Effects of turbulent dispersion of atmospheric balance motions of planetary boundary layer

New Reynolds' mean momentum equations including both turbulent viscosity and dispersion are used to analyze atmospheric balance motions of the planetary boundary layer. It is pointed out that turbulent dispersion with r 0 will increase depth of Ekman layer, reduce wind velocity in Ekman layer and produce a more satisfactory Ekman spiral lines fit the observed wind hodograph. The wind profile in the surface layer including tur-bulent dispersion is still logarithmic but the von Karman constant k is replaced by k1 = 1 -2/k, the wind increasesa little more rapidly with height.     

Stability and wind shear effects on meso-scale cellular convection

A three-dimensional numerical model has been used to assess the effects of vertical stability and wind shear on the nature and form of meso-scale cellular convection (MCC). The model was shown to be capable of simulating a real occasion of MCC before it was used in idealised cases. These cases revealed different regimes in MCC: open cells, longitudinal bands and closed cells/transverse bands. Open cells were favoured by the existence of instability in the surface layer and a lack of wind shear in the Ekman layer. Longitudinal bands were favoured by similar conditions in the surface layer plus wind shear in the Ekman layer. A near-neutral surface layer favoured the occurrence of closed cells/transverse bands. The depth of convection in the longitudinal bands was a function of the stability in both the surface and Ekman layers and of the wind shear in the Ekman layer. The regimes are related to the instability and shear through bulk Richardson numbers in the surface and Ekman layers.     

BAROCLINITY AND NONLINEAR EKMAN PUMPING DYNAMICS

With the Ekman momentum approximation,the influence of atmospheric baroclinity on the dynamics of boundarylayer is studied.Some new results are obtained.These results show that the atmospheric baroclinity plays an importantrole in altering the horizontal velocity of Ekman boundary layer and its angle with the horizontal wind velocity compo-nent near the surface.There are three different physical factors affecting the nonlinear Ekman suction,the vertical mo-tion at the top of boundary layer:first,barotropic geostrophic relative vorticity at the ground;second,the thermal windvorticity induced by the baroclinity;and third,the nonlinear interaction between the barotropic geostrophic relativevorticity and the baroclinic thermal wind vorticity.These results may provide a better physical basis for theparameterization of boundary layer and the interpretation of the numerical modeling results.     

边界层动力学中的Ekman动量近似

自由大气中,大气运动的基本状态是地转风,近年来发展的地转动量近似,是为了进一步研究非均匀地转流的动力学问题,然而,在边界层大气中,运动的基本状态是经典的Ekman流,所以对边界层运动来说,地转动量近似是不合适的,需作一推广。本文提出了一种所谓Ekman动量近似,它相似于自由大气中的地转动量近似,并讨论了Ekman动量近似的物理基础,对边界层的风场结构及边界屋顶部的垂直速度也作了详细分析。     

Ekman动量近似下中间边界层模式中的风场结构

发展了一个准三维的、中等复杂的边界层动力学模式，该模式包含了EKman动量近似下的惯性加速度和Blackadar的非线性湍流粘性系数，它进一步改进了Tan和Wu(1993）提出的边界层理论模型。该模型在数值计算复杂性上与经典Ekman模式相类似，但由于包含了Ekman动量近似下的惯性项，使得该模式比传统Ekman模式更近于实际过程。中详细地比较了该模式与其他简化边界层模式在动力学上的差异，结果表明：在经典的Ekman模式中，由于忽略了流动的惯性项作用，导致在气旋性切变气流（反气旋性切变气流）中风速和边界层顶部的垂直速度的高估（低估），而在半地转边界层模式中，由于高估了流动惯性项的作用，结果与经典Ekman模式相反。同样，该模式可以应用于斜压边界层，对于Ekman动量下的斜压边界层风场同时具有经典斜压边界层和Ekman动量近似边界层的特征。     

An Analytical Study of the Diurnal Variations of Wind in a Semi-geostrophic Ekman Boundary Layer Model

