Interpolation between Businger–Dyer Formulae and Free Convection Forms: A Revised Approach

In this study, profile functions for flux calculations during unstable conditions are proposed and examined. These functions are based on a direct interpolation for the dimensionless wind speed and temperature gradients between the standard Businger–Dyer formulae, , , and free convection forms, , being the Monin–Obukhov stability parameter. A previously presented interpolation between the corresponding profile relationships, in attempting to provide a general relationship for the whole unstable regime, leads to serious restrictions for the values of in the free convection forms. These restrictions rendered available experimental data almost inapplicable, since the behaviour of the formulae in the near-neutral range controls the values of those parameters. The proposed interpolation provides functions that, firstly, fit the standard Businger–Dyer forms for near-neutral conditions and, secondly, satisfy the asymptotic behaviour as , permitting wider ranges of possible values. This step is very important, taking into account the large spread of the experimental data. Thus, as further and more accurate observations at strong instability become available, this approach could prove very efficient in fitting these data while retaining correct near-neutral behaviour.     

Estimation of the Exchange Coefficient of Heat During Low Wind Convective Conditions

For the first time, the exchange coefficient of heat C_H has been estimated from eddy correlation of velocity and virtual temperature fluctuations using sonic anemometer measurements made at low wind speeds over the monsoon land atJodhpur (26°18' N, 73°04' E), a semi arid station. It shows strong dependence on wind speed, increasing rapidly with decreasing wind speed, and scales according to a power law C_H = 0.025U₁₀ ^-0.7 (where U₁₀ is the mean wind speed at 10-m height). A similar but more rapid increase in the drag coefficient C_Dhas already been reported in an earlier study. Low winds (<4 m s^-1) are associated with both near neutral and strong unstable situations. It is noted that C_H increases with increasing instability. The present observations best describe a low wind convective regime as revealed in the scaling behaviour of drag, sensible heat flux and the non-dimensional temperature gradient. Neutral drag and heat cofficients,corrected using Monin–Obukhov (M–O) theory, show a more uniform behaviour at low wind speeds in convective conditions, when compared with the observed coefficients discussed in a coming paper.At low wind convective conditions, M-O theory is unable to capture the observed linear dependence of drag on wind speed, unlike during forced convections. The non-dimensional shear inferred from the present data shows noticeable deviations from Businger's formulation, a forced convection similarity. Heat flux is insensitive to drag associated with weak winds superposed on true free convection. With heat flux as the primary variable, definition of new velocity scales leads to a new drag parameterization scheme at low wind speeds during convective conditionsdiscussed in a coming paper.     

Flux-profile Relationships for Wind Speed and Temperature in the Stable Atmospheric Boundary Layer

Wind and temperature profiles in the stable boundary layer were analyzed in the context of MoninObukhov similarity. The measurements were made on a 60-m tower in Kansas during October 1999 (CASES-99). Fluxprofile relationships, obtained from these measurements in their integral forms, were established for wind speed and temperature. Use of the integral forms eliminates the uncertainty and accuracy issues resulting from gradient computations. The corresponding stability functions, which were nearly the same for momentum and virtual sensible heat, were found to exhibit different features under weakly stable conditions compared to those under strongly stable conditions. The gradient stability functions were found to be linear, namely _m = 1+ 5.8 and _h = 1 + 5.4 up to a limit of the MoninObukhov stability parameter = 0.8; this is consistent with earlier findings. However, for stronger stabilities beyond a transition range, both functions were observed gradually to approach a constant, with a value of approximately 7. To link these two distinct regimes, a general but pliable functional form with only two parameters is proposed for the stability functions, covering the entire stability range from neutral to very stable conditions.     

