An Investigation of Higher-Order Closure Models for a Forested Canopy

Simultaneous triaxial sonic anemometer velocity measurements vertically arrayed at six levels within and above a uniform pine forest were used to examine two parameterization schemes for the triple-velocity correlation tensor employed in higher-order closure models. These parameterizations are the gradient-diffusion approximation typically used in second-order closure models, and the full budget for the triple-velocity correlation tensor typically employed in third-order closure models. Both second- and third-order closure models failed to reproduce the measured profiles of the triple-velocity correlation within and above the canopy. However, the Reynolds stress tensor profiles (including velocity variances) deviated greatly from the measurements only within the lower levels of the canopy. It is shown that the Reynolds stresses are most sensitive to the parameterization of the triple-velocity correlation in these lower canopy regions where local turbulent production is negligible and turbulence is mainly sustained by the flux transport term. The failure of the third-order closure model to reproduce the measured third moments in the upper layers of the canopy-top contradicts conclusions from a previous study over shorter vegetation but agrees with another study for a deciduous forest. Whether the third-order closure model failure is due to the zero-fourth-cumulant closure approximation is therefore considered. Comparisons between measured and predicted quadruple velocity correlations suggest that the zero-fourth-cumulant approximation is valid close to the canopy-atmosphere in agreement with recent experiments.     

The ejection-sweep cycle over bare and forested gentle hills: a laboratory experiment

Progress on practical problems such as quantifying gene flow and seed dispersal by wind or turbulent fluxes over nonflat terrain now demands fundamental understanding of how topography modulates the basic properties of turbulence. In particular, the modulation by hilly terrain of the ejection-sweep cycle, which is the main coherent motion responsible for much of the turbulent transport, remains a problem that has received surprisingly little theoretical and experimental attention. Here, we investigate how boundary conditions, including canopy and gentle topography, alter the properties of the ejection-sweep cycle and whether it is possible to quantify their combined impact using simplified models. Towards this goal, we conducted two new flume experiments that explore the higher-order turbulence statistics above a train of gentle hills. The first set of experiments was conducted over a bare surface while the second set of experiments was conducted over a modelled vegetated surface composed of densely arrayed rods. Using these data, the connections between the ejection-sweep cycle and the higher-order turbulence statistics across various positions above the hill surface were investigated. We showed that ejections dominate momentum transfer for both surface covers at the top of the inner layer. However, within the canopy and near the canopy top, sweeps dominate momentum transfer irrespective of the longitudinal position; ejections remain the dominant momentum transfer mode in the whole inner region over the bare surface. These findings were well reproduced using an incomplete cumulant expansion and the measured profiles of the second moments of the flow. This result was possible because the variability in the flux-transport terms, needed in the incomplete cumulant expansion, was shown to be well modelled using “local” gradient-diffusion principles. This result suggests that, in the inner layer, the higher-order turbulence statistics appear to be much more impacted by their relaxation history towards equilibrium rather than the advection-distortion history from the mean flow. Hence, we showed that it is possible to explore how various boundary conditions, including canopy and topography, alter the properties of the ejection-sweep cycle by quantifying their impact on the gradients of the second moments only. Implications for modelling turbulence using Reynolds-averaged Navier Stokes equations and plausible definitions for the canopy sublayer depth are briefly discussed.     

Z-Less stratification under stable conditions

Turbulent fluctuations were measured at a height of 2.5 m in stable conditions over grass to investigate the variability of the second- and third-order moments involving temperature, humidity and vertical wind velocity. With the exception of the normalized second moment of temperature, very little variation of the normalized moments was found with changes in the dimensionless stability parameter =z/L, whereL is the Obukhov stability length. Such limited variation is expected for stable conditions, and the normalized second moment of temperature might have been affected by nonstationary conditions. In addition, the variability of the normalized moments was lessened by computing the turbulence statistics over 56 min, instead of 26 min. Values of third-order moments involving the vertical velocity were all close to zero.     

