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Spectral analysis of turbulence in an unstable suburban atmosphere Roth, Matthias 1988

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S P E C T R A L A N A L Y S I S O F T U R B U L E N C E IN A N U N S T A B L E S U B U R B A N A T M O S P H E R E By M A T T H I A S R O T H Dipl. Natw., The Swiss Federal Institute of Technology, Zurich, Switzerland, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Geography) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1988 © Matthias Roth, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geography  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Feb . 2 6 , 1 9 8 8  DE-6(3/81) A B S T R A C T To measure variances and covariances of meteorological variables over very rough surfaces such as tall crops, forests and urbanized areas, that are consistent with Monin-Obukhov similarity theory, it is essential to avoid the roughness sub-layer, where the fluxes are affected by wake effects introduced by individual roughness elements. It is also necessary to work in the surface layer where height above the effective surface is the only length scale and the semi-logarithmic profile laws are obeyed. This thesis provides turbulence spectra of temperature, the vertical and longitudinal wind components as well as the fluxes of sensible heat and momentum measured in an unstable surface layer over a suburban surface. The sensor height employed in the field programme is at or below recommended values for the top of the roughness sub-layer, hence there is reason to question whether the observations are influenced by wake effects. This is of importance since the same measurement height has been used in computations of turbulent fluxes for energy balance studies at this site. The (co)spectra, normalized with the (co)variances, for all quantities investigated show very good agreement with reference data from smoother surfaces and the few studies available from other urban turbulence programmes. This study provides the first composite cospectra of sensible heat and momentum over urban terrain. It is concluded that the site used is suitable in regard to turbulence and energy balance measurements representing suburban terrain. An ancillary result shows that the recommended averaging times for the turbulent fluxes can be relaxed. T A B L E O F C O N T E N T S A B S T R A C T i i T A B L E O F C O N T E N T S i i i LIST O F T A B L E S v LIST O F F I G U R E S vi LIST O F S Y M B O L S A N D A B B R E V I A T I O N S vii i A C K N O W L E D G E M E N T S xii 1. I N T R O D U C T I O N 1 1.1 Background 1 1.2 Statement of the problem 2 1.3 Research objectives 7 2. T H E N A T U R E O F T U R B U L E N T S P E C T R A A N D T H E F I E L D  P R O G R A M M E 9 2.1 Spectral representation 9 2.2 Instrument requirements and measurement techniques 11 2.3 Measurements of turbulence from urban environments 13 2.4 Observation site 16 2.5 Instrumentation 23 2.5.1 Gill anemometer 23 2.5.2 Sonic anemometer/thermometer 25 2.6 Observation programme 27 2.7 Data acquisition and processing 29 i v 3. RESULTS 33 3.1 Time series representation 33 3.2 Spectral characteristics 40 3.2.1 Vertical velocity spectra 40 3.2.2 Horizontal velocity spectra 45 3.2.3 Temperature spectra 48 3.2.4 Heat flux cospectra 51 3.2.5 Momentum flux cospectra 55 3.3 Averaging times 58 4. SUMMARY AND CONCLUSIONS 62 4.1 Summary 62 4.2 Conclusions 65 REFERENCES 67 APPENDICES 71 A Modified Gill anemometer 71 A . l Derivation of u and w from the Gill VI and V2 signals 71 A. 2 Correction for non-linear response 74 B Spectral analysis 7 7 C Test of stationarity 79 V LIST OF TABLES 2.1 Dates and duration of turbulence measurements 28 4.1 Summary of spectral peak frequencies 64 C l Summary of results from run test of stationarity 80 V i LIST OF FIGURES 1.1 Possible framework for urban climate classification 3 2.1 Location of the observation site (Sunset) 19 2.2 Photographic view from top of the tower 19 2.3 Schematic representation of the tower 20 2.4 Photographic view of the tower 22 2.5 Photographic view of top part of tower 22 3.1 Time series (60 min) of w, T and wT (Sonic) for run 234.2 34 3.2 Time series (60 min) of w, u and uw (Gill) for run 234.2 36 3.3 Time series (60 min) of w, T and wT (Sonic) for run 245.1 38 3.4 Time series (60 min) of w, u and uw (Gill) for run 245.1 39 3.5 Individual and composite spectra of vertical velocity normalized by variance 41 3.6 Composite spectra of vertical velocity from different studies 43 3.7 Same as Figure 3.5 but for the along-wind component 46 3.8 Composite spectrum of the along-wind velocity component compared with Steyn, 1982 47 3.9 Same as Figure 3.5 but for temperature 49 3.10 Composite spectrum of air temperature 50 3.11 Individual and composite cospectra of sensible heat flux in area-preserving form 52 3.12 Composite cospectrum of sensible heat flux 54 3.13 Same as Figure 3.11 but for the momentum flux 56 3.14 Composite momentum flux cospectra 57 3.15 Comparison of sensible heat fluxes averaged over 60 and 15 min, respectively 61 v i i A. 1 Diagram of Gill propeller system to define angles 75 A. 2 Correction of Gill sensors for non-linear response 75 v i i i L I S T O F S Y M B O L S A N D A B B R E V I A T I O N S Abbreviations: ABL Atmospheric boundary layer GTVA Gill twin propeller-vane anemometer PC Personal Computer MOST Monin-Obukhov similarity theory RC Resistor-capacitor SAT Sonic anemometer/thermometer UBL Urban boundary layer UCL Urban canopy layer Symbols: A j Kolmogorov constant for momentum A 2 Kolmogorov constant for temperature Ac Altocumulus As Altostratus A .