Taylor’s hypothesis and two-point coherence measurements

Experimentally obtained time coherence has traditionally been interpreted as streamwise one-dimensional spatial coherence through Taylor’s hypothesis. We calculate corrections to the highwavenumber part of the coherence to account for the errors caused by the deviation from Taylor’s hypothesis in high-intensity turbulent flows. The small-scale turbulence is assumed to be frozen and convected by a fluctuating convection velocity. Both Lumley’s two-term approximation and the Gaussian approximation are used in the calculations. In general, we find that the coherence for crossstream separations is significantly overestimated by the direct use of Taylor’s hypothesis, the error increasing with wavenumber; that for streamwise separations is underestimated. The analyses are compared with cross-stream coherence measurements in the atmospheric surface layer. Our results indicate that predictions from Lumley’s approximation yield better agreement with experimental data for cross-stream separations than those from the Gaussian model. Our study suggests that reliable measurement of two-point spatial coherence can be achieved only for scales not too small compared to the sensor separation.     

Attenuation of Scalar Fluxes Measured with Spatially-displaced Sensors

Observations from the Horizontal Array Turbulence Study (HATS) field program are used to examine the attenuation of measured scalar fluxes caused by spatial separation between the vertical velocity and scalar sensors. The HATS data show that flux attenuation for streamwise, crosswind, and vertical sensor displacements are each a function of a dimensionless, stability-dependent parameter n _m multiplied by the ratio of sensor displacement to measurement height. The scalar flux decays more rapidly with crosswind displacements than for streamwise displacements and decays more rapidly for stable stratification than for unstable stratification. The cospectral flux attenuation model of Kristensen et al. agrees well with the HATS data for streamwise sensor displacements, although it is necessary to include a neglected quadrature spectrum term to explain the observation that flux attenuation is often less with the scalar sensor downwind of the anemometer than for the opposite configuration. A simpler exponential decay model provides good estimates for crosswind sensor displacements, as well as for streamwise sensor displacements with stable stratification. A model similar to that of Lee and Black correctly predicts flux attenuation for a combination of streamwise and crosswind displacements, i.e. as a function of wind direction relative to the sensor displacement. The HATS data for vertical sensor displacements extend the near-neutral results of Kristensen et al. to diabatic stratification and confirm their finding that flux attenuation is less with the scalar sensor located below the anemometer than if the scalar sensor is displaced an equal distance either horizontally or above the anemometer.     

DCT ANALYSIS DISTRIBUTION FEATURES OF NEAR-SURFACE WIND FIELDS DURING THE LANDFALL OF VONGFONG

1 INTRODUCTION The typhoon is a destructive weather phenomenon that stands at the top of ten major natural disasters[1]. Typhoon-related damages are the immediate consequences of weather it brings forth, which include heavy rain, strong winds and storm surges (water gain). They cause flash floods, bring down houses and break through dams[1]. Of the research on typhoons for the recent 10 years[2], new advances have been reported on abrupt changes in the motion, structure and intensity of…     

A Variable Mesh Spacing for Large-Eddy Simulation Models in the Convective Boundary Layer

A variable vertical mesh spacing for large-eddy simulation (LES) models in a convective boundary layer (CBL) is proposed. The argument is based on the fact that in the vertical direction the turbulence near the surface in a CBL is inhomogeneous and therefore the subfilter-scale effects depend on the relative location between the spectral peak of the vertical velocity and the filter cut-off wavelength. From the physical point of view, this lack of homogeneity makes the vertical mesh spacing the principal length scale and, as a consequence, the LES filter cut-off wavenumber is expressed in terms of this characteristic length scale. Assuming that the inertial subrange initial frequency is equal to the LES filter cut-off frequency and employing fitting expressions that describe the observed convective turbulent energy one-dimensional spectra, it is feasible to derive a relation to calculate the variable vertical mesh spacing. The incorporation of this variable vertical grid within a LES model shows that both the mean quantities (and their gradients) and the turbulent statistics quantities are well described near to the ground level, where the LES predictions are known to be a challenging task.     