A time-dependent semi-geostrophic Ekman boundary-layer model based on the geostrophic momentum approximation is used to study the diurnal wind variation in the planetary boundary layer (PBL) and the evolution of the low-level nocturnal jet (LLJ). The coefficient of eddy viscosity varies periodically with time, varies linearly with height in the surface layer and is constant above the surface layer. The influence of horizontal advection of momentum on the diurnal wind variation in the PBL, the development of inertial oscillations (IOs) and the formation of the LLJ are examined.In comparison with the Ekman solutions, the diurnal wind variation in semi-geostrophic Ekman boundary-layer dynamics has the following features: (1) the phase angle of the diurnal wind wave shifts with height, the rate of shifting is increased in anticyclonic regions and decreased in cyclonic regions, (2) the time of occurrence of the low-level maximum wind speed is later in anticyclonic regions and earlier in cyclonic regions, (3) the height of occurrence of the maximum wind speed is higher in the anticyclonic and lower in cyclonic regions, (4) the wind speed maximum and the amplitude of the diurnal wind variation are larger in anticyclonic and smaller in cyclonic regions, (5) the period of IOs is larger in anticyclonic regions and smaller in cyclonic regions, (6) anticyclonic vorticity is conducive to the generation of LLJ in the PBL. These features are interpreted by means of the physical properties of semi-geostrophic Ekman boundary-layer dynamics and inertial oscillation dynamics.     

斜压性与非线性Ekman抽吸的动力特征

本文利用Ekman动量近似研究了斜压性对Ekman层动力学的影响,得到了一些新的结果。大气斜压性对Ekman层的水平风速分布及近地面的风速矢的水平分量夹角有重要的改变作用。斜压边界层顶部的非线性Ekman抽吸(垂直运动)由三个不同的物理因子决定,第一、正压性的地面地转涡度,第二、斜压性作用产生的热成风涡度,第三、正压性的地面地转涡度与斜压性的热成风涡度的非线性相互作用。这些理论结果为边界层的参数化及数值模拟结果的解释提供物理基础。     

Vorticity,divergence, and vertical velocity in a baroclinic boundary layer with a linear variation of the geostrophic wind

The Ekman-Taylor problem for the planetary boundary layer is solved in the case of a thermal wind which varies linearly with height. The upper boundary condition is a vanishing ageostrophic wind, while the lower boundary condition is continuity of the stress vector across the interface between the planetary boundary layer and the surface layer. The latter condition is used to determine the magnitude and the direction of the wind at the bottom of the Ekman layer.Theoretical hodographs are compared with observed hodographs based on five years of ohservations from Ship N in the Pacific, giving fair agreement.The divergence, the vorticity, and the vertical velocity are calculated through the Ekman layer with emphasis on differences between the classical barotropic and the baroclinic cases; these differences are significant, especially in the vertical velocities as compared to the standard approximation.An extension of the present study to include thermal stratification is desirable.     

An Approximate Analytical Solution For The Baroclinic And Variable Eddy Diffusivity Semi-Geostrophic Ekman Boundary Layer

The WKB method has been used to develop an approximate solutionof the semi-geostrophic Ekman boundary layer with height-dependenteddy viscosity and a baroclinic pressure field. The approximate solutionretains the same simple form as the classical Ekman solution. Behavioursof the approximate solution are discussed for different eddy viscosityand the pressure systems. These features show that wind structure inthe semi-geostrophic Ekman boundary layer depends on the interactionbetween the inertial acceleration, variable eddy viscosity and baroclinicpressure gradient. Anticyclonic shear has an acceleration effect on theair motion in the boundary layer, while cyclonic shear has a decelerationeffect. Decreasing pressure gradient with height results in a super-geostrophicpeak in the wind speed profile, however the increasing pressure gradient withheight may remove the peak. Anticyclonic shear and decreasing the variableeddy viscosity with height has an enhanced effect on the peak.Variable eddy viscosity and inertial acceleration has an important role in thedivergence and vorticity in the boundary layer and the vertical motion at the top of the boundary layer that is called Ekman pumping. Compared to the constanteddy viscosity case, the variable eddy diffusivity reduces the absolute value ofEkman pumping, especially in the case of eddy viscosity initially increasing with height. The difference in the Ekman pumping produced by different eddy diffusivity assumptions is intensified in anticyclonic flow and reduced in cyclonic flow.     

THE ADJUSTMENT OF WIND TO EKMAN FLOW WITHIN THE PLANETARY BOUNDARY LAYER

By using simple barotropic boundary layer equations with constant eddy viscosity,the analytical solution is obtained under the initial condition that the distribution of wind for a given pressure is not the well-known Ekman flow.We have found that the wind will finally adjust to the Ekman flow at a rate faster than that of geostrophic adjustment.We have also found that the thinner the boundary layer,the faster the rate of adjustment.     

