A simple model of neutrally stratified boundary-layer flow over real terrain incorporating wavenumber-dependent scaling

Our experience in applying earlier versions of a model of boundary-layer flow over low hills to real terrain (see Walmsley et al., 1982) has led to the development of a new version which we designate MS3DJH/3. The main improvements are the use of terrain-dependent length and velocity scales and the blending of inner and outer layer results into a single universally valid solution for the velocity perturbation field. MS3DJH/3 was carefully calibrated against alternative computations of flow over idealized two-dimensional terrain features using more detailed models prior to its application to real, three-dimensional terrain. It still provides high spatial resolution with low computing cost and is applicable to flow over terrain features with horizontal scales from 10 m to 10 km.Contractor: 24 Heslop Drive, Toronto.     

A mixed spectral finite-difference model for pollutant concentrations over a hill

Given incident logarithmic profiles of wind and pollutant concentration above a rough, absorbing surface, the three-dimensional distribution of pollutant concentration over a hill of gentle slope is determined from a linearized model. The model is applied in neutrally stratified flow, without chemistry, and is integrated using spectral methods in the horizontal and a finite-difference scheme in the vertical. This approach allows for flexibility in choosing a closure scheme and a variety of surface boundary conditions. This was not possible in the analytic approach of Padro (1987) who added pollutant concentration and flux to the MS3DJH/1 model of Walmsley et al. (1980). The present model requires as input the turbulent kinetic energy, E, dissipation, , and the perturbation vertical velocity, w, from the three-dimensional boundary-layer flow model of Beljaars et al. (1987), hereinafter referred to as MSFD, The latter model also supplies wind velocity perturbations at the upper boundary, as input to upper boundary conditions on the pollutant flux perturbations.The present study describes applications of the model to idealized terrain features: isolated two- and three-dimensional hills and ridges and an infinite series of ridges. (Application to real terrain, however, presents no difficulties.) Comparisons were made with different (though uniform) surface roughnesses. Tests were performed to examine the effect of upstream terrain features in the periodic domain and to illustrate the importance of the vertical resolution of the output for interpreting results from the sinusoidal terrain case.     

On the parameterization of drag over small-scale topography in neutrally-stratified boundary-layer flow

Predictions of the surface drag in turbulent boundary-layer flow over two-dimensional sinusoidal topography from various numerical models are compared. For simple 2D terrain, the model results show that the drag increases associated with topography are essentially proportional to (slope)² up to the steepness at which the flow separates. For the purposes of boundary-layer parameterisation within larger-scale models, we propose a representation of the effects of simple 2D topography via an effective roughness length, z ₀ ^eff. The form of the varation of z ₀ ^eff with terrain slope and topographic wavelength is established for small slopes from the model results and a semi-empirical formula is proposed.     

Wind flow over ridges in simulated atmospheric boundary layers

The flows over four two-dimensional triangular hills and three two-dimensional bell-shaped hills have been investigated in a simulated rural atmospheric boundary layer modelled to a scale of 1:300: Further measurements were made over two of the triangular hills in a simulated rural boundary layer of 1: 3000 scale and in a simulated urban boundary layer modelled to a scale of 1:400. The effect of the model hill surface roughness was also investigated. Flow measurements were restricted to the mean velocity U, RMS velocity fluctuations u and the energy spectra for the streamwise velocity component Measurements were made at a number of longitudinal positions in the approach flow, over the model hills and downstream of the model hills. For each model hill, the crest was the region of largest mean velocity and smallest velocity fluctuations. The largest mean velocities over the model hills occurred for hills of intermediate slope rather than for the steepest hills. A decrease in the scale of the simulated atmospheric boundary layer led to a reduction in the amplification factors at the hill crests, whereas an increase in the surface roughness of the approach flow resulted in increased amplification factors at the hill crests.     