Power Spectra and Cospectra for Wind and Scalars in a Disturbed Surface Layer at the Base of an Advective Inversion

This paper reports power spectra and cospectra of windspeed and several scalars measured at two heights nearthe base of an advective inversion. The inversion hadformed over a paddy field downwind of an extensive dryregion. Winds over the paddy field were variable instrength and direction, as a result of convectivemotions in the atmospheric boundary layer passing overfrom the dry region upwind. Fetch over the rice waslarge enough that advective effects on the transportprocesses were small at the upper level and negligibleat the lower level. Results from the lower level areinterpreted in terms of a horizontally homogeneous,but disturbed, surface layer.Power spectra of longitudinal and lateral velocitywere substantially enhanced at low frequencies. Theresulting vertical motions added only a small amountto the spectrum of vertical velocity but this stronglyaffected scalar power spectra and cospectra. Thesewere all substantially enhanced over a range of lowfrequencies. We also found that differences in lowerboundary conditions cause differences among scalarspectra at low frequencies.Our analysis shows that the spectra and cospectra havethree components, characterized by different scalingregimes. We call these the ILS (inner-layer scaling),OLS (outer-layer scaling) and CS (combined scaling)components. Of these, the CS component had notpreviously been identified. We identify CS componentsof spectra by their independence of height andfrequency. Spectra with these characteristics had beenpredicted by Kader and Yaglom for a layer of theatmosphere where spectral matching between ILS and OLSwas proposed. However, we find that the velocity andscalar scales used by Kader and Yaglom do not fit ourresults and that their concept of a matching layer isincompatible with our application. An alternativebasis for this behaviour and alternative scales areproposed.We compare our decomposition of spectra into ILS, CSand OLS components with an extended form of Townsend'shypothesis, in which wind and scalar fluctuations aredivided into active and inactive components. Wefind the schemes are compatible if we identify all OLSspectral components as inactive, and all CS and ILScomponents as active.By extending the implications of our results toordinary unstable daytime conditions,we predict that classical Monin–Obukhovsimilarity theory should be modified. We find that theheight of the convective boundary layer is animportant parameter when describing transportprocesses near the ground, and that the scalar scalein the ILS part of the spectrum, which includes theinertial subrange, is proportional to observationheight times the local mean scalar gradient, and notthe Monin–Obukhov scalar scale parameter. The formerdepends on two stability parameters: the Monin–Obukhovstability parameter and the ratio of the inner-layerand outer-layer velocity scales. The outer-layer scalecan reflect disturbances by topographically-inducededdying as well as by convective motions.     

A spatial length scale analysis of turbulent temperature and velocity fluctuations within and above an orchard canopy

Turbulent flow within and above an almond orchard was measured with three-dimensional wind sensors and fine-wire thermocouple sensors arranged in a horizontal array. The data showed organized turbulent structures as indicated by coherent asymmetric ramp patterns in the time series traces across the sensor array. Space-time correlation analysis indicated that velocity and temperature fluctuations were significantly correlated over a transverse distance more than 4m. Integral length scales of velocity and temperature fluctuations were substantially greater in unstable conditions than those in stable conditions. The coherence spectral analysis indicated that Davenport's geometric similarity hypothesis was satisfied in the lower frequency region. From the geometric similarity hypothesis, the spatial extents of large ramp structures were also estimated with the coherence functions.     

Cross-Spectra Over the Sea from Observations and Mesoscale Modelling

Cospectra and quadrature spectra are calculated for six pairs of tall offshore measurement masts near the Horns Rev I wind farm in the Danish North Sea and the Nysted wind farm in the Baltic sea. The mast-pairs are separated from one another by horizontal distances of 2.13–12.4 km. Cospectra and quadrature spectra for the two sites are classified in terms of the angle between the mean wind direction and the line connecting each pair of masts. The frequency axes of the spectra are normalized to remove the effect of mean wind speed and separation distance. Results indicate a larger contribution to the quadrature spectrum for flow from the sea than for flow from the land, and the patterns in the spectra are clearer and better defined for Horns Rev I (which has a long uninterrupted sea-fetch from the west) than for Nysted (which is surrounded by a more complicated coastline). The analysis is replicated based on 3-month simulations using the weather research and forecasting (WRF) numerical model with a horizontal grid spacing of 2 km. For the sea-fetch directions, good agreement in spectral properties between the model and observations is found. Analytical expressions based on the properties of the cross-correlation function and an exponentially decaying coherence function are fitted to the normalized cospectra and quadrature spectra. The expressions are shown to be a good fit to the spectra calculated from the WRF simulations and to the observed spectra for directions with a long sea-fetch, which suggests that to a good approximation, the average cospectra and quadrature spectra over the sea can be written as functions of frequency, mean wind speed, separation distance and the angle between the wind direction and the orientation of the masts.     