Surface-Layer Fluxes in Stable Conditions

Micrometeorological tower data from the Microfronts experiment are analyzed. Scale-dependencies of the flux and flux sampling error are combined to automatically determine Reynolds turbulence cut-off time scales for computing fluxes from time series. The computed downward heat flux at the 3 m height averaged over nine nights with 7.3 hours each night is 20% greater than the downward heat flux computed at the 10 m height. In contrast, there is only a 1.2% difference between 3 m and 10 m heat fluxes averaged over daytime periods, and there is less than a 2% difference between 3 m and 10 m momentum fluxes whether averaged over nighttime or daytime periods.Stability functions, M(z/L) and H(z/L) are extended to z/L up to 10, where z is the observational height and L is the Obukhov length. For 0.01 < z/L < 1 the estimated functions generally agree with Businger-Dyer formulations, though the H estimates include more scatter compared to the M estimates. For 1 < z/L < 10, the flux intermittency increases, the flux Richardson number exceeds 0.2, and the number of flux samples decreases. Nonetheless the estimates of the stability function M based on 3-m fluxes are closer to the formula proposed by Beljaars and Holtslag in 1991 while the M functions based on 10-m fluxes appears to be closer to the formula proposed by Businger et al. in 1971. The stability function H levels off at z/L = 0.5.     

The Effect of Chessboard Variability of the Surface Fluxes on the Aggregated Turbulence Fields in a Convective Atmospheric Surface Layer

To what degree the variability of surface features can be identified in the turbulent signals observed in the atmospheric boundary layer is still an unresolved problem. This was investigated by conducting an analytical experiment for a one-dimensional 'chessboard'-type surface-flux distribution on the basis of local free convection scaling. The results showed that, due to their nonlinear dependency on the surface fluxes, the dimensionless gradients of the mean quantities and the dimensionless standard deviations are altered by the surface-flux variability. Furthermore, passive scalars, such as humidity, are considerably more sensitive to surface variability than the main active scalar, temperature. However, the response of the gradients of the mean quantities is fairly negligible in the range of variability studied herein as compared to that of the standard deviations, which were found to be more sensitive to the surface-flux variability. In addition, the phase difference between the active and the passive scalar flux distribution strongly affects the passive scalar turbulence. This dissimilarity between passive and active scalars, or between passive scalars when their source distributions are different, brings into question the use of variance methods for the measurement of a scalar flux, such as evaporation, over variable surfaces. The classical Bowen ratio method, which depends on the validity of the Reynolds analogy for the vertical gradients of the mean quantities, was shown to be relatively more robust. However, under conditions of strong surface variability, it can also be expected to fail.     

Diurnal Variations of Thermal Roughness Height over a Grassland

The thermal roughness height associated with the surface radiation temperature has been previously found to vary between different surface types. This study finds that the thermal roughness height varies diurnally even over a homogeneous senescent grassland. The corresponding roughness length for momentum is relatively constant.Both the aerodynamic temperature and the surface radiation temperature are found to be closely related to the air temperature in the middle of the grass canopy. However, the aerodynamic temperature is strongly influenced by the horizontally integrated heat transfer, while the surface radiation temperature represents the integrated thermal emission through the grass depth within the field of view of the radiometer. The aerodynamic temperature is less sensitive to variations and measurement errors in sensible heat flux, wind speed, and air temperature than the thermal roughness height. We find that formulating the aerodynamic temperature in terms of the surface radiation temperature is better posed for use in the bulk formula than using the surface radiation temperature directly and adjusting the thermal roughness length.     

Measuring Sensible Heat Flux Density over Pasture Using the ct2-Profile Method

Two large-aperture infrared scintillometers were positioned at heights (z) of 10 and 1.5 m with beams propagated horizontally over pasture for distances of 3.1 km and 141 m, respectively. From each scintillometer a half-hourly average value of the path-averaged, temperature-structure parameter (CT2) was obtained in unstable atmospheric conditions. The result suggested CT2 to scale with height as z-2/3. Using the CT2-profile method a path-averaged measure of the Obukhov length (L0) was calculated for each half-hour period, whence L0 was used to determine the friction velocity, u* and the surface-layer temperature scaling parameter, T*. The scintillometer-based sensible heat flux density Hsc was then calculated from Hsc = Cpu*T*. A time series of half-hourly averaged Hsc compared to Hec obtained by the eddy covariance method agreed to within 10%, with R2 = 0.67, for a range of unstable conditions (- 0.2 z/L0 - 0.01).     

Structure of the atmospheric surface layer over an industrialized equatorial area

This paper is written to report observations of the structure of the atmospheric surface layer over a coastal industrialized equatorial area. The observations were recorded at Prai Industrial Park, Penang (5° 22′ N, 100° 23′ E) a relatively simple terrain area during the south-west monsoon season in the period of three months using slow response systems. The limitations of the instruments used and its effects on the results are discussed. Wind turbulence and temperature were measured on a 10 m tower and analyzed using eddy correlation method and Monin–Obukhov similarity relations to obtain the normalized standard deviation of longitudinal (σ_u/u), lateral (σ_v/u) and vertical wind velocity fluctuations (σ_w/u) with respect to stability parameter z/L. From the results of the analysis, we found that most of turbulence is generated by shear or mechanical force. It was found that the average neutral value of σ_u/u is 2.35, 1.98 for σ_v/u and 1.47 for σ_w/u with a significantly lower than the proportionality to the power of 1/3 during unstable atmospheric conditions, and thus do not obey Monin–Obukhov similarity theory. It was observed that σ_u/u and σ_v/u values increase linearly in the range of 0 < z/L < 2 and fairly well correlated while σ_w/u does not.     

Aerodynamic Variables in the Bulk Formulation of Turbulent Fluxes

Aerodynamic variables are required to apply Monin–Obukhov similarity theory in the bulk formulation of surface fluxes. In the literature, these aerodynamic variables are commonly misinterpreted. In this paper, we review the concept of the aerodynamic variable, its connection to surface-layer similarity theory and how and why the aerodynamic variable is replaced with other variables.Observed mean variables below the surface layer, such as the surface radiation temperature, or the air temperature at canopy height, are often used in place of the extrapolated aerodynamic variables in the bulk formula, requiring empirical relationships between aerodynamic and observed variables, or requiring empirical adjustments of bulk resistances. The present study examines the validity of these relationshi Experiment (CODE). The results indicate that using a measured substitute for an aerodynamic variable can lead to significant errors in estimates of turbulent surface fluxes.     

On Monin–Obukhov Similarity In The Stable Atmospheric Boundary Layer

Atmospheric measurements from several field experiments have been combined to develop a better understanding of the turbulence structure of the stable atmospheric boundary layer. Fast response wind velocity and temperature data have been recorded using 3-dimensional sonic anemometers, placed at severalheights (1 m to 4.3 m) above the ground. The measurements wereused to calculate the standard deviations of the three components of the windvelocity, temperature, turbulent kinetic energy (TKE) dissipation andtemperature variance dissipation. These data were normalized and plottedaccording to Monin–Obukhov similarity theory. The non-dimensional turbulencestatistics have been computed, in part, to investigate the generalapplicability of the concept of z-less stratification for stable conditions. From the analysis of a data set covering almost five orders ofmagnitude in the stability parameter = z/L (from near-neutral tovery stable atmospheric stability), it was found that this concept does nothold in general. It was only for the non-dimensional standard deviation oftemperature and the average dissipation rate of turbulent kinetic energythat z-less behaviour has been found. The other variables studied here(non-dimensional standard deviations of u, v, and w velocity components and dissipation of temperature variance) did not follow the concept of z-less stratification for the very stable atmospheric boundary layer. An imbalance between production and dissipation of TKE was found for the near-neutral limit approached from the stable regime, which matches with previous results for near-neutral stability approached from the unstable regime.     

Townsend's Hypothesis, Coherent Structures And Monin–Obukhov Similarity

Townsend's hypothesis states that turbulence near a wall can be divided into an activepart that transports momentum, and an inactive part that does not, and that these twokinds of turbulence do not interact. Active turbulence is generated by wind shear and has properties that scale on local parameters of the flow, while inactive turbulence isthe product of energetic processes remote from the surface and scales on outer-layerparameters. Both kinds of motion can be observed in the atmospheric surface layer, soMonin–Obukhov similarity theory, which is framed in terms of local parameters only,can apply only to active motions. If Townsend's hypothesis were wrong, so that activeand inactive motions do interact in some significant way, then transport processes nearthe ground would be sensitive to outer-layer parameters such as boundary-layer depth,and Monin–Obukhov theory would fail.Experimental results have shown that heat transport near the ground does depend onprocesses in the outer layer. We propose a mechanism for this whereby inactive motionsinitiate active, coherent ejection/sweep structures that carry much of the momentum andheat. We give evidence that the inactive motions take the form of streak patterns of fasterand slower air, and argue that these are induced by the pressure effects of large eddiespassing overhead. The streak pattern includes regions where faster streams of air overtakeand engulf slower-moving streaks. Transverse vortices form across the spines of the streaksat these places and some of them develop into horseshoe vortices. These horseshoe vorticesgrow rapidly and are rotated forward in the sheared flow so they soon contact the ground,squirting the air confined between the legs of the horseshoe vortex outwards as a forcefulejection. This model is consistent with a wide range of results from the field and laboratoryexperiments. Heat transport is significantly affected, so undermining the dimensionalassumptions of Monin–Obukhov similarity theory.     

Large Aperture Scintillometer Used Over A Homogeneous Irrigated Area, Partly Affected By Regional Advection

Scintillometer measurements were collected over an irrigated wheat field ina semi-arid region in northwest Mexico. Conditions were unstable in the morning andstable during the afternoon, while latent heat fluxes remained high throughout the day.Regional advection was observed during near-neutral conditions. Monin–Obukhovsimilarity relationships for the structure parameter of temperature were verified in both unstable and stable conditions, but were violated close to near-neutral conditions. We found that, using additional measurements of radiation, soil heat flux and windspeed, areally averages of both sensible and latent heat fluxes can be reliably predicted by large aperture scintillometer measurements, as long as the net radiation is greater than zero.     

Determination Of The Surface Drag Coefficient

This study examines the dependence of the surface drag coefficienton stability, wind speed, mesoscale modulation of the turbulent flux and method of calculation of the drag coefficient. Data sets over grassland, sparse grass, heather and two forest sites are analyzed. For significantly unstable conditions, the drag coefficient does not depend systematically on z/L but decreases with wind speed for fixed intervals of z/L, where L is the Obukhov length. Even though the drag coefficient for weak wind conditions is sensitive to the exact method of calculation and choice of averaging time, the decrease of the drag coefficient with wind speed occurs for all of the calculation methods. A classification of flux calculation methods is constructed, which unifies the most common previous approaches.The roughness length corresponding to the usual Monin–Obukhovstability functions decreases with increasing wind speed. This dependence on wind speed cannot be eliminated by adjusting the stability functions. If physical, the decrease of the roughness length with increasing wind speed might be due to the decreasing role of viscous effectsand streamlining of the vegetation, although these effects cannot be isolated from existing atmospheric data.For weak winds, both the mean flow and the stress vector often meander significantly in response to mesoscale motions. The relationship between meandering of the stress and wind vectors is examined. For weak winds, the drag coefficient can be sensitive to the method of calculation, partly due to meandering of the stress vector.     

Bulk Formulation of the Surface Heat Flux

An interpretive literature survey examines different approachesfor applying the bulk aerodynamic formulato predict the surface heat flux. The surface heat flux is often predicted in terms of the surface radiation temperature, which is also used to predict the upward longwave radiation and the heat flux into the soil. In models, the thermal roughness length based on the surface radiation temperature (radiometric roughness length) is often specified to be smaller than the roughness length for momentum for a number of distinct reasons. The definition of the radiometric roughness length depends on the way that the surface temperature is measured, the choice of stability functions and displacement height and inclusion of any additional resistances.Using airborne eddy correlation data collected over eight different sites including bare soil, crops and grassland and several types of forests, the radiometric roughness length is found to vary by orders of magnitude in a manner that is difficult to formulate. Alternatively, we evaluate the approach where the thermal roughness length is equated with the better behaved roughness length for momentum and the corresponding aerodynamic surface temperature is modelled in terms of the surface radiation temperature, solar radiation, and vegetation index. The influence of wind speed and soil moisture on the difference between the aerodynamic and surface radiation temperatures is also examined.     