Evaluation of an alternative method for numerically modeling nonhydrostatic flows over irregular terrain

Modeling nonhydrostatic atmospheric flow requires the solution of the vertical equation of motion and a prognostic or diagnostic equation for pressure. If the nonhydrostatic components of the flow are relatively small, they can be approximated and incorporated into a purely hydrostatic model, which usually is conceptually simpler and computationally more efficient. A method to do this for a linear model of local thermally-induced circulations is further developed and adapted to a non-linear numerical model of the neutral atmospheric boundary layer. A hydrostatic model and the quasi-nonhydrostatic version were used to simulate neutral flow over simple terrain features. One set of observations taken over a simple change in roughness and another set taken over a change in both roughness and terrain were simulated by both models to assess the capabilities of the quasi-nonhydrostatic technique.It is found that (as expected) the pressure deviation from the hydrostatic state is negligible for the roughness change, but it is an important aspect of neutral flow over terrain. Thus, for flow encountering a simple roughness change, the hydrostatic approximation is good, even for small horizontal scales. However, the quasi-nonhydrostatic model qualitatively produces the features in the observations for flow over a terrain change that the hydrostatic model cannot produce.Journal Paper No. J-12737 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 2779.     

The perturbed structure of the neutral atmospheric boundary layer over irregular terrain

The first-order (linear) response of the planetary boundary layer is calculated for flow over periodic terrain, for variations in both surface roughness and terrain elevation. Calculations are made for horizontal wavenumbers varying from 10^–4m^–1 to 3 × 10^–3m^–1. A simple second-order closure model of the turbulence is used, and Coriolis and buoyancy forces are neglected. As expected, flow over a periodic terrain produces corresponding periodic structure in all meteorological fields above the surface. The periodic structure consists of two components. The first is very nearly evanescent with height, showing little vertical structure. It corresponds to the motion that would be observed were the atmosphere inviscid. The second component, introduced by turbulent viscosity, exhibits considerable vertical structure, with vertical wavelengths the order of 100 m, and thus could be responsible for the layering often seen on acoustic sounder observations of the atmospheric boundary layer.Wave Propagation Laboratory.Environmental Science Group.     

Effective Roughness Calculated from Satellite-Derived Land Cover Maps and Hedge-Information used in a Weather Forecasting Model

In numerical weather prediction, climate and hydrologicalmodelling, the grid cell size is typically larger than the horizontal length scales of variations in aerodynamicroughness, surface temperature and surface humidity. These local land cover variations give rise to sub-gridscale surface flux differences. Especially the roughness variations can give a significantly differentvalue between the equilibrium roughness in each of the patches as compared to the aggregated roughness value,the so-called effective roughness, for the grid cell. The effective roughness is a quantity that secures thephysics to be well-described in any large-scale model. A method of aggregating the roughness step changesin arbitrary real terrain has been applied in flat terrain (Denmark) where sub-grid scale vegetation-drivenroughness variations are a dominant characteristic of the landscape. The aggregation model is a physicaltwo-dimensional atmospheric flow model in the horizontal domain based on a linearized version of theNavier Stoke equation. The equations are solved by the Fast Fourier Transformation technique, hence the codeis very fast. The new effective roughness maps have been used in the HIgh Resolution Limited Area Model(HIRLAM) weather forecasting model and the weather prediction results are compared for a number of casesto synoptic and other observations with improved agreement above the predictions based on currentstandard input. Typical seasonal springtime bias on forecasted winds over land of +0.5 m s^-1 and-0.2 m s^-1 in coastal areas is reduced by use of the effective roughness maps.     

Estimates of wind speed perturbation in boundary-layer flow over isolated low hills — Solutions

A computationally efficient model — the MS3DJH — was used to predict the near-surface wind-speed variations over a given idealised twin hill. Results given compare the model predictions with those of trained meteorologists. Further application of the model to assessment of e.g. power-producing turbine sites is discussed.     

A method for solving the planetary boundary-layer equations

A method for solving the nonlinear three-dimensional steady-state equations which govern planetary boundary layer flow is described. The method is applicable to air motions over terrain with horizontally varying surface roughness, temperature and moisture. It can also be applied to a physical system consisting of the air and the sea (or earth) boundary layers taken together. Examples of calculations for selected cases of terrain variations are presented. Convergence of the method has been assessed by determining the degree to which the solutions satisfy the set of equations.The results of the calculations show that the method produces qualitatively realistic distributions of the different meteorological variables.Contribution No. 1251 from the University of Miami, Rosenstiel School of Marine and Atmospheric Sciences.     