Two-Point Correlation Analysis Of Neutrally Stratified Flow Within And Above A Forest From Large-Eddy Simulation

Two-point space-time correlations ofvelocities, a passive scalar and static pressure arecalculated using the resolvable flow fields computedby large-eddy simulation (LES) of neutrally stratifiedflow within and above a sparse forest. Zero-time-lagspatial auto-correlation contours in thestreamwise-vertical cross-section for longitudinal andlateral velocities and for a scalar are tilted fromthe vertical in the downstream direction, as istypical in near-wall sheared flow. On the other hand,auto-correlations of vertical velocity and of staticpressure are vertically coherent. Zero-time-lagspatial auto-correlations in the spanwise-verticalcross-section show no distinct tilt, and those forboth longitudinal and vertical velocities demonstratedistinct negative side lobes in the middle forest andabove, while longitudinal velocity in the subcrowntrunk space is laterally in-phase. Static pressureperturbations appear to be spatially coherent in thespanwise direction at all heights, especially insidethe forest. Near the forest floor, longitudinalvelocity is found to be in-phase with static pressureperturbation and to be closely linked to theinstantaneous streamwise pressure gradient, supportinga previous proposal that longitudinal velocity in thisregion is dominantly modulated by the pressurepatterns associated with the coherent sweep/ejectionevents. Near treetop height, a lack of linkage betweenthe pressure gradient and the local time derivative ofthe longitudinal velocity supports the hypothesis ofadvection dominating turbulent flow.The major phase characteristics of the two-pointcorrelations essentially remained the same from fourLES runs with different domain size and/or gridresolution. A larger LES domain yielded betteragreement with field observations in a real forest onboth the magnitudes of the correlations and thesingle-point integral time scales. A finer gridresolution in the LES led to a faster rate of decreaseof correlation with increasing separation in space ortime, as did the higher frequency fluctuations in theturbulent records from field measurements. Convectivevelocities estimated from the lagged two-pointauto-correlations of the calculated flow fields werecompared with similar calculations from wind-tunnelstudies. At the canopy top, estimates from thecorrelation analyses agree with the translationvelocity estimated from instantaneous snapshots of ascalar microfront using both LES and field data. Thistranslation velocity is somewhat higher than the localmean wind speed. Convective velocities estimated fromlagged correlations increase with height above thecanopy. It is suggested that an appropriate filteringprocedure may be necessary to reduce the effects ofsmall-scale random turbulence, as was reported in astudy over an orchard canopy. The mean longitudinalvelocity near the treetops is found to be moreappropriate than the local mean longitudinal velocityat each height to link single-point integral timescales with directly calculated spatial integralstreamwise length scales.     

Boundary Conditions for Scalar (Co)Variances over Heterogeneous Surfaces

The problem of boundary conditions for the variances and covariances of scalar quantities (e.g., temperature and humidity) at the underlying surface is considered. If the surface is treated as horizontally homogeneous, Monin–Obukhov similarity suggests the Neumann boundary conditions that set the surface fluxes of scalar variances and covariances to zero. Over heterogeneous surfaces, these boundary conditions are not a viable choice since the spatial variability of various surface and soil characteristics, such as the ground fluxes of heat and moisture and the surface radiation balance, is not accounted for. Boundary conditions are developed that are consistent with the tile approach used to compute scalar (and momentum) fluxes over heterogeneous surfaces. To this end, the third-order transport terms (fluxes of variances) are examined analytically using a triple decomposition of fluctuating velocity and scalars into the grid-box mean, the fluctuation of tile-mean quantity about the grid-box mean, and the sub-tile fluctuation. The effect of the proposed boundary conditions on mixing in an archetypical stably-stratified boundary layer is illustrated with a single-column numerical experiment. The proposed boundary conditions should be applied in atmospheric models that utilize turbulence parametrization schemes with transport equations for scalar variances and covariances including the third-order turbulent transport (diffusion) terms.     

