Stephen E. Belcher

Adjustment of a turbulent boundary layer to a canopy of roughness elements

Am odel is developed for the adjustment of the spatially averaged time-mean flow of ad eep turbulent boundary layer over small roughness elements to a canopy of larger three-dimensional roughness elements. Scaling arguments identify three stages of the adjustment. First, the drag and the finite volumes of the canopy elements decelerate air parcels; the associated pressure gradient decelerates the flow within an impact region upwind of the canopy. Secondly, within an adjustment region of length of order Lc downwind of the leading edge of the canopy, the flow within the canopy decelerates substantially until it comes into a local balance between downward transport of momentum by turbulent stresses and removal of momentum by the drag of the canopy elements. The adjustment length, Lc ,i s proportional to(i) the reciprocal of the roughness density (defined to be the frontal area of canopy elements per unit floor area) and (ii) the drag coefficient of individual canopy elements. Further downstream, within a roughness-change region ,t hecanopy is shown to affect the flow above as if it were a change in roughness length, leading to the development of an internal boundary layer. A quantitative model for the adjustment of the flow is developed by calculating analytically small perturbations to a logarithmic turbulent velocity profile induced by the drag due to a sparse canopy with L/Lc � 1, where L is the length of the canopy. These linearized solutions are then evaluated numerically with a nonlinear correction to account for the drag varying with the velocity. A further correction is derived to account for the finite volume of the canopy elements. The calculations are shown to agree with experimental measurements in a fine-scale vegetation canopy, when the drag is more important than the finite volume effects, and a canopy of coarse-scale cuboids, when the finite volume effects are of comparable importance to the drag in the impact region. An expression is derived showing how the effective roughness length of the canopy, z eff 0 ,i s related to the drag in the canopy. The value of z eff varies smoothly with fetch through the adjustment region from the roughness length of the upstream surface to the equilibrium roughness length of the canopy. Hence, the analysis shows how to resolve the unphysical flow singularities obtained with previous models of flow over sudden changes in surface roughness.

The International Urban Energy Balance Models Comparison Project: First Results from Phase 1

A large number of urban surface energy balance models now exist with different assumptions about the important features of the surface and exchange processes that need to be incorporated. To date, no comparison of these models has been conducted; in contrast, models for natural surfaces have been compared extensively as part of the Project for Intercomparison of Land-surface Parameterization Schemes. Here, the methods and first results from an extensive international comparison of 33 models are presented. The aim of the comparison overall is to understand the complexity required to model energy and water exchanges in urban areas. The degree of complexity included in the models is outlined and impacts on model performance are discussed. During the comparison there have been significant developments in the models with resulting improvements in performance (root-mean-square error falling by up to two-thirds). Evaluation is based on a dataset containing net all-wave radiation, sensible heat, and latent heat flux observations for an industrial area in Vancouver, British Columbia, Canada. The aim of the comparison is twofold: to identify those modeling approaches that minimize the errors in the simulated fluxes of the urban energy balance and to determine the degree of model complexity required for accurate simulations. There is evidence that some classes of models perform better for individual fluxes but no model performs best or worst for all fluxes. In general, the simpler models perform as well as the more complex models based on all statistical measures. Generally the schemes have best overall capability to model net all-wave radiation and least capability to model latent heat flux.

Constructing design weather data for future climates

We develop a method, here called ‘morphing’, to produce design weather data for building thermal simulations that accounts for future changes to climate. Morphing combines present-day observed weather data with results from climate models. The procedure yields weather time series that encapsulate the average weather conditions of future climate scenarios, whilst preserving realistic weather sequences. In this sense the method ‘downscales’ coarse resolution climate model predictions to the fine spatial and temporal resolutions required for building thermal simulations. The morphing procedure is illustrated by application to CIBSE design weather years and climate change scenarios for the UK. Heating degree days calculated from the weather series morphed to future climates show a marked reduction compared to present day, by an amount that agrees well with results calculated directly from the climate model. This agreement gives confidence that the morphing technique faithfully transforms the weather sequences. Practical application: There is overwhelming consensus amongst the scientific community that the Earths climate is warming. This warming will have implications for building services in the UK that should be considered now. This article describes a method for producing weather data with best current estimates of future climate that can be used to quantify the risk of building overheating.

