射其中一部分以感热的形式来加热大气;绿化后地气之间的湍流交换增强,增加了大气不稳定度,减弱了白天高空逆温;本文还讨论了不同绿化布局对白天高空逆温的影响以及人为热源和城市绿化之间的非线性相互作用.     

The role of thermals in the convective boundary layer

Detailed measurements of the structure of thermals throughout the convective boundary layer were obtained from the NCAR Electra aircraft over the ocean during the Air Mass Transformation Experiment (AMTEX). Humidity was used as an indicator of thermals. The variables were first high-pass filtered with a 5 km cutoff digital filter to eliminate mesoscale variations. Segments of the 5 min (30 km length) horizontal flight legs with humidity greater than half the standard deviation of humidity fluctuations for that leg were defined as thermals. This was found to be a better indicator of thermals than temperature in the upper part of the boundary layer since the temperature in a thermal is cooler than its environment in the upper part of the boundary layer. Using mixed-layer scaling, the normalized length and number of thermals were found to scale with the 1/3 and -1/3 powers, respectively, of normalized height, while vertical velocity and temperature scaled according to similarity predictions in the free convection region of the surface layer. The observational results presented here extend throughout the entire mixed layer. Using these results in the equation for mean updraft velocity of a field of thermals, the sum of the vertical pressure gradient and edge-effect terms can be estimated. This residual term is found to be important throughout most of the boundary layer. The magnitude of the divergence of vertical velocity variance within a thermal is found to be larger than the magnitude of the mean updraft velocity term throughout most of the mixed layer.Part of this work was completed while visiting Risø National Laboratory, Denmark.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

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.     

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.     

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.     

An analytical study on the urban boundary layer

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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 embedded mixed-layer-ocean circulation model

The rationale and numerical technique of embedding an oceanic bulk mixed-layer model with a multi-level primitive equation model is presented. In addition to the usual prognostic variables that exist in a multi-level primitive equation model, the embedded model predicts the depth of the well-mixed layer as well as the jumps in temperature and velocity that occur at the base of that layer. The depth of the mixed layer need not coincide with any of the fixed-model levels used in the primitive equations calculations.In addition to advective changes, the mixed layer can deepen by entrainment and it can reform at a shallower depth in the absence of entrainment. When the mixed layer reforms at a shallower depth, the vertical profile of temperature below the new, shallower mixed layer is adjusted to fit the fixed-level structure used in the primitive equations calculations using a method which conserves heat, momentum and potential energy. Finally, a dynamic stability condition, which includes a consideration of both the vertical current shear and the vertical temperature gradient, is introduced in place of the traditional ‘convective adjustment’.A two-dimensional version of the model is used to test the embedded model formulations and to study the response of the ocean to a stationary axisymmetric hurricane. The model results indicate a strong interdependence between vertical turbulent mixing and advection of heat.     

森林下垫面陆面物理过程及局地气候效应的数值模拟试验

文中基于大气边界层和植被冠层微气象学基本原理 ,建立了一个森林植被效应的陆面物理过程和二维大气边界层数值模式。并应用该模式进行了植被和土壤含水量等生物和生理过程在陆面过程和局地气候效应方面的数值模拟试验。所得数值模拟试验结果与实际情况相吻合。结果表明 ,应用该模式可获得植被温度、植被冠层内空气温度、地表温度日变化特征 ;森林下垫面大气边界层风速、位温、比湿、湍流交换系数的时空分布和日变化特征。该模式还可应用于不同下垫面 ,模拟陆面物理过程与大气边界层相互作用机制及其局地气候效应的研究 ,这将为气候模式与生物圈的耦合研究奠定一个良好的基础。     

下垫面特性突变时的内边界层数值研究

本文建立了一个考虑层结的二维内边界层数值模式,研究了下垫面粗糙度、温度、湿度突变引起的内边界层。结果表明:下垫面温度突变引起的温度内边界层与下垫面粗糙度突变引起的速度内边界层具有明显的差异。     

Stratified flow over non-uniform surface conditions: Mixing-length model

The response of a neutrally stratified atmospheric surface layer to sudden changes in surface roughness and temperature is investigated by solving the steady-state boundary-layer equations. Near the discontinuities, the distributions of velocity and shear stress are found to be critically dependent upon whether or not the pressure and buoyancy terms are retained in the governing equations. Computed distributions of velocity and shear stress are found to be in good agreement with windtunnel and field data. The growth of the internal boundary layer under neutral conditions for smooth to rough transitions follows the 0.8 power law; however, rough to smooth transitions are associated with a 0.7 power law.     

Feedbacks in the Land-Surface and Mixed-Layer Energy Budgets

A mixed-layer model of the surface energy budget and the planetary boundary layer (PBL) is developed, based on the prognostic equations for soil temperature, mixed layer potential temperature and specific humidity and the growth and abrupt collapse of the PBL. Detailed parameterizations of the longwave radiative fluxes are included. The feedbacks in the uncoupled (i.e. surface energy budget with non-responding PBL) and coupled land surface and atmospheric mixed-layer energy budgets are examined. A simplified, time continuous, version of the model, in which the specific humidity budget is the balance of evapotranspiration and dry-air entrainment, and the PBL height is given by the lifted condensation level, is shown to be in good agreement with the complete model. By forcing the simplified model with daily mean rather than periodic solar radiation, an equilibrium model state is achieved where the fluxes are in close agreement with the daily mean fluxes corresponding to the periodic forcing. The model also agrees favorably with measurements from the FIFE field experiment. Feedbacks are examined using the equilibrium model state. The uncoupled and coupled model sensitivities with respect to the minimal stomatal resistance and the atmospheric specific humidity not only differ in magnitude, but in sign as well. This results puts into question the extent to which uncoupled land-surface models that are forced with atmospheric variables may be used in sensitivity studies.