^ real part of FC at harmonic index k AV averaging time B arbitrarily chosen variable describing the turbulence in the surface layer B^ imaginary part of FC at harmonic index k B(<x) correction for non-cosine response c specific heat of air at 10 'C (1010 J kg^K"1) ix Cirrus degree Centigrade cospectral density of momentum flux cospectral density of sensible heat flux distance constant zero-plane displacement length decibel spacing of transducers of the SAT discrete time interval inter-element spacing of rougness elements accuracy height of embankment non-dimensional frequency non-dimensional peak frequency Nyquist-frequency Fourier coefficient height of main roughness elements harmonic index (used as a subscript) wavenumber peak wavelength Monin-Obukhov length frequency (Hz) half power point (frequency) dissipation rate of T 12 sensible heat flux density response function X sec second S spectral density of velocity components Srp spectral density of temperature SD spectral density t time T turbulent fluctuation of temperature with respect to the mean u turbulent fluctuation of the longitudinal wind component with respect to the mean U mean horizontal wind speed uw kinematic momentum flux v turbulent fluctuation of the lateral wind component with respect to the mean VI horizontal sensor of the GTVA V2 tilted sensor of the GTVA w turbulent fluctuation of vertical wind component with respect to the mean wT kinematic sensible heat flux Xj set of real numbers z height above ground zQ surface roughness length z-x height of atmospheric boundary layer z' effective height of measurement z* height of interface between the roughness and surface layers Greek symbols: «x angle between horizontal and tilted sensor in the GTVA S dissipation rate for turbulent energy TT mathematical constant (3.14159...) X i 0 density of air at 10 " C (1.2 kg m" 3 ) ° u v w T variances of u ,v ,w and T fluctuations 0"s variance of velocity components i n general TQ ver t ical flux density of horizontal momentum 1 (co)spectral density x i i ACKNOWLEDGEMENTS I would like to express my thanks to the persons who have contributed to the success of this work. The project was initiated by my supervisors to whom I am most grateful for their continuous support. In particular Dr. Timothy R. Oke provided guidance and critical yet constructive remarks and had always an attentive ear whenever I wished to discuss matters. Dr. Douw G. Steyn, inspired me through his continuous efforts and enthusiasm which were not just confined to purely academic work. I am indebted to Dr. William G. Large who kindly lend the modified Gill propeller anemometer, to Dr. Steve Pond whose helpful advise resulted in the proper operation of this instrument and to Dr. Gordon McBean. Special thanks go to Dr. Andy T. Black who provided software for the 2 I X data logger and to Dr. Mike Church for the use of his Rockland filter. Research grants by the Natural Sciences and Engineering Research Council of Canada and the Atmospheric Environment Service of Environment Canada to Dr. T. R. Oke funded the field work. Thanks are also due to the British Columbia Hydro and Power Authority for making their Main waring substation available as a research site. Finally I would like to express appreciation for the contributions of many friends and fellow graduate students who made the two years of a graduate program a, after all, interesting and exciting experience. In particular I would like to thank Helen Cleugh (she was never shy to climb to the top of the tower), Scott Robeson (how many courses did we struggle through together) and Hans Peter Schmid who also, as a fellow countryman, made sure that an appropriate level of continental culture could be sustained. During the course of this thesis I was financially supported by a University of British Columbia Graduate Fellowship and Teaching and Research Assistantships from the Department of Geography. 1 1. INTRODUCTION 1.1 Background In the atmosphere, the flow near the ground and up to the top of the atmospheric boundary layer (ABL) is almost always turbulent. There are many reasons why atmospheric scientists are concerned with the structure of the ABL and the properties of turbulence. The ABL and its processes are important firstly because most human activities take place in this part of the troposphere, and secondly, the ABL serves as the link between the surface and the free atmosphere above, thereby combining small-scale processes with large-scale dynamics. Turbulence, which is the main characteristic of the ABL, is of immediate interest because turbulence imposes forces on buildings, bridges, towers, airplanes and other structures. It is also responsible for the fluxes of important physical quantities such as momentum, heat, moisture and pollutants. The mixing ability of turbulence is fundamental to the diffusive power of a fluid and is therefore of obvious interest in air pollution applications. The study of turbulence is basic to understanding and predicting the structure of the surface and mixed layers, which are both part of the ABL. Given the importance of turbulence parameters in the derivation and validation of turbulence models and the computation of energy balances, it is essential to have measurement techniques which give useful estimations of the desired characteristics. Since turbulent velocities and other properties are apparently random functions in time and space, a statistical approach to their description is usually adopted. Therefore observations are usually analyzed using 2 statistical measures such as integral statistics, the shape of energy spectra and integral length scales. Most of our understanding of the turbulent atmospheric surface layer is based on Monin-Obukhov similarity theory (MOST) (e.g., Monin and Obukhov, 1954; Monin and Yaglom, 1965) which states that the structure of turbulence over an 'ideal' (low roughness, horizontally-homogeneous) surface depends on a small number of scaling parameters. 