Numerical experiments with a one-dimensional higher order turbulence model: Simulation of the Wangara Day 33 case

A one-dimensional stratocumulus model is developed and incorporated into a cloud/mesoscale model to simulate the evolution of the marine stratocumulus-capped mixed layer. This paper describes the formulation of the higher-order turbulence model. In a companion paper (Chen and Cotton, 1983), the formulation of the atmospheric radiation model, the partial condensation and the cloud fractional parameterization are described. The second-order moments of this model are partially diagnosed. In order to close the system, the parameterization for the third-order moments given by Zeman and Lumley (1976) is adopted and is generalized to include total water and cloud water. A new scheme to parameterize the skewness terms is proposed in order to satisfy the enforced realizability. Those skewness terms are used to close the third-order moments. In this paper, experiments are carried out to test the turbulence model by using the Wangara Day 33 data, which represents a ‘dry’ case study. Sensitivity experiments using the turbulence length scale parameterizations formulated by Andréet al. (1978) and Sun and Ogura (1980) are reformed and are compared.     

Instability in Lagrangian stochastic trajectory models,and a method for its cure

A number of authors have reported the problem of unrealistic velocities (“rogue trajectories”) when computing the paths of particles in a turbulent flow using modern Lagrangian stochastic (LS) models, and have resorted to ad hoc interventions. We suggest that this problem stems from two causes: (1) unstable modes that are intrinsic to the dynamical system constituted by the generalized Langevin equations, and whose actual triggering (expression) is conditional on the fields of the mean velocity and Reynolds stress tensor and is liable to occur in complex, disturbed flows (which, if computational, will also be imperfect and discontinuous); and, (2) the “stiffness” of the generalized Langevin equations, which implies that the simple stochastic generalization of the Euler scheme usually used to integrate these equations is not sufficient to keep round-off errors under control. These two causes are connected, with the first cause (dynamical instability) exacerbating the second (numerical instability); removing the first cause does not necessarily correct the second, and vice versa. To overcome this problem, we introduce a fractional-step integration scheme that splits the velocity increment into contributions that are linear (U _i ) and nonlinear (U _i U _j ) in the Lagrangian velocity fluctuation vector U, the nonlinear contribution being further split into its diagonal and off-diagonal parts. The linear contribution and the diagonal part of the nonlinear contribution to the solution are computed exactly (analytically) over a finite timestep Δt, allowing any dynamical instabilities in the system to be diagnosed and removed, and circumventing the numerical instability that can potentially result in integrating stiff equations using the commonly applied explicit Euler scheme. We contrast results using this and the primitive Euler integration scheme for computed trajectories in a drastically inhomogeneous urban canopy flow.     

A Parameterization of Third-order Moments for the Dry Convective Boundary Layer

We describe one-dimensional (1D) simulations of the countergradient zone of mean potential temperature observed in the convective boundary layer (CBL). The method takes into account the third-order moments (TOMs) in a turbulent scheme of relatively low order, using the turbulent kinetic energy equation but without prognostic equations for other second-order moments. The countergradient term is formally linked to the third-order moments and , and a simple parameterization of these TOMs is proposed. It is validated for several cases of a dry CBL, using large-eddy simulations that have been realized from the MESO-NH model. The analysis of the simulations shows that TOMs are responsible for the inversion of the sign of in the higher part of the CBL, and budget analysis shows that the main terms responsible for turbulent fluxes and variances are now well reproduced.     

A Re-Evaluation of Long-Term Flux Measurement Techniques Part II: Coordinate Systems

To convert measurements of windspeed, eddy flux and scalar concentration into estimates of surface scalar exchange, we implicitly or explicitly assimilate the measurements into mathematical statements of the mass balance in a control volume on a representative patch of the surface. The form of this statement depends on the coordinate system in which it is written and the coordinate system should be chosen so that measurements can be used optimally. This requirement imposes a set of conditions on the coordinates. Here we perform a comparative analysis of some candidate coordinate systems, concentrating on the Cartesian and physical streamline systems. We show that over gentle topography there are definite advantages in working in streamline coordinates. Transforming measurements of vector and tensor quantities measured in the reference frame, s _i, of the anemometer into the reference frame, e _i, of the chosen coordinate system involves using the measured statistics of the wind field to define three Euler rotation angles. We compare the method in most common use, which employs the components of the mean wind vector and the Reynolds stress tensor to define these angles, with the more recent planar-fit method that uses instead an ensemble of mean wind vectors to define the rotations. We find that, in real flows, the standard method has a previously unrecognized closure problem that ensures that the third rotation angle defined using the stress tensor or scalar flux vector will always be in error and often give unphysical results. An alternative procedure is recommended. Finally, the relationships between measurements and model outputs are discussed.     