Turbulent shear flow over slowly moving waves

We investigate the changes to a fully developed turbulent boundary layer caused by the presence of a two-dimensional moving wave of wavelength L = 2π/ k and amplitude a. Attention is focused on small slopes, ak , and small wave speeds, c , so that the linear perturbations are calculated as asymptotic sequences in the limit ( u * + c )/ U B ( L ) → 0 ( u * is the unperturbed friction velocity and U B ( L ) is the approach-flow mean velocity at height L ). The perturbations can then be described by an extension of the four-layer asymptotic structure developed by Hunt, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill. When ( u * + c )/ U B ( L ) is small, the matched height, z m (the height where U B equals c ), lies within an inner surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the inviscid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influenced by the matched height and the region of reversed flow below z m : they are similar to the perturbations due to a static undulation, but the ‘effective roughness length’ that determines the shape of the unperturbed velocity profile is modified to z m = z 0 exp ( kc/u * ). The solutions for the perturbations to the boundary layer are used to calculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At first order in ( u * + c )/ U B ( L ), however, there are three new effects which, numerically, contribute significantly to the growth rate, namely: the asymmetries in both the normal and shear Reynolds stresses associated with the leeside thickening of the boundary layer, and asymmetric perturbations induced by the varying surface velocity associated with the fluid motion in the wave; further asymmetries induced by the variation in the surface roughness along the wave may also be important.

Structure of turbulent flow over regular arrays of cubical roughness.

The structure of turbulent flow over large roughness consisting of regular arrays of cubical obstacles is investigated numerically under constant pressure gradient conditions. Results are analysed in terms of first- and second-order statistics, by visualization of instantaneous flow fields and by conditional averaging. The accuracy of the simulations is established by detailed comparisons of first- and second-order statistics with wind-tunnel measurements. Coherent structures in the log region are investigated. Structure angles are computed from two-point correlations, and quadrant analysis is performed to determine the relative importance of Q2 and Q4 events (ejections and sweeps) as a function of height above the roughness. Flow visualization shows the existence of low-momentum regions (LMRs) as well as vortical structures throughout the log layer. Filtering techniques are used to reveal instantaneous examples of the association of the vortices with the LMRs, and linear stochastic estimation and conditional averaging are employed to deduce their statistical properties. The conditional averaging results reveal the presence of LMRs and regions of Q2 and Q4 events that appear to be associated with hairpin-like vortices, but a quantitative correspondence between the sizes of the vortices and those of the LMRs is difficult to establish; a simple estimate of the ratio of the vortex width to the LMR width gives a value that is several times larger than the corresponding ratio over smooth walls. The shape and inclination of the vortices and their spatial organization are compared to recent findings over smooth walls. Characteristic length scales are shown to scale linearly with height in the log region. Whilst there are striking qualitative similarities with smooth walls, there are also important differences in detail regarding: (i) structure angles and sizes and their dependence on distance from the rough surface; (ii) the flow structure close to the roughness; (iii) the roles of inflows into and outflows from cavities within the roughness; (iv) larger vortices on the rough wall compared to the smooth wall; (v) the effect of the different generation mechanism at the wall in setting the scales of structures.

The Role of Wave-Induced Coriolis–Stokes Forcing on the Wind-Driven Mixed Layer

The interaction between the Coriolis force and the Stokes drift associated with ocean surface waves leads to a vertical transport of momentum, which can be expressed as a force on the mean momentum equation in the direction along wave crests. How this Coriolis-Stokes forcing affects the mean current profile in a wind-driven mixed layer is investigated using simple models, results from large-eddy simulations, and observational data. The effects of the Coriolis-Stokes forcing on the mean current profile are examined by reappraising analytical solutions to the Ekman model that include the Coriolis-Stokes forcing. Turbulent momentum transfer is modeled using an eddy-viscosity model, first with a constant viscosity and second with a linearly varying eddy viscosity. Although the Coriolis-Stokes forcing penetrates only a small fraction of the depth of the wind-driven layer for parameter values typical of the ocean, the analytical solutions show how the current profile is substantially changed through the whole depth of the wind-driven layer. It is shown how, for this oceanic regime, the Coriolis-Stokes forcing supports a fraction of the applied wind stress, changing the boundary condition on the wind-driven component of the flow and hence changing the current profile through all depths. The analytical solution with the linearly varying eddy viscosity is shown to reproduce reasonably well the effects of the Coriolis-Stokes forcing on the current profile computed from large-eddy simulations, which resolve the three-dimensional overturning motions associated with the turbulent Langmuir circulations in the wind-driven layer. Last, the analytical solution with the Coriolis-Stokes forcing is shown to agree reasonably well with current profiles from previously published observational data and certainly agrees better than the standard Ekman model. This finding provides evidence that the Coriolis-Stokes forcing is an important mechanism in controlling the dynamics of the upper ocean.

On the distortion of turbulence by a progressive surface wave

A rapid-distortion model is developed to investigate the interaction of weak turbulence with a monochromatic irrotational surface water wave. The model is applicable when the orbital velocity of the wave is larger than the turbulence intensity, and when the slope of the wave is suciently high that the straining of the turbulence by the wave dominates over the straining of the turbulence by itself. The turbulence suers two distortions. Firstly, vorticity in the turbulence is modulated by the wave orbital motions, which leads to the streamwise Reynolds stress attaining maxima at the wave crests and minima at the wave troughs; the Reynolds stress normal to the free surface develops minima at the wave crests and maxima at the troughs. Secondly, over several wave cycles the Stokes drift associated with the wave tilts vertical vorticity into the horizontal direction, subsequently stretching it into elongated streamwise vortices, which come to dominate the flow. These results are shown to be strikingly dierent from turbulence distorted by a mean shear flow, when ‘streaky structures’ of high and low streamwise velocity fluctuations develop. It is shown that, in the case of distortion by a mean shear flow, the tendency for the mean shear to produce streamwise vortices by distortion of the turbulent vorticity is largely cancelled by a distortion of the mean vorticity by the turbulent fluctuations. This latter process is absent in distortion by Stokes drift, since there is then no mean vorticity. The components of the Reynolds stress and the integral length scales computed from turbulence distorted by Stokes drift show the same behaviour as in the simulations of Langmuir turbulence reported by McWilliams, Sullivan & Moeng (1997). Hence we suggest that turbulent vorticity in the upper ocean, such as produced by breaking waves, may help to provide the initial seeds for Langmuir circulations, thereby complementing the shear-flow instability mechanism developed by Craik & Leibovich (1976). The tilting of the vertical vorticity into the horizontal by the Stokes drift tends also to produce a shear stress that does work against the mean straining associated with the wave orbital motions. The turbulent kinetic energy then increases at the expense of energy in the wave. Hence the wave decays. An expression for the wave attenuation rate is obtained by scaling the equation for the wave energy, and is found to be broadly consistent with available laboratory data.

A global perspective on Langmuir turbulence in the ocean surface boundary layer

The turbulent mixing in thin ocean surface boundary layers (OSBL), which occupy the upper 100 m or so of the ocean, control the exchange of heat and trace gases between the atmosphere and ocean. Here we show that current parameterizations of this turbulent mixing lead to systematic and substantial errors in the depth of the OSBL in global climate models, which then leads to biases in sea surface temperature. One reason, we argue, is that current parameterizations are missing key surface-wave processes that force Langmuir turbulence that deepens the OSBL more rapidly than steady wind forcing. Scaling arguments are presented to identify two dimensionless parameters that measure the importance of wave forcing against wind forcing, and against buoyancy forcing. A global perspective on the occurrence of wave-forced turbulence is developed using re-analysis data to compute these parameters globally. The diagnostic study developed here suggests that turbulent energy available for mixing the OSBL is under-estimated without forcing by surface waves. Wave-forcing and hence Langmuir turbulence could be important over wide areas of the ocean and in all seasons in the Southern Ocean. We conclude that surface-wave-forced Langmuir turbulence is an important process in the OSBL that requires parameterization. Citation: Belcher, S. E., et al. (2012), A global perspective on Langmuir turbulence in the ocean surface boundary layer, Geophys. Res. Lett., 39, L18605, doi: 10.1029/2012GL052932.

Mixing and transport in urban areas

Concern over terrorist releases of harmful material has generated interest in short-range air-borne dispersion in urban areas. Here, we review the important fluid dynamical processes that control dispersion in the first kilometre, the neighbourhood scale, when much of the material remains within the urban canopy. Dispersion is then controlled by turbulent mixing and mean flow transport through the network of streets. We consider mixing and transport in a long straight street, street intersections and then a network of streets connected by intersections. The mixing and transport in these systems are illustrated with results from recent fine-resolution numerical simulations and laboratory models, which then inform simpler scaling estimates and modelling schemes. Finally, we make some tentative steps to pull the process studies together to begin to understand results from full-scale observations. In particular, it is shown that the positions of ‘shear layers’ and ‘dividing streamlines’ largely control the patterns of mixing and transport. It is also shown that neighbourhood-scale dispersion follows one scaling in the near field and another in the far field after passage through many intersections. The challenge for the future is to bring these threads together into a coherent mathematical model.

A global climatology of wind-wave interaction

Abstract Generally, ocean waves are thought to act as a drag on the surface wind so that momentum is transferred downward, from the atmosphere into the waves. Recent observations have suggested that when long wavelength waves—which are characteristic of remotely generated swell—propagate faster than the surface wind, momentum can also be transferred upward. This upward momentum transfer acts to accelerate the near-surface wind, resulting in a low-level wave-driven wind jet. Previous studies have suggested that the sign reversal of the momentum flux is well predicted by the inverse wave age, the ratio of the surface wind speed to the speed of the waves at the peak of the spectrum. Data from the 40-yr ECMWF Re-Analysis (ERA-40) have been used here to calculate the global distribution of the inverse wave age to determine whether there are regions of the ocean that are usually in the wind-driven wave regime and others that are generally in the wave-driven wind regime. The wind-driven wave regime is found to o...