经典Ekman流,Ekman层的平衡风场及其扰动稳定性

本文首先利用变分方法,考察了边界层运动能量的变化,指出经典Ekman流是在不可压缩条件下,能量积分达最小值时的一种平衡运动。这对Ekman层运动的物理本质有了进一步的认识。其次,讨论了Ekman动量近似下的Ekman层的平衡风场特征,研究了该平衡风场附近扰动的稳定性问题,结果表明,在自由大气气压场不发生扰动条件下,Ekman层中存在一类新的与惯性不稳定相类似的动力不稳定,且其不稳定性可与Ekman抽吸相联系,还讨论了一般性条件下的扰动不稳定性问题。     

The wind field in the nonlinear multilayer planetary boundary layer

By use of the small parameter expansion method, the nonlinear planetary boundary layer (PBL) is studied in this paper. The PBL is divided into the surface layer and the Ekman layer, which is divided into several sublayers. In the surface-layer, the eddy coefficient K is taken as a linear function of height; in the Ekman layer, different constant K values are taken within different sublayers: these values are determined from O'Brien's formula (O'Brien, 1970) approximately. Under the upper and lower boundary conditions and the continuity conditions of the wind velocities and turbulent stresses at each boundary between sublayers, analytical expressions for wind velocity in all sublayers and the vertical velocity at the top of the PBL are obtained. A specific example of steady axisymmetrical circular high and low pressure areas is analysed, and some new conclusions are obtained. The results are in better agreement with reality than previous results. This example also shows that the vertical velocity at the top of the PBL caused by friction approaches zero near the center of a high or low pressure system for this model, but attains its maximum absolute values near the center of the high or low pressure area for Wu's (1984) model. This is due to the fact that in our model, the geostrophic wind speed near the center of this specific vortex approaches zero, which causes the wind shear and the friction effect to be very weak. Therefore the wind distribution in the PBL is very sensitive to the type of eddy coefficient.     

The wind in the baroclinic boundary layer with three sublayers incorporating the weak non-linear effect

In considering the weak non-linear effect, and using the small parameter expansion method, the analyt-ical expressions of the wind distribution within PBL (planetary boundary layer) and the vertical velocity at the top of the PBL are obtained when the PBL is divided into three layers and different eddy transfer coefficients K are adopted for the three layers. The conditions of barotropy and neutrality for the PBL are extended to that of baroclinity and non-neutral stratification. An example of a steady circular vortex is used to display the characteristics of the horizontal wind within the PBL and the vertical velocity at the top of the PBL. Some new results have been obtained, indicating that the magnitude of the speed in the lower height calculated by the present model is larger than that by the model in which k is a constant within the whole boundary layer, for example, in the classical Ekman boundary layer model and the model by Wu (1984). The angle between the wind at the top of the PBL and the wind near the surface calculated by the present model is less than that calculated by the single K model. These results are in agreement with the observations.     

The influences of orography upon the flow within Ekman boundary layer under the approximation of geostrophic momentum

In this paper, the influences of orography on the boundary layer flow with the approximation of geostrophic momentum are studied. The wind velocity at the lower boundary will not always be zero when the orography exists. So the structure of the boundary layer flow, as well as the vertical velocity at the top of the boundary layer, is affected. There are three factors affecting the vertical motion at the top of the boundary layer: lifting due to orography; divergence due to Ekman flow, and advection of the geostrophic momentum. These effects and the features of the flow within the boundary layer are discussed in detail.     

变化涡动沾性系数的Ekman动量近似模式风场结构

在边界层动力学中，涡动粘性系数是影响边界层风场结构的一个重要参数。本文利用边界层动力学中的Ekman动量近似理论，给出了涡动粘性系数随高度缓变条件下的Ekman动量近似边界层模式解，着重讨论了边界层的风场结构、水平散度、垂直涡度以及边界层顶部的垂直速度。结果分析表明：与常值涡动粘性系数情况相比，在边界层低层随高度增加的涡动粘性系数可以导致低层边界层风速随高度迅速增加，即风速垂直切变增加，同时风速矢与地转风之间的夹角减小。惯性项作用可以导致上述作用在气旋性区域减小、而在反气旋性区域增大。随高度增加的涡动粘性系数导致水平散度绝对值、垂直涡度绝对值以及边界层顶部的垂直速度绝对值在气旋性区域减小，而在反气性旋区域增大。涡动粘性系数与惯性之间的非线性相互作用是边界层动力学中重要过程。     

Similarity theory and resistance laws for the planetary boundary layer

Methods are developed for the determination of parameters of the atmospheric planetary boundary layer, within the framework of similarity theory based on the external parameters — wind velocity at the upper boundary of the layer, its thickness, air temperature difference between the upper and the lower boundaries, roughness of the underlying surface, and buoyancy forces. The form of the resistance laws is discussed. Determination of the thickness of the stationary and horizontally homogeneous (Ekman) boundary layer is analyzed and generalizations of the latter are suggested for non-stationary and inhomogeneous boundary layers.     