The perturbed structure of the neutral atmospheric boundary layer over irregular terrain

The differential equations for first-order (linear) response of the planetary boundary layer are formulated for flow over periodic terrain, for variations in both surface roughness and terrain elevation. A simple second-order closure model of the turbulence is used, and Coriolis forces are neglected. Flow over a periodic terrain produces corresponding periodic structure in all meteorological fields above the surface. The periodic structure consists of two components. The first is very nearly evanescent with height. It corresponds to the motion that would be observed were the atmosphere inviscid. The second component, introduced by turbulent viscosity, exhibits a phase variation with height in addition to a decay in amplitude. W.K.B. [Wentzel-Kramers-Brillouin] approximations for the two components are developed, and the coupling between the components is discussed. The formulation for calculating solutions by numerical integration is developed, including specification of appropriate boundary conditions. Calculations are presented in a companion paper.Wave Propagation Laboratory.Environmental Science Group.     

Surface-layer flow in complex terrain: Comparison of models and full-scale observations

Four models of surface boundary-layer flow in complex terrain are compared with observations made at Blashaval Hill, North Uist, Scotland. The field experiment is described by Mason and King (1985). Three of the models are derived from the two-dimensional theory of Jackson and Hunt (1975) and are described in Mason and King (1985), Walmsley et al. (1986) and Troen and Petersen (1989). The fourth is a mass-consistent code based on Traci et al. (1979). The model results are in good agreement with each other and are generally within the observed range of variation ( ~ ± 16%) in normalized wind speed. For most wind direcions (7 of 11), model results of normalized wind speed at the summit were within 7% of the observed mean values. For some wind directions, calculations using the Guidelines of Walmsley et al. (1989) suggested that variations in surface roughness were important. This led us to apply one of our models incorporating nonuniform surface roughness. The lack of significant improvement for cases when water lay upstream of Blashaval Hill is attributed to compensating changes at summit and reference sites and to very local effects on the wind data. Sensitivity to topography lying to the west and northwest of Blashaval was also investigated. Results suggested an influence from those distant topographic features for some wind directions. When those features were incorporated, maximum errors in normalized wind speed at the summit were reduced from 18 to 13%.     

Boundary-layer flow over analytical two-dimensional hills: A systematic comparison of different models with wind tunnel data

Two mass consistent models (MATHEW and MINERVE) and two dynamic linearized models (MS3DJH/3R and FLOWSTAR) are used to simulate the mean flow over two-dimensional hills of analytical shape and of varying slope. The results are compared with detailed wind tunnel data (RUSHIL experiment at US EPA). Different numerical experiments have been performed, varying input data and control parameters, to test the data-processing methodology and to evaluate the minimum input data (for mass consistent models only) necessary to obtain a reliable flow field. The models behave differently according to the physical assumptions made and numerical procedure used: an assessment is then made in order to identify the proper solution for the different conditions of topography and wind data.     