1.2 Statement of the problem The lowest 5 or 10% of the ABL constitutes the surface layer wherein the vertical fluxes of entities are assumed to be approximately constant with height, and horizontal advection due to changes in surface character in the upwind zone is absent. In this layer turbulent fluxes can be evaluated using standard micrometeorological approaches such as the aerodynamic, Bowen ratio-energy balance and eddy correlation methods. Several experimental results (Thom et al., 1975; Garratt, 1978a,b and 1980; Raupach, 1979) have shown, that conventional flux-profile relationships and the validity of the non-dimensional Monin-Obukhov functions for momentum and heat transfer must be questioned over horizontally uniform but very rough natural surfaces such as tall crops and forests. Results from field studies over these surfaces reinforce the idea that the surface layer over a very rough surface must be considered in two parts: usually referred to as the inertial sub-layer (Tennekes, 1973) and the roughness sub-layer (Raupach, 1979). In the former, under adiabatic conditions, height above the effective surface is the only length scale and the semi-logarithmic profile laws are obeyed. On the 3 other hand, in the roughness sub-layer which is adjacent to the surface itself, the flow is affected by wake effects introduced by individual roughness elements. In this region surface-defined length scales become important. Based on such findings Oke (1984) suggested a possible framework for the complex and high roughness environment of urban terrain. Figure 1.1 shows the urban atmosphere sub-divided into an urban canopy layer (UCL) which is dominated by (a) Figure 1.1: Idealized arrangement of boundary layer structures over a city (from 1984). microscale features associated with individual roughness elements and an overlying urban boundary layer (UBL). The transition between the UCL and UBL is not characterized by a sharp discontinuity rather the microscale features of the UCL meld into local or meso-scale ones in the roughness layer (called the 'roughness sub-layer' by Raupach, 1979 and 'transition layer' by Garratt, 1980). The integrated effects of the urban 'surface' form the horizontally homogeneous turbulent surface layer and the mixed layer of the UBL. To measure variances and covariances of meteorological variables that are consistent with MOST it is essential to work in the surface layer and to avoid the roughness layer. Tennekes (1973) and Townsend (1976, pp. 139-143) suggest that the minimum height for the application of the logarithmic wind profile above large roughness elements (z*) is of the order of 50 to 100zQ, where zQ is the surface roughness length. According to Pasquill (1974) the wake effects of single buildings do not extend higher above ground than about 2.5 times the building height. A more recent analysis from a wind tunnel study over a regularly arrayed rough surface (Raupach et al., 1980) defines the criterion for the lower limit of the surface layer as z* = h + 1.5Di for momentum flux in neutral stratification, where h is the height of roughness elements and Di is the inter-element spacing. Garratt (1978a) defines the same criterion, based on tower measurements over tree-covered terrain (zQ = 0.4 m) with z* = 4.5h for momentum and 3h for sensible heat under neutral conditions. In the above recommendations z* is measured from ground level upwards. Using the same site as in the 1978 study, Garratt (1980) derives z* = 3Di (above zero-plane displacement). In terms of the roughness length, Garratt (1980) decides on the minimum height criterion for wind (momentum flux) as z* = 150z0 and 35zQ respectively for a sparse (zQ = 0.4 m) and a denser forest (z = 0.9 m) he investigated, again indicating the 5 importance of the spacing of the roughness elements. In the case of air temperature (heat flux) Garratt (1980) found z* = 100zQ. Further, by incorporating Raupach's (1979) observations, Garratt (1980) concluded that for wind z*lzQ = 100 would be a useful generalization and the ratio probably decreases to about 10 for high density vegetation. For temperature over a dense forest z*/zQ is about 65. The Garratt (1980) recommendations are valid for unstable atmospheric conditions and refer to heights measured from the zero-plane displacement upwards. Considering the fact that, for surfaces with tall roughness elements, the sensors have to be placed tens of metres above the surface and that a fully adjusted boundary layer only develops very slowly, the upwind fetch has to be in the order of kilometres to avoid advective effects (Munro and Oke, 1975). Hence we conclude that the height of sensor operation, z, has to meet the condition of z/zQ > > 1 to avoid wake effects, however, it has to be small enough to ensure that the vertical fluxes can be assumed constant with height. These requirements deserve special attention in environments with tall roughness elements like cities and forests. Associated with increasing height of measurement is a need for larger averaging times when sampling turbulent characteristics, because the dominant eddy scale in the streamwise direction increases with height and so does the integral time scale. Lumley and Panofsky (1964) show that required averaging times for turbulence moments increase with the order of the moment. Based on Lumley and Panofsky's averaging time expression, Wyngaard (1973) demonstrated how the required averaging time for second moments depends on the variables involved and on the stability state. For local free convection he approximated the averaging time AV by: AV = (zVUe2) (<B' 2>/<B> 2) (1) 6 where z', U and e are the height above surface layer datum, the mean wind speed and the desired accuracy, respectively, < B > is the mean of the variable of interest and <B > is the ensemble variance of B about its ensemble mean. He approximated the second factor on the right side of (1) on the basis of results from 'ideal' surfaces (e.g. extensive, flat grasslands). For neutral and unstable conditions the following averaging time relations for the second moments of the vertical wind speed (w), air temperature (T) and the kinematic fluxes (covariances) of momentum (uw) and sensible heat (wT) are derived: AV, dw,flT 4/e 2 (zYU) (2) neutral uw 207e z(z'/U) (3) AV, Cw,oT 4/e 2 (z'/U) (4) AV, uw 100/ez(z'/U) y unstable (z7L = - l ) (5) 12/e 2 (z7U) (6) where tfw and <£p are the variances of vertical windspeed and temperature and L is the Monin-Obukhov length. 