Two-Dimensional Scalar Spectra in the Deeper Layers of a Dense and Uniform Model Canopy

The turbulent flow inside dense canopies is characterized by wake production and short-circuiting of the energy cascade. How these processes affect passive scalar concentration variability in general and their spectral properties in particular remains a vexing problem. Progress on this problem is frustrated by the shortage of high resolution spatial concentration measurements, and by the lack of simplified analytical models that connect spectral modulations in the turbulent kinetic energy (TKE) cascade to scalar spectra. Here, we report the first planar two-dimensional scalar concentration spectra (ϕ _cc ) inside tall canopies derived from flow visualization experiments. These experiments were conducted within the deeper layers of a model canopy composed of densely arrayed cylinders welded to the bottom of a large recirculating water channel. We found that in the spectral region experiencing wake production, the ϕ _cc exhibits directional scaling power laws. In the longitudinal direction (x), or the direction experiencing the largest drag force, the ϕ _cc (k _x ) was steeper than and followed an approximate at wavenumbers larger than the injection scale of wake energy, where k _x is the longitudinal wavenumber. In the lateral direction (y), the spectra scaled as up to the injection scale, and then decayed at an approximate power law. This departure from the classical inertial subrange scaling (i.e., k ^−5/3) was reproduced using a newly proposed analytical solution to a simplified scalar spectral budget equation. Near the velocity viscous dissipation range, the scalar spectra appear to approach an approximate k ⁻³, a tantalizing result consistent with dimensional analysis used in the inertial-diffusive range. Implications to subgrid modelling for large-eddy simulations (LES) inside canopies are briefly discussed.     

用QuikSCAT资料分析"黄蜂"登陆前后近地层风场的分布特征

利用QuikSCAT资料对"黄蜂"近地层风场分布的演变特征进行了诊断分析.并采用了离散余弦转换(DCT)对其方差的波谱结构进行了讨论.结果表明台风近地层风场分布具有明显的不对称性,其方差大部分产生于波数1和2的气流中,并且与波的方向有关.当台风沿纬向西移时,有两个纬向排列的波谱中心,主要是纬向波数2和经向波数1的气流;而当台风沿径向北上时,有两个经向排列的波谱中心,主要是纬向波数1和经向波数2的气流.波数1的气流对切向风的方差做主要贡献,波数2的气流对径向风的方差做主要贡献.不对称性分布的特征随大尺度环境场和台风自转气旋环流的改变而改变.当台风靠近大陆时台风大风出现的在台风前进方向前沿,类似于陈联寿等分析的第二类台风大风模式,在左前方是西北偏北大风,在右前方是东南大风.     

Scalar Transport over Forested Hills

Numerical simulations of scalar transport in neutral flow over forested ridges are performed using both a 1.5-order mixing-length closure scheme and a large-eddy simulation. Such scalar transport (particularly of CO₂) has been a significant motivation for dynamical studies of forest canopy–atmosphere interactions. Results from the 1.5-order mixing-length simulations show that hills for which there is significant mean flow into and out of the canopy are more efficient at transporting scalars from the canopy to the boundary layer above. For the case with a source in the canopy this leads to lower mean concentrations of tracer within the canopy, although they can be very large horizontal variations over the hill. These variations are closed linked to flow separation and recirculation in the canopy and can lead to maximum concentrations near the separation point that exceed those over flat ground. Simple scaling arguments building on the analytical model of Finnigan and Belcher (Q J Roy Meteorol Soc 130:1–29, 2004) successfully predict the variations in scalar concentration near the canopy top over a range of hills. Interestingly this analysis suggests that variations in the components of the turbulent transport term, rather than advection, give rise to the leading order variations in scalar concentration. The scaling arguments provide a quantitative measure of the role of advection, and suggest that for smaller/steeper hills and deeper/sparser canopies advection will be more important. This agrees well with results from the numerical simulations. A large-eddy simulation is used to support the results from the mixing-length closure model and to allow more detailed investigation of the turbulent transport of scalars within and above the canopy. Scalar concentration profiles are very similar in both models, despite the fact that there are significant differences in the turbulent transport, highlighted by the strong variations in the turbulent Schmidt number both in the vertical and across the hill in the large-eddy simulation that are not represented in the mixing-length model.     

Turbulence in an almond orchard: Spatial variations in spectra and coherence

The spatial variability of turbulence in a fully-leafed almond orchard was studied. Two three-dimensional sonic anemometers were used to measure turbulence spectra and coherence at different vertical and lateral separations inside the canopy. Peak frequencies of the horizontal velocity components, normalized by local horizontal wind speed, are greater in the canopy crown than in the trunkspace. Peak-normalized frequencies for the vertical velocity power spectra are similar in the canopy crown and in the subcanopy trunkspace. Spectral slopes in the inertial subrange are more negative than those predicted with Kolmogorov's -2/3 theory. It is thought that the foliage elements act to short-circuit the eddy cascade. Lateral separation of the instruments in the subcanopy trunkspace has little effect on the shape of the velocity spectra. On the other hand, lateral and vertical velocity coherences between spatially separated sensors are low inside the canopy. These low coherences are due to the Eulerian length scales being of the same order of magnitude as the separation distances of the anemometers. Phase angles between velocity components are about zero for small separation distances. When the two instruments are separated by 9 m and one instrument is positioned in a row while the other is between two rows, vertical velocities are about 180 deg out of phase and the streamwise velocities are about 40 to 60 deg out of phase. These data support the contention that preferred differences occur between within- and between-row wind flow regimes.     

On Heavy Particle Dispersion in Turbulent Shear Flows: 3-D Analysis of the Effects of Crossing Trajectories

In the approaches used to predict the dispersion of discrete particles moving in a turbulent flow, the effects of crossing trajectories due to gravity (or any other external force field) are generally accounted for by modifying the integral time scales according to the well-known analysis of Csanady (J Atmos Sci 20:201–208, 1963). Here, an alternative theoretical analysis of the time correlation of the fluid velocity fluctuations along a particle trajectory is presented and applied in a turbulent shear flow. The study is carried out in the frame of three-dimensional Langevin-type stochastic models, where the main unknowns are the drift tensor components rather than the conventional integral time scales of the fluid seen by the particles. Starting from a model for the space-time velocity covariance tensor of the turbulence under the assumption of homogeneous shear flow, the various components of the time correlation tensor of the fluid seen are expressed in the asymptotic case of large mean relative velocity (between the particles and the flow) compared to the particle velocity fluctuations. In order to provide comparison with the generally used expressions arising from isotropic turbulence assumption, we examine also the conventional integral time scales of the fluid seen in the directions parallel and perpendicular to the mean relative velocity. The most prominent deviations from isotropic turbulence are observed when the external force field is in the direction of the mean velocity gradient: in this case the loss of correlation in the mean flow direction is significantly lower than expected in a uniform flow, an observation that is in qualitative agreement with the few available data.     

Multi-resolution Hybrid Data Assimilation Core on a Cubed-sphere Grid (HybDA)

This study illustrates the characteristics of the data assimilation system at the Korea Institute of Atmospheric Prediction Systems (KIAPS), based on the cubed-sphere grid system. The most interesting feature is the use of spherical harmonic functions defined on cubed-sphere grid points, which makes it possible to control the allowable physical wavenumber for the analysis increments. The relevant computational costs and parallel scalability are represented. The multiple-resolution approach is a distinguishable aspect of this data assimilation system. The wavenumber, up to which the analysis is conducted, increases as the outer iteration progresses. This multiresolution strategy is based on an investigation into the change of spectral components of analysis increments. The multi-resolution outer-loop provides cost-effective analysis-improvement, by explicitly controlling the analysis increments entered into the observation operator. To utilize the high-resolution deterministic forecast as a background state, it is subtracted from the forecast ensemble, to produce ensemble forecast perturbation that is hybridized with static background error covariance. Based on the cycled analysis experiments, the higher-resolution deterministic forecast is shown to preserve the high-frequency feature of the analysis increment relative to the ensemble mean forecast.     

Correction Of Eddy-Covariance Measurements Incorporating Both Advective Effects And Density Fluxes

Equations are presented to correct eddy-covariancemeasurements for both fluctuations in density andnon-zero mean advection, induced by convergence ordivergence of flow, and spatial source/sinkinhomogeneity, under steady-state and transientconditions. This correction collapses to theWebb–Pearman–Leuning expression ifthe mean vertical velocity is zero, and formally addsthe Webb–Pearman–Leuning expression to the correctionssuggested by Lee for conditions ofnon-zero vertical velocity and source/sink and meanscalar horizontal homogeneity. The equation requiresmeasurement of the mean vertical gradients of thescalar concentration of interest (air temperature,humidity, CO₂) as well as an accurateestimation of the mean vertical velocity, in additionto the vertical eddy covariance of the scalar. Simplemethods for the approximation of sensor tilt andcomplex terrain flow angle are presented, to allowestimation of non-zero mean vertical velocities. Theequations are applied to data from a maize crop and aforest to give examples of when the correction issignificant. In addition, a term for thethermodynamic expansion energy associated with watervapour flux is derived, which implies that the sonictemperature derived sensible heat flux will accuratelyinclude this contribution.     