Modelling Boundary-Layer Clouds With a Statistical Cloud Scheme and a Second-Order Turbulence Closure

We present a second-order turbulence model for the cloudy planetary boundary layer (PBL), which includes a statistical scheme of the sub-grid scale condensation. The model contains prognostic equations for the turbulent kinetic energy, total water, and liquid water temperature, the latter two being assumed to be conservative variables. Using these conservative thermodynamic variables the condensation process is formulated as a function of the departure of the total water from saturation and its variance. The computation of the variance requires second moment correlations which are modelled through the parameterization of the third-order moments using a convective mass-flux formulation. The inclusion of these third moments and new assumptions on heat flux transport lead to a nonlocal turbulence scheme with counter-gradient effects. The final form for the heat flux turns out to be a linearized version of a previously established result. For the statistical cloud formulation, a linear combination of a Gaussian and a positively skewed distribution function is used with a modified liquid water flux expression to account fornon-Gaussian behaviour.The effect of the turbulence scheme on the boundary-layer cloud structure is discussed and the performance of the model is tested by comparing it against the large eddy simulation (LES) of the undisturbed period of the Atlantic Stratocumulus Transition Experiment (ASTEX). The model is able to produce both mean and turbulent quantities that are in reasonable agreement with the LES output of ASTEX.     

Numerical Simulation of Flow over Two-Dimensional Hills Using a Second-Order Turbulence Closure Model

We study turbulent flow over two-dimensional hills. The Reynolds stresses are represented by a second-order closure model, where advection, diffusion, production and dissipation processes are all accounted for. We solve a full set of primitive non-hydrostatic dynamic equations for mean flow quantities using a finite-difference numerical method. The model predictions for the mean velocity and Reynolds stresses are compared with the measured data from a wind-tunnel experiment that simulates the atmospheric boundary layer. The agreement is good. The performance of the second-order closure model is also compared withthat of lower level turbulence models, including the eddy-viscositymodel and algebraic Reynolds stress models. It is concluded that thepresent closure is a considerable improvement over the other modelsin representing various physical effects in flow over hills. Thefeasibility of running a finite-difference numerical simulationincorporating a full second-order closure model on an IBM workstationis also demonstrated.     

Detection of Flow Distortions and Systematic Errors in Sonic Anemometry Using the Planar Fit Method

We describe the coordinate transformations that can be used to convert the velocity components measured by a set of sonic anemometers with time-dependent tilt fluctuations into a single, time-independent coordinate system. By applying the planar fit method (PFM) to each anemometer dataset, it is possible, for planar flows, to locate the flow plane at each measurement point and compare its orientation with the topography. Installation on a ship is also considered. An application of this method to intercomparison data has led to the detection of an instrument error due to a misalignment between the assembly of the sonic transducers and the anemometer pedestal. If this error occurs, pedestal levelling does not guarantee that measurements are unbiased. A correction method is proposed and the results of two experiments are shown. Flow planarity at different levels and flow distortion caused by the mast are highlighted. The influence of the error on the evaluation of the Reynolds stresses using PFM or the double rotation method and the triple rotation method is discussed and the tilt corrected stresses calculated using the three methods compared.     

REYNOLDS EXCHANGE IN THE ATMOSPHERIC MULTI-SCALE MOTIONS

In this paper the system of Reynolds equations of the multi-scaled atmospheric motions is set up based on the con-cept of decomposing the meteorological elements into multi-scale disturbances.It is proved to be true that the Reynoldsexchange term in the averaged motion is equal to the sum of averaged nonlinear terms in all sub-averaged motions.Inorder to avoid the higher order closure in Eulerian approaches,a new K-theory based on the multi-scaled Reynoldsequations is given in which the subscale motions are described by Langevin equation as the air particles are moving inthe Eulerian average background.From the new K-theory are derived the momentum,heat and mass exchangecoefficients as the functions of statistical variables such as variances and Lagrangian time scales of velocity,temperatureand other meteorological elements in disturbances.The new K-theory also expounds the causes for the differences be-tween the exchange coefficients of one element and another and gives the ambient conditions in which the buoyancyand/or Coriolis force Will build the chaotic disturbances into the orderly gradient of mean values of the correspondingelements.In consequence the K-theory can be used to explain some of negative viscosity phenomena in atmospheric mo-tions.     