7 1.3 Research objectives This thesis presents the results of a project designed to measure and analyse the turbulent fluctuations of u, w, T and the covariances of uw and wT at one height over an urbanized area. The purpose of the study is to present the turbulence measurements in a form so as to arrive at conclusions concerning the observation height involved and the averaging times used. The analysis should provide information regarding the suitability of this site for turbulence measurements and their use in energy balance computations. This is of interest since the same site has been used in a number of suburban energy balance studies (e.g. Steyn, 1980 and 1982; Kalanda et al., 1980; Oke et al., 1981; Oke and McCaughey, 1983; Cleugh and Oke, 1986 and Oke and Cleugh, 1987). The approach taken is to compute the energy spectra and cospectra which are very powerful tools in analysing turbulence fluctuations since they represent the distribution of energy or variance (covariance) with respect to frequency. Information about the dominant eddy sizes involved in the turbulent transfer can be gained from the shape of the spectra (cospectra). If, as suggested by the studies cited earlier, an extra length scale due to influences of individual roughness elements is introduced, deviations from 'ideal' reference spectra should be observed in the shape of the spectra (cospectra). If these are present it means that the sensors are being operated too low so that results relate to processes in the roughness layer (given the observation height used there is little possibility that such deviations are due to advective effects i.e. the sensor is too high). The question of averaging time is approached by inspecting the low frequency end of the spectra (cospectra). If, for instance, the averaging time chosen is too short, low frequency contributions are lost resulting in a less prominent roll-off on that side of the spectra (cospectra). Unfortunately some of the variables (e.g. horizontal wind and temperature) 8 are characterized by only slightly decreasing spectral density with decreasing frequency, which makes analysis of this kind difficult. Turbulence studies over 'ideal' surfaces have been performed at a few sites in the past, the most extensive one being the 1968 AFCRL (Air Force Cambridge Research Laboratories) experiment in Kansas (e.g. Haugen et al. 1971) which was an attempt to obtain a comprehensive set of data on wind and temperature fluctuations over a flat and uniform site. The results from this 'ideal' boundary layer experiment showed the applicability of MOST. Therefore the empirical results from the AFCRL study (e.g. Kaimal et al., 1972) and another 'ideal' boundary layer experiment performed in Minnesota (Kaimal, 1978) are used as a standard against which the suburban results are compared. In the following the phrase 'reference spectra' will refer to the spectral results from these two studies. They are also referred to hereinafter as the 'reference' or 'Kaimal' data. 9 2. THE N A T U R E OF TURBULENT SPECTRA AND THE FIELD P R O G R A M M E 2.1 Spectral representation The representation of spectra and cospectra is based on turbulent energy and temperature variance budgets and the Kolmogorov-Obukhov hypotheses (e.g. Champagne et al., 1977). According to the second Kolmogorov hypothesis for the inert ial subrange (between the production and dissipation scales) the spectral densities (S) of velocity components are given: S ( k x ) = A x t ^ k f 5 7 3 (7) where k^ is the wavenumber in the x-direction; 6 is the dissipation rate of turbulent energy and A ^ is a universal constant for the inert ial subrange (Kolmogorov constant). A s a consequence of local isotropy A ^ for the vert ical and lateral velocity components differ by a factor of 4/3 from A-^ for the longitudinal component. S imi l a r ly , for the temperature fluctuations, the one-dimensional spectrum can be represented: S T ( k j ) = A 2 £ ' 1 / 3 N k j " 5 7 3 (8) where N is the rate of destruction by molecular conductivity of T^ /2 . A 2 is a constant analogous to A j in (7). In both, (7) and (8), the spectral densities are in units of velocity squared, or temperature squared, per unit wavenumber. In the field we measure frequency, not wavenumber and the conversion from one to the other is made through the use of Taylor ' s frozen turbulence hypothesis. It is assumed that, in practice the wavenumber can be replaced by: 10 k1 = 2tm/U (9) where n is the frequency in Hertz. With kj S(k1) = n S(n) (10) and by using (9), equations (7) and (8) can be rewritten: nS(n) = A 6 2 / 3 (2im/U) " 2 / 3 (11) nST(n) = B £ ' 1 / 3 N (2irnAJ)-2 / 3 (12) Surface layer similarity predicts that the frequency spectra expressed in dimensionless form are functions only of stability and the dimensionless frequency: f = nz/U (13) Since dissipation rates are not easily measured, the spectra are often normalized with the total variances leading to the forms: n S(n)/o*s = S(f, stability) (14) n ST(n)/(TT = S(f, stability) (15) where the spectral density is in units of velocity, or temperature, squared per unit frequency. o"s and Orp are the variances of the velocity and temperature fluctuations 11 respectively. In this form, if the left-hand side of (14) and (15) is plotted against the logarithm of f, the area beneath the spectra is proportional to the variance for a specific frequency band. This normalization also results in the spectra being proportional to f in the inertial subrange. Similarly for the cospectra, based on similarity arguments by Wyngaard and Cote (1972), the dimensionless forms (by normalizing with the covariances as in (14) and (15)) are: n Couw(n)/uw = Co u w(f, stability) (16) n Cow T(n)/wf = Co w T (f, stability) (17) and the cospectra are proportional to f " 4 7 3 in the inertial subrange. 