Turbulent Helicity in the Atmospheric Boundary Layer

We consider the assumption postulated by Deusebio and Lindborg (J Fluid Mech 755:654–671, 2014) that the helicity injected into the Ekman boundary layer undergoes a cascade, with preservation of its sign (right- or alternatively left-handedness), which is a signature of the system rotation, from large to small scales, down to the Kolmogorov microscale of turbulence. At the same time, recent direct field measurements of turbulent helicity in the steppe region of southern Russia near Tsimlyansk Reservoir show the opposite sign of helicity from that expected. A possible explanation for this phenomenon may be the joint action of different scales of atmospheric flows within the boundary layer, including the sea-breeze circulation over the test site. In this regard, we consider a superposition of the classic Ekman spiral solution and Prandtl’s jet-like slope-wind profile to describe the planetary boundary-layer wind structure. The latter solution mimics a hydrostatic shallow breeze circulation over a non-uniformly heated surface. A 180°-wide sector on the hodograph plane exists, within which the relative orientation of the Ekman and Prandtl velocity profiles favours the left rotation with height of the resulting wind velocity vector in the lowermost part of the boundary layer. This explains the negative (left-handed) helicity cascade toward small-scale turbulent motions, which agrees with the direct field measurements of turbulent helicity in Tsimlyansk. A simple turbulent relaxation model is proposed that explains the measured positive values of the relatively minor contribution to turbulent helicity from the vertical components of velocity and vorticity.     

地形与Ekman边界层中的气流

利用σ坐标讨论地形与边界层气流是有很多方便的地方,因为,在此坐标中,下边界条件较为简单。在本工作中,首先将混合长理论加以推广并将它用于σ坐标,于是导得了用以描述地形上空边界层气流的控制方程,对边界层气流的特征,特别是对于Ekman抽吸作用进行了详细分析。指出有三种因子影响边界层顶部的垂直运动,第一种因子是边界层内涡度分布,这是与边界层中由于摩擦作用所引起的辐合辐散有直接联系;第二种因子是由于边界层顶部的气流爬坡运动所引起的;第三种是由于边界层中跨越等压线的分量爬坡所引起的,它出现于当等压线与地形等高线相平行时,或地转风呈现绕流情况时,这一作用最为明显。     

Observations of the barotropic Ekman layer over an urban terrain

Data from low-level soundings over Cambridge, U.S.A. were selected on the basis of an Ekman-like variation of the wind vector with altitude combined with evidence of a barotropic atmosphere. The method of geostrophic departure was used to determine the shear-stress distribution. The analysis yields the dimensionless properties of the barotropic Ekman layer under neutral and stable stratification. Some important results include: the geostrophic drag coefficient displays no dependence on the degree of static stability; the dimensionless height of the boundary layer decreases with increasing stability in agreement with the prediction of Zilitinkevich; the properties of the urban surface layer, where the roughness elements are multistory buildings, show no dependence on atmospheric stability under the moderate wind conditions which display the Ekman-like wind profile; and the directions of the horizontal shear stress and the vertical derivative of the velocity vector usually tend to be parallel only near the surface layer. Values of the two constants of the Rossby number similarity theory are found for the neutral barotropic Ekman layer at a surface Rossby number equal to 2 × 10⁵. The implications of the work with respect to wind-tunnel simulation of the flow over models of urban areas are discussed.     

An Improved Ekman Layer Approximation for Smooth Eddy Diffusivity Profiles

The Ekman boundary-layer model is extended analytically for a gradually varying eddy diffusivity K(z) ≥ 0, z ≥ 0. A solution for the Ekman layer is provided having similar structure to the constant-K case; that is, exponentially decaying sine functions for the two horizontal wind components. The analytical asymptotic solution compares well with its numerical counterpart for various K(z). The result can be useful in theoretical studies such as Ekman pumping, for efficient estimation of the Ekman layer profiles in various analyses with near-neutral stratifications, or for a rapid initialization of mesoscale models.