Wind-Flow Dynamics Over a Vineyard

Wind-flow dynamics has been extensively studied over horizontally uniform canopies, but agricultural plantations structured in rows such as vineyards have received less attention. Here, the wind flow over a vineyard is studied in neutral stratification from both large-eddy simulation (LES) and in situ measurements. The impact of row structure on the wind dynamics is investigated over a range of wind directions from cross-row to down-row, and a typical range of row aspect ratio (row separation/height ratio). It is shown that the mean flow over a vineyard is similar to that observed in uniform canopies, especially for wind directions from cross-row to diagonal. For down-row winds, the mean flow exhibits noticeable spatial variability across each elementary row-gap pattern, as the wind is channeled in the inter-row. This spatial variability increases with the aspect ratio. With down-row winds the turbulent structures are also more intermittent and generate larger turbulent kinetic energy and momentum flux. The displacement height and roughness length of the vineyard vary with the aspect ratio in a way similar to their variation with canopy density in uniform canopies. Both parameters take smaller values in down-row wind flow, for which the canopy appears more open. The analysis of velocity spectra and autocorrelation functions shows that vineyard canopies share similar features to uniform canopies in terms of turbulent coherent structures, with only minor changes with wind direction.     

Spectral analysis of the AVHRR sea surface temperature variability off the west coast of Vancouver Island

Abstract

From 16 AVHRR infrared satellite images of the west coast of Vancouver Island, British Columbia, collected during the five summers of 1984–1988, 4 alongshore temperature transects were sampled. Upon Fourier transforming the transect data, we found that the energy spectra of the temperature variance in alongshore wavenumber space in general followed a –2.1 power law, which agreed with previous observations from other parts of the world.

Summer images may be divided into 2 types: upwelling dominated and non‐upwelling dominated. When a strong upwelling‐induced alongshore cold front was observed, the regimes shoreward and seaward of the front had distinctly different spectra. Cross‐spectral analysis of transect data between images taken a day apart in the presence of strong upwelling events revealed significant coherence at the low wavenumber regime (wavelength 300 km and above, corresponding to the large eddies) and often at the high wavenumber regime (wavelength 30 km or below, corresponding to the fine structures of the eddies). The coherence dropped for images taken 2 or more days apart, suggesting a decorrelation time‐scale of about 2 days. In the absence of strong upwelling and associated eddies, summer transect temperature data from different years often showed a similar alongshore linear trend in addition to possible large differences in the mean temperature.     

Large-eddy simulations of thermally forced circulations in the convective boundary layer. Part I: A small-scale circulation with zero wind

We have conducted large-eddy simulations (LES) of the atmospheric boundary layer with surface heat flux variations on a spatial scale comparable to the boundary layer depth.We first ran a simulation with a horizontally homogeneous heat flux. In general the results are similar to those of previous large-eddy simulations. The model simulates a field of convective eddies having approximately the correct velocity and spatial scales, and with the crucial property that kinetic energy is transported vigorously upwards through the middle levels. However, the resolved temperature variance is only about half what is observed in the laboratory or the atmosphere. This deficiency — which is shared by many other large-eddy simulations — has dynamic implications, particularly in the pressure/temperature interaction terms of the heat flux budget. Recent simulations by other workers at much higher resolution than ours appear to be more realistic in this respect.The surface heat flux perturbations were one-dimensional and sinusoidal with a wavelength equal to 1.3 times the boundary-layer depth. The mean wind was zero. Results were averaged over several simulations and over time. There is a mean circulation, with ascent over the heat flux maxima (vertical velocity ~0.1w _*) and descent over the heat flux minima. Turbulence is consistently stronger over the heat flux maxima. The horizontal velocity variance components (calculated with respect to the horizontal average) become unequal, implying that convective eddies are elongated parallel to the surface heat flux perturbations.A consideration of the budgets for temperature and velocity suggests several simplifying concepts.The research reported in this paper was conducted while the first author was on study leave at Colorado State University.