关于不同尺度大气运动中的雷诺交换

本文从多尺度分解概念出发,建立了多尺度Reynolds方程组.证明了平均运动的Reynolds交换项应为各级子平均运动非线性项的平均和.文中将子尺度运动处理为描述质点个别运动的Langevin形式以避免Euler方式导致的高阶闭合困难,简单地得到了K闭合表示,给出了动量、热量、质点交换系数的表达式并阐明了它们之间差异的形成原因.此外本文还给出了表现无序脉动量在导向力(如浮力、科氏力)作用下建立有序平均量梯度的负K值及其存在条件,解释了大气中大尺度运动的一些负粘性现象.     

Incorporating topography into the multiscale systems for the atmosphere and oceans

Recently, new hyperbolic systems of equations that can be used to describe smooth flows accurately in both the atmosphere and oceans have been developed. These ‘approximate systems’ are derived by slowing down the speed of the fast waves instead of increasing their speed to infinity as in the primitive equations. The approximate systems have a number of theoretical advantages over the traditional systems. The practical implications of some of these advantages have already been demonstrated for the oceanic case. There is another advantage of the new systems that has not been discussed extensively. A model based on either of the new systems can be used to describe different scales of motion, e.g. the large, medium, or small scale. In addition, a mechanism is provided for a smooth transition between these scales. The incorporation of topography into the approximate systems has also not been discussed. To demonstrate the multiscale nature of the transformed systems in the presence of topography, numerical results from a model based on the approximate system for meteorology are compared with analytic solutions for three topographic scales. Removing the horizontal means of the density and pressure, which was necessary to obtain the proper scaling of the equations in the original papers, reduces the truncation error associated with a transformed system near steep mountains. For example, in the atmospheric case a second-order method requires only approximately 10 points across the base of the mountain to achieve a 1% relative error for any of the three topographic solutions during the relevant time scale of the associated motion.     

Accuracy of moments of velocity and scalar fluctuations in the atmospheric surface layer

A detailed accuracy analysis is presented for moments, up to order four, of both velocity (horizontal u and vertical w) and scalar (temperature and humidity q) fluctuations, as well as of the products uw, w and wq, in the atmospheric surface layer. The high-order moments and integral time scales required for this analysis are evaluated from data obtained at a height of about 5 m above the ocean surface under stability conditions corresponding to Z/L \- –0.05. Measured moments and probability density functions of some of the individual fluctuations show departures from Gaussianity, but these are sufficiently small to enable good estimates to be obtained using Gaussian instead of measured moments. For the products, the assumption of joint Gaussianity for individual fluctuations provides a reasonable, though somewhat conservative, estimate for the integration times required. The concept of Reynolds number similarity implies that differences in integration time requirements for flows at different Reynolds numbers arise exclusively from differences in integral time scales. A first approximation to the integral time scales relevant to atmospheric flows is presented.     

The relative importance of ejections and sweeps to momentum transfer in the atmospheric boundary layer

Using an incomplete third-order cumulant expansion method (ICEM) and standard second-order closure principles, we show that the imbalance in the stress contribution of sweeps and ejections to momentum transfer (ΔS _o ) can be predicted from measured profiles of the Reynolds stress and the longitudinal velocity standard deviation for different boundary-layer regions. The ICEM approximation is independently verified using flume data, atmospheric surface layer measurements above grass and ice-sheet surfaces, and within the canopy sublayer of maturing Loblolly pine and alpine hardwood forests. The model skill for discriminating whether sweeps or ejections dominate momentum transfer (e.g. the sign of ΔS _o ) agrees well with wind-tunnel measurements in the outer and surface layers, and flume measurements within the canopy sublayer for both sparse and dense vegetation. The broader impact of this work is that the “genesis” of the imbalance in ΔS _o is primarily governed by how boundary conditions impact first and second moments.     

A new large-eddy simulation model for simulating air flow and warm clouds above highly complex terrain. Part I: The dry model

This paper presents the dry version of a new large-eddy simulation (LES) model, which is designed to simulate air flow and clouds above highly complex terrain. The model is three-dimensional and nonhydrostatic, and the governing equations are sound filtered by use of the anelastic approximation. A fractional step method is applied to solve the equations on a staggered Cartesian grid. Arbitrarily steep and complex orography can be accounted for through the method of viscous topography. The dynamical model core is validated by comparing the results for a spreading density current against a benchmark solution. The model accuracy is further assessed through the simulation of turbulent flow across a quasi two-dimensional ridge. The results are compared with wind-tunnel data. The method of viscous topography is not restricted to moderately sloped terrain. Compared to models using curvilinear grids, it allows this model to be applied to a much wider range of flows. This is illustrated through the simulation of an atmospheric boundary-layer flow over a surface mounted cube. The results show that the dry model version is able to accurately represent the complex flow in the vicinity of three-dimensional obstacles. It is concluded that the method of viscous topography was successfully implemented into a micrometeorological LES model. As will be shown in Part II, this allows the detailed study of clouds in highly complex terrain.     

Numerical Study of Dual-Plume Interference in a Turbulent Boundary Layer

Direct numerical simulation is used to investigate the interference arising from the dispersion of passive scalar plumes released from a pair of point sources in a fully-developed wall-bounded shear flow. Four different lateral separations of the two sources for both near ground-level and elevated releases are considered. The downwind evolution of the correlation between the plume concentrations along the centreline between the two sources and the behaviour of the lateral profiles of the correlation at various locations downwind of the two sources are examined in detail. Differences in the exceedance probability over a high concentration level for a single plume and the total plume are highlighted and studied, and the effects of destructive and constructive interferences on the exceedance probabilities for the total plume are used to explain these differences. One significant result is that all higher-order (third-order and above) moments of the total concentration can be inferred from the application of a clipped-gamma distribution using the information embodied in only the first- and second-order concentration moments of each single plume, and in the cross-correlation coefficient of the instantaneous concentration of the two plumes.     

Turbulence in the Stable Boundary Layer at Higher Richardson Numbers

We present some algebraic and numerical simulations of the stable boundary layer. We also discuss the problem of the existence of a critical Richardson number (Ri), beyond which the turbulence is suppressed. We compare the results of a second-order algebraic model with those of a third-order numerical model and, to this purpose, numerical simulations of a wind-tunnel flow, which is characterized by various Richardson numbers, were performed. As far as the second-order model is concerned, solutions, for the Richardson number greater than any critical value, can be obtained by modifying the time scales of the second-order equation pressure correlation terms in order to account for a buoyancy damping factor. We show that using a third-order model allows the same results (no critical Richardson number) to be obtained without modifications to the time scales. It is suggested that the non-locality, accounted for by the third-order moments, could allow the turbulence to persist also for Ri > 1.     

Scaling Range Exponents From X-Wire Measurements In The Atmospheric Surface Layer

This paper focuses on the behaviour of moments, up to order 6, of longitudinal and vertical velocity increments, measured in the atmospheric surface layer, at a height of 1.7 m. The local derivatives of these moments with respect to the spatial separation between two points indicate that inertial range power-law exponents cannot be determined unambiguously. This is supported by the local slopes of moments of the locally averaged energy dissipation rate but contrasts with the extensive power-law ranges indicated by spectra of longitudinal and vertical velocity fluctuations. The third-order longitudinal velocity structure function provides stronger evidence of anisotropy in the inertial range than either the second-order velocity structure functions or the velocity spectra.     

Water exchange in fjords induced by tidally generated internal lee waves

Tidal processes are examined that control the water exchange between two basins of the Trondheimsfjord through a narrow channel with sills. For this purpose, a non-hydrostatic numerical model based on the laterally averaged Reynolds equations in the Boussinesq approximation was developed. The model takes into account the real vertical fluid stratification, variable bottom topography and variable cross-section of the fjord. Numerical experiments were performed to investigate tidally generated internal waves and their influence on the water exchange.The model produces both baroclinic tides and tidally generated lee waves. It was found that, for the Skarnsund strait which connects the Middle Fjord and the Beitstadfjord, the internal tides generated over the Skarnsund sills are very weak. Their amplitudes do not exceed 1 m.The intense short internal waves, which are identified as unsteady lee waves, comprise the basic input of the total internal wave field. These waves are generated by tidal currents at sill breaks, are trapped by topography in the generation area and grow by continuing feedback into large-amplitude waves. As the tidal flow slackens, they move upstream as freely propagating waves.As essentially nonlinear responses, the lee waves cause a nonlinear water transport. The detailed analysis of the residual currents produced by unsteady lee waves (which are propagating in both directions from the Scarnsund sills) has shown, in particular, that the residual currents can reach values as high as 0.27 m s⁻¹.It was also found that such currents exert a considerable effect on the water exchange through the Skarnsund strait between the adjacent basins. This mechanism can play an important role in water renewal and formation of the Beitasdfjord waters.