2.2 Instrument requirements and measurement techniques Turbulent transport in the surface layer involves a wide range of eddy sizes. Therefore measurement requires not only sensors with sufficient frequency response but also the capacity to record and process large volumes of data. Fast response sensors ensure that the instrument responds to the higher frequencies at which a substantial energy content can be found. The high frequency cut-off can be chosen so as not to exclude significant energy. This limit depends on the stability of the atmosphere and the variable of interest. On the low frequency end the cut-off is determined by the length of the record. In the context of atmospheric turbulence, however, it is not a- simple task to decide on the appropriate cut-off frequency on this side of the spectrum where the 12 transition between microscale turbulence and mesoscale processes occur. If the averaging time is too long (greater than 1 hour) the stationarity of the time series, which is one requirement of surface layer theory, is in doubt. McBean (1972) discusses errors associated with inadequate choices of low and high frequency cut-offs. One of the most direct techniques for measuring turbulent quantities and fluxes is the eddy-correlation method. It involves recording the time variation of the fluctuations of variables at a fixed-point in space. The mean of the instantaneous fluctuations is subsequently removed (Reynold's decomposition) and the time-means of the respective products of the remaining turbulent fluctuations are evaluated. This leads to the following definitions of the variances, covariances and flux densities: The averages can be considered as the superposition of the oscillation of the respective fluctuations over all frequencies such that the spectral form for the variances and the kinematic fluxes of momentum and sensible heat are: ru,v,w,T = u 2> v 2> w 2 > T 2 (18) (19) Q H = oc p wT (20) ru,v,w,T 5 I u,v,w,T W (21) CO UW J l (n) dn (22) o oo wT $ J w T (n) dn (23) o 13 In practice the frequency limits of zero and infinity at the low and the high frequency end respectively are replaced by the real limits imposed by the sensor response and the averaging time as discussed earlier. The spectra and cospectra are determined by digital Fourier transform techniques, which give discrete values of f (n) at intervals of n such that i (n)dn gives the covariance in a band, centred at n, of width dn. In terms of frequencies the high and the low frequency cut-offs are at the Nyquist-frequency fjsj which is set by the digitization period dt such that fj^ = l/2dt and the reciprocal of the duration of the measurement. As Large (1979) points out, contributions from frequencies higher than fj^ can be allowed to alias back below fj^ which increases the effective upper limit of the integration. 2.3 Measurements of turbulence from urban environments In comparison with typical rural terrain turbulence over urban areas is modified due to mechanical and thermal effects induced by the large roughness and the presence of a 'heat island' leading to greater turbulent activity. In addition, it is possible that significant scales of turbulent fluctuations are generated due to the regular geometry of cities and also pressure effects may be important in momentum transfer. The following section is a review of the relatively few studies of spectral measurements of atmospheric turbulence above an urbanized surface. Most studies lack a complete set of parameters for the evaluation of the similarity functions used for proper MOST scaling. In this respect the studies conducted by Clarke et al. (1982) and Hogstrom et al. (1982) who normalized their results with the dissipation functions, are notable 14 exceptions. Most spectral results are only normalized by the variances or covariances and are therefore comparable with the results in this thesis. Comparison between studies is made difficult because of the inter-site variability in the characteristics of the surfaces, and differences in measurement height. The studies most comparable to that presented here are those by Coppin (1979), Steyn (1980) and Clarke et al. (1982). Their programmes were very similar concerning measuring heights, instruments employed and the nature of the surface. Most studies have been conducted using sensor heights below the recommended z* values. For the vertical velocity component over urban terrain Ramsdell, 1975 (6.7 m above roughness elements), Coppin, 1979 (height 33.8 m), Steyn, 1982 (same site as used in this study, height; 29 m) and Hogstrom et al., 1982 (6 m above roof top) find more energy at lower frequencies than the reference spectra, with a slight shift of the peak towards lower frequencies, for unstable to near neutral stratification. This feature could not be confirmed by Clarke et al., 1982 (height: 31 m) whose spectra did not show a deviation from the Kaimal reference curves for unstable stratification. Clarke et al., however, do report more low frequency energy in the vertical component for stable and neutral conditions. For the horizontal components Steyn (1982) and Hogstrom et al., 1982 (v-component only) again report more low frequency energy than the reference standard for unstable stratification. On the other hand, for the u-component Coppin (1979), Clarke et al. (1982) and Hogstrom et al. (1982) show agreement, or lower, spectral densities than measured by Kaimal. In neutral and stable conditions Clarke et al. (1982) report, similarly to their vertical component spectra, more low frequency energy for the horizontal component due to the developement of a peak at low frequencies as a result of mesoscale processes. 