Flow Over Partially Forested Ridges

Numerical simulations of flow over hills that are partially covered with a forest canopy are performed. This represents a much more realistic situation than previous studies that have generally concentrated on hills that are fully-forested. The results show that the flow over the hill is sensitive to where on the hill the forest is positioned. In particular, for low slopes flow separation is predominantly located within the forest on the lee slope. This has implications for the transport of scalars in the forest canopy. For large hills the results show more variability in scalar concentrations within the canopy compared to either a fully-forested hill or a patch of forest over flat terrain. These results are likely to have implications for a range of applications including the siting and interpretation of flux measurements over forests in complex terrain, predicting wind damage to trees and wind-farm developments. Calculation of the hill-induced pressure drag and canopy-plus-surface stress shows a strong sensitivity to the position of the forest relative to the hill. Depending on the position of the forest the individual drag terms may be strongly enhanced or reduced and may even change sign. The net impact is generally to reduce the total drag compared to an equivalent fully-forested hill, but the amount of the reduction depends strongly on the position of the forest canopy on the hill. In many cases with large, wide hills there is a clear separation of scales between the adjustment of the canopy to a forest edge (of order 6 ? 8L _c, where L _c is the canopy adjustment length scale) and the width of the hill. This separation means that the hill-induced pressure and flow fields and the forest-edge induced pressure and flow fields can in some sense be considered as acting separately. This provides a means of explaining the combined effects of partial forestation and terrain. It also offers a simple method for modelling the changes in drag over a hill due to partial forest cover by considering the impact of the hill and the partial canopy separately. Scaling arguments based on this idea successfully collapse the modelled drag over a range of different hill widths and heights and for different canopy parameters. This offers scope for a relatively simple parametrization of the effects of partial forest cover on the drag over a hill.     

Risø 1978: Further investigations into the effects of local terrain irregularities on tower-measured wind profiles

Observations of flow over complex terrain taken at Risø during June–July 1978 and numerical studies confirm earlier findings that small variations in surface elevation have significant effects on mean wind profiles. Measured shear stresses in the nonequilibrium region of the flow are consistent with theory but quite different from those obtained assuming simple flux-profile relationships. These findings imply that flux-profile relationships can be quite complicated over other than simple homogeneous terrain.     

An improved method for integrating the Mixed Spectral Finite Difference (MSFD) model equations

In their Mixed Spectral Finite Difference (MSFD) model for flow over complex terrain, Beljaars et al. (1987) solve a set of coupled, second-order ordinary differential equations (ODEs) for the first-order perturbations to the logarithmic velocity profile caused by nonuniform surface roughness and topography. To solve this set of ODEs, they employ a Forward Euler Shooting Method. It is demonstrated here that the shooting method is computationally unstable for this problem. An absolutely stable finite-difference method based on a block tridiagonal LU factorization of the finite-difference matrix is presented. The advantages of the present algorithm over the method used by Beljaars et al. are demonstrated both by theoretical argument and numerical experiment.     

Neutral Flow Over A Series Of Rough Hills: A Laboratory Experiment

This paper presents laboratory experiments of aerodynamically fully rough, neutral flow over a series of sinusoidal hills. Two sets of hills, with maximum gradients (slopes) of 0.2 (10°) and 0.4 (20°), were considered.The flow remained attached in the former case while separation occurredin the latter. Characteristics of the mean flow and turbulence statistics are discussed and compared with profiles over a flat surface covered with the same roughness as the hills. Comparisons are made with linear theory predictions for the flow in the inner region and aloft. Accurate measurements of the surface pressure were also made, enabling the comparison between the measured pressure drag and predictions from theoretical and computational work with different turbulent closure schemes. Organised secondary flow in the spanwise direction, observed previously in both experimental and computational studies, was also observed here over the small hills.     

Note on a Parametric Relation for Separating Flow over a Rough Hill

A parametrization method used to account for the effects of flow separation and wall roughness on the lower boundary condition for turbulent boundary layers is investigated against direct numerical simulation and laser Doppler anemometry data. The numerical simulation represents flow over a smooth, flat surface with a prescribed external adverse pressure gradient. The water-channel experiments cover flow over smooth and rough hills for two specified Reynolds numbers. Global optimization algorithms based on four different direct search methods are used to assess the parametrization function, C, in terms of local mean velocity profiles and the parametrization parameters u _* (friction velocity), ∂ _x p (local pressure gradient), z ₀ (effective roughness) and d (zero-plane displacement). The study investigates regions of attached and reversed flows, and forty-two velocity profiles are compared with the proposed expression for the function C, including two profiles that satisfy the solution of Stratford.     

Comments and further analysis on effective roughness lengths for use in numerical three-dimensional models

Simple calculations of the apparent roughness length for the areally averaged flow over flat but heterogeneous terrain are presented. These results could be used to specify effective roughness lengths for use in large-scale models. Some of our conclusions differ significantly from those reached recently by André and Blondin (1986).