The Near-Surface Stress Profile and Conditional Averaging of Turbulence

Using the conditional average formulation, we suggest a new explanation for why the stress in the atmospheric surface layer is often observed to vary with height. In essence, because turbulence series are always correlated for small lags, the steady-state equations of motion with negligible viscous terms that traditionally require vertical fluxes to be constant with height accordingly now require the vertical fluxes to vary with height. This result has implications for interpreting and validating Monin–Obukhov similarity theory.     

Heat and Water Vapour Diffusivities Near the Base of a Disturbed Stable Internal Boundary Layer

We present results from an experiment that wasdesigned to investigate turbulent transportrelationships in a nearly homogeneous boundary layerdisturbed by unsteady wind swings, as found at thebase of an advective inversion with a convectiveboundary layer overhead. In such a situation wemeasured vertical gradients and eddy fluxes of temperature andhumidity at two heights. From these, the turbulentdiffusivities of heat and water vapour are obtained,and compared to the predictions of Monin–Obukhovsimilarity theory and those of a numericalsecond-order closure model. It is found that themeasured diffusivities exceed both predictions. Thisis interpreted as a consequence of the unsteadyconditions. It is also found that the diffusivity forheat is roughly 10% larger than that for watervapour. This is in agreement with a theoreticaltreatment of the unsteadiness effects that wedeveloped in an earlier publication. This result isnot reproduced by the numerical model because themodel has no provision for unsteady conditions. Ourresult disagrees with that from an earlier, verysimilar, field experiment, which may be due to asystematic underestimation of sensible heat flux inthe older experiment.     

Heat Flux in the Coastal Zone

Various difficulties with application of Monin–Obukhov similarity theory are surveyed including the influence of growing waves, advection and internal boundary-layer development. These complications are normally important with offshore flow. The transfer coefficient for heat is computed from eddy correlation data taken at a mast two kilometres off the Danish coast in RASEX. For these coastal zone data, the thermal roughness length shows no well-defined relation to the momentum roughness length or roughness Reynolds number, in contrast to previous theories. The variation of the momentum roughness length is dominated by wave state. In contrast, the thermal roughness length shows significant dependence on wave state only for small values of wave age where the mixing is apparently enhanced by wave breaking. The development of thin internal boundary layers with offshore flow substantially reduces the heat transfer and thermal roughness length but has no obvious influence on momentum roughness length. A new formulation of the thermal roughness length based on the internal boundary-layer depth is calibrated to the RASEX data. For the very stable case, the turbulence is mainly detached from the surface and existing formulations do not apply.As an alternative to adjusting the thermal roughness length, the transfer coefficient is related directly to the stability and the internal boundary-layer depth. This avoids specification of roughness lengths resulting from the usual integration of the non-dimensional temperature function. The resulting stability function is simpler than previous ones and satisfies free convection similarity theory without introduction of the gustiness factor. The internal boundary layer also influences the moisture transfer coefficient.     

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.     

Stratified Atmospheric Boundary Layers

Various features of different stability regimes of the stable boundary layer are discussed. Traditional layering is examined in terms of the roughness sublayer, surface layer, local similarity, z-less stratification and the region near the boundary-layer top. In the very stable case, the strongest turbulence may be detached from the surface and generated by shear associated with a low level jet, gravity waves or meandering motions. In this case, similarity theory and the traditional concept of a boundary-layer break down. The elevated turbulence may intermittently recouple to the surface. Inability to adequately measure turbulent fluxes in very stable conditions limits our knowledge of this regime.     

On The Temperature-Humidity Correlation And Similarity

The correlation coefficient between temperature and specific humidity (r_Tq) is considered. A brief literature review as well as experimental data reveals that there is evidence that the eddy diffusivities for heat and water vapour are not equal if |r_Tq| 1. Several conditions under which this can occur are discussed. It is concluded that if non-local effects are of the same order as local effects T and q will behave in a non-similar fashion. Moreover, it is shown that if |r_Tq| = 1, temperature and specific humidity both will obey Monin–Obukhov similarity. In addition, based on that derivation, a new method is presented to determine the Bowen ratio from the T - q correlation. A first test of this new Bowen ratio regression method is presented. It is recommended to use r_Tq as a diagnostic tool to judge whether T-q similarity holds.