15 The scarce information that is available about the spectra of scalars and the cospectra of fluxes over urban surfaces, show similarities to the rural reference results. Coppin (1979) observed a stability dependence of the low frequency end of his temperature spectra (normalized by the total temperature variance) over a suburban area. The stable and neutral temperature spectra of Clarke et al. (1982) show a shift of the peak to lower frequencies. Their unstable temperature spectra have a broad peak and suggest no differences compared to the corresponding Kaimal spectra. Stress cospectra reported by Bowne and Ball, 1970 (height: 53.5 m) exhibit more low frequency energy than is usually observed, whereas Coppin's (1977) results coincide with the reference spectra from Kaimal et al., (1972). The heat flux cospectra measured by Coppin show more low frequency energy with increasing instability. Apart from the horizontal velocity component results of Ramsdell (1975), there is general agreement (in the few studies) that the spectral length scale derived from the frequency of the peak amplitude of urban spectra increases with height of measurement in agreement with those observed over 'ideal' sites. The stability dependence of the vertical length scale also conforms with similarity theory showing an inverse relationship with stability. For convective conditions Wamser and Miiller, 1977 (height: 50 and 250 m) observed a decrease in the vertical length scale with increasing roughness. This could not be confirmed by Clarke et al. (1982) whose spectral length scale for the vertical component was generally larger in comparison with reference data and showed no dependence on roughness. The latter result is in agreement with similarity theory. The spectral results, in general, support the applicability of Monin-Obukhov theory in the surface layer over urban terrain. The shape of all spectral components is preserved, 16 but minor changes in the location of the peak frequencies compared with reference spectra (e.g. Kaimal) and in the amount of low frequency energy are reported. The fact that the overall shape seems to be preserved is surprising for those measurements gathered at heights clearly within the roughness layer, where individual roughness elements are assumed to influence the turbulence statistics. For example the measured spectra of Hogstrom et al. (1982) and Ramsdell (1975) show virtually no change in their overall shape or even in the location of the peak frequency. 2.4 Observation site The data for this study were gathered over a suburban area of Vancouver, British Columbia, Canada as part of a suburban surface and boundary layer research programme during the summer of 1986. Other results from this micrometeorological programme are presented in analyses by Steyn and McKendry (1988), Cleugh (1988), Grimmond (1988), Schmid (1988) and Steyn (1988). Since this study is concerned with measurement and analysis of turbulence statistics in the surface layer above tall roughness elements, site selection becomes very important. A prime criterion is the fetch required for the atmospheric layer of interest to adjust to changes in surface properties upstream. The adjustment must completely propagate through the layer, resulting in no height dependence of the fluxes. To probe the surface layer, the instruments are mounted on a tower. The general requirements leading to the selection of this particular site are fully described and discussed in Kalanda (1979) and Steyn (1980), so only a synopsis of the relevant information is given here. 17 The general climatology of Vancouver, B.C. is discussed in Hay and Oke (1976). Briefly, the large scale flow is predominately from the westerly quadrant. Embedded in this flow are disturbances (cyclones) which are best developed in winter. Several physical features of the landscape combine to produce local climate variations. Chief among these are the influences of orography, proximity to water bodies and urbanization. Vancouver is located at the mouth of a coastal valley (lower Fraser valley) which extends from the Strait of Georgia in the west to the Fraser canyon in the east. It is bounded by the Coastal Mountains, which are dissected by fjiords, to the north and the Cascade Range to the south east. Because of this setting, both land/sea and mountain/valley circulations occur as shown by Steyn and Faulkner (1986). The effect of the land/sea breeze is to cause mainly westerly winds during daytime and weaker easterly flow during the night. Due to the sea breeze and the associated advection of air, the height of the ABL is reduced to only 500m typically (e.g. Steyn and Oke, 1982; Steyn and McKendry, 1988). The suburban site, called Sunset, is located in south Vancouver (Figure 2.1) and consists mainly of residential housing. Within a circumference of more than 1km 1-2 storey dwellings prevail with a mean height of 8.5m and an inter-element spacing of approximately 23m (Cleugh, 1988). The surrounding area is 64% greenspace (lawns, gardens, parks etc.) and 36% built up (25% houses and other buildings, 11% paved). The tower site itself which is located in a transformer substation has an albedo of 0.12-0.14. The otherwise residential land-use also includes the following special features: - between 0.75-2 km NW of the site there is a park and a cemetery both of which are irrigated with water during dry weather. 18 - 100m SE of the tower is a 20 m high school building. - 1-2 km S and NE of the site there are scattered commercial and industrial buildings. The aerodynamic roughness of the surrounding suburban surface is assessed by Steyn (1980) using Lettau's (1969) land-use/roughness element analysis over sixteen sectors. This gives a roughness length ranging from 0.4 to 0.7 m with a mean of 0.5 m. Despite the spatial variability in the structure of the surroundings no dependence of roughness length on wind direction could be found, indicating reasonable surface homogeneity. Steyn (1980) computed a zero-plane displacement length, d, in the western sector of approximately 3.5 m based on estimations from land-use analysis. This length was used to adjust all height measurements to a more realistic datum for the surface layer. Figure 2.2 provides a view from the top of the tower to the west, the direction of the prevailing daytime winds. Instruments to probe the surface layer were mounted on a triangular-section, steel lattice, free-standing tower erected in the south-east corner of the Mainwaring substation (Figures 2.3 and 2.4). The tower was fitted with an external climbing ladder and the instruments for this study were mounted on top of the tower. Steyn (1980) calculated a shadow factor of 0.14 for the upper section of the tower. A trailer at the base of the tower housed the recording and data logging equipment. Figure 2.3 shows a schematic view of the tower, the instrument locations and the relation of the tower to the immediate surroundings. To avoid tower effects on the variables to be measured, the sonic anemometer/thermometer system (SAT) was mounted at the end of a 1.3 m long boom facing to the south-west at a height of 27.5 m Figure 2.2: Photographic view from the top of the tower to the west. 20 > 4) N E SW > 6 - 4 J " ' U _ ,0) 0) 5 &b • Q -5 4 instrument trailer a 22.0 19.0 18.4 13.9 12.1 8.9 0.0 d (30.5) (27.5) (26.9) (22.4) (20.6) (17.4) (8.5) (5.0) •(0.0) Figure 2.3: Schematic of the Sunset tower and instrument locations. Instruments include: 2-propeller Gill anemometer (level 6); net pyrradiometer, sonic anemometer/thermometer, Krypton hygrometer (all at level 5 SW); relative humidity probe, Met-101 wind vane, Met-101 cup anemometer (all at level 5 NE); Bowen ratio system (levels 1 and 4) and two 2-dimensional sonic anemometers (u,v) (levels 2 and 3). 21 above ground. This arrangement ensured that the SAT was located upwind of the tower and was facing into the predominant wind direction. A 2-propeller Gill anemometer (see Chapter 2.5.1 for details) was mounted at the top of the tower at 30.5 m above ground where it enjoyed an unobstructed 'view' in all directions. The mean temperature was derived from dry-bulb thermometers which were part of a Bowen ratio system with sensors at 17.4 and 26.9 m above ground level. Other instruments are also indicated on Figure 2.3. They were part of a parallel micrometeorological study (Cleugh, 1988). Figure 2.4 shows a photographic view of the entire tower from the west, and Figure 2.5 provides a close-up of the top part of the tower. Since the base of the tower was 5 m below the base of the surrounding land (E = 5 m), and allowing an additional 3.5 m for the displacement length d, the effective height for the turbulence measurements z' = z - (E + d) which is 22 and 19 m for the Gill and SAT systems, respectively and 13.7 m for the measurement of the mean temperature. Using the dimensions of the Sunset tower, and the recommended methods to calculate z* given in Chapter 1.2, we find values for wind (above ground level) of 25-50 m (Tennekes/Townsend), 21 m (Pasquill), 43 m (Raupach et al), 38 and 26 m for momentum and sensible heat flux, respectively (Garratt, 1978a) and 73 m (Garratt, 1980; using the inter-element spacing approach). Taking Garratt's (1980) recommendations expressed in terms of zQ, z* is between 21 and 79 m (above ground) for the momentum flux, depending on the density of the surface roughness elements and about 54 m for the sensible heat flux. Hence, in this study with measurement heights of 30.5 and 27.5 m only, there is reason to question whether the instruments are operated at a sufficient height or if the observations are likely to be influenced by wake effects. Figure 2.4: Photographic view of the Sunset tower. Figure 2.5: Photographic view of top part of Sunset tower. 23 Under unstable conditions, equations (4) to (6) from Chapter 1.2 yield averaging times of 19 min (second moments of vertical wind speed and temperature), 543 min (momentum flux) and 56 min (heat flux) using z' values of 22 and 19m, a typical mean wind speed of 3 m s"* and a degree of uncertainty of 15%. As one approaches neutral conditions the averaging time for the heat flux increases and for the momentum flux decreases. Apart from the momentum flux the averaging values are intuitively reasonable. At time scales larger than about one hour or so non-stationary processes begin to affect the flow field thereby making it impossible to apply the recommended values for the momentum flux. In this study an averaging time of 60 min was chosen for all second moments measured. 2.5 Instrumentation 2.5.1 G i l l anemometer The turbulent fluctuations of the longitudinal (u) and vertical (w) velocities, the covariance of the kinematic momentum flux (uw) and the mean wind direction were measured with a modified Gill twin propeller-vane anemometer (Pond and Large, 1978). The Gill propeller anemometer system possesses a number of advantages including good calibration stability (not only with a particular propeller but also from propeller to propeller), it is simple and easy to service, has very low power requirements and is reliable for fairly long periods. However, there exist some drawbacks resulting in errors. These deficiencies are discussed by Horst (1973) and concern the cosine, frequency and threshold responses. 2k Although these response problems pose no serious difficulties in determining the horizontal components, they affect the measurement of the vertical velocity. Due to the small magnitude of the vertical wind and its large angle-of-attack, the vertical sensor is often operating in its non-linear calibration region and has a large distance constant. McBean (1975) concluded that the vertical anemometer is operating in its linear calibration region for less than 5% of the time therefore most probably reducing variance and covariance measurements. Based on this argument several authors have suggested rotating the vertical sensor by some amount to introduce some of the horizontal wind component into the vertical one. The system used for this study was a Gill twin propeller-vane anemometer (GTVA) as described by Pond and Large (1978). Further description and results can be found in Large (1979), Pond et al. (1979) and Large and Pond (1982). The system uses a propeller-vane for the horizontal velocity components and a propeller which is tilted at an angle of about 60° relative to the horizontal plane for the vertical component. The vane ensures that the whole system is always directed into the wind. By using a tilted propeller approximately one half of the horizontal component is introduced into the vertical one. This reduces the problems associated with non-linear response and stall and improves the frequency response. The mathematical derivation of the u and w components from such signals is given in Appendix A . l . Calibration adjustment to account for the non-linear response is described in Appendix A.2. The propellers used were 4-bladed with a diameter of approximately 0.227 m. On the basis of adjusting the high frequency end in the measured spectra to the required -5/3 slope the distance constants (D) were determined to be 0.75 m and 0.75 m / B(«) = 0.86 m (where B(«) is a factor to account for the angle in the tilted sensor) for the VI (horizontal), and V2 (tilted), propeller respectively. If we assume that the sensors behave 25 as simple RC filters then the 3dB down frequency n Q (half power point) can be found by n Q = U/2wD, demonstrating that the cut-off frequency decreases with increasing D. For a wind speed of 3 m s"1 and the respective D values, n Q is 0.64 and 0.56 Hz for the VI and V2 propeller, respectively. Pond and Large (1978) list possible error sources in the measurement of the momentum flux with the GTVA system. Careful measurement of the angle between the horizontal and the tilted sensors is essential. An error of * 1° in this angle results in a *5% error in the momentum flux. Calibration errors are in the order of 4% and errors in the compensation for the non-cosine response are * 3%. Pond and Large (1978) conclude that if everything goes wrong, and if the leveling of the instrument is off the horizontal plane by *5*, the kinematic momentum flux may be in error by as much as 18%. However, since errors may tend to cancel each other a lesser overall error can be assumed. The accuracy of this system is comparable with most other systems for measuring momentum flux densities. 2.5.2 Sonic anemometer/thermometer The turbulent fluctuations of the vertical velocity (w) and the temperature (T) and hence the covariance of the kinematic heat flux (wT)' were measured with a sonic anemometer & fine-wire thermocouple (SAT) (Campbell Scientific, Model CA27T). The general theory of a sonic system is discussed in detail by Kaimal and Businger (1963). The original form of the anemometer part of the SAT used in this study was first introduced by Campbell and Unsworth (1979). 26 Sonic anemometers are particularly well suited for high frequency turbulence measurements because sonic techniques do not depend on a mechanical response to changes in the wind field. Frequency response is only limited by the spatial resolution along the acoustic path between transducers. The basic principle of operation is that one transducer oscillates at 40 KHz while the opposite transducer oscillates in response to this 40 KHz oscillation, with the phase difference between receiver and transmitter dependent on the vertical component of the wind vector. The square waves from the transmitter and receiver are then compared and the phase difference is integrated over one signal period and later converted to wind speed by further circuitry at the end. The signal direction and the roles of the transducers are then reversed, hence, each transducer is capable of acting either as a receiver or as a transmitter. The output of this particular sonic anemometer is 1 V/m s*1 over a range of -4.0 V. The characteristics of the anemometer which are most important for eddy correlation work are its resolution, the drift of the anemometer output at zero wind speed, the cosine response and the frequency response (Campbell and Unsworth, 1979). The noise levels on the output indicated a maximum wind speed resolution of better than 0.01 m s"1. The drift at zero wind speed is less than 3 mm s"1 K" 1 . Campbell and Unsworth point out that since the mean value of w is removed in eddy correlation calculations this degree of drift is unimportant unless the temperature of the electronics is allowed to change rapidly. The cosine response shows errors of less than ±10% over the range of -30° to + 30° from horizontal. McBean (1972) demonstrates that this range of angles accounts for almost all eddy flux transfer even in unstable conditions. The frequency response of the sonic anemometer depends on the path length. According to Mitsuta (1966), the signal amplitude is reduced by 10% at a frequency of 0.26U/d^, where d^ is the path length. In this case, with d-^  being 0.1 m, the 90% cut-off frequency would be at 2.6U Hz (m s"1)"1. 2 7 A fine-wire thermocouple probe (diameter 12.7/»m is mounted within the vertical axis of the sonic anemometer, 20 to 30 mm from the sonic path. The frequency response of the junction exceeds 30 Hz (Biltoft and Gaynor, 1987). The output is 0.25 V ' C ' 1 over a range of *4.0 V. The probe is referenced to the mounting base the thermal mass of which is presumed to be large enough to provide a constant reference temperature. Temperature variance is computed from the temperature difference between the ambient air temperature and this slowly varying thermal mass. Tanner (1987; personal communication) confirms a reference junction time constant of about 20 min. In a recent paper, however, Biltoft and Gaynor (1987) compare high frequency temperature measurements from a SAT system with temperature readings from fast-response platinum wire temperature probes. They conclude, by analysing data from non-stationary (advective) conditions, that the reference junction time constant may be considerably less than that quoted by Tanner. 2.6 Observation programme Observations were taken on eight days at the end of August and the beginning of September 1986. The observation period concluded a period of exceptionally good weather. The summer was dominated by an anti-cyclonic regime bringing very dry and warm weather to the Vancouver area. These conditions affected the first four days of measurements. The second half of the observation period was under the influence of a disturbance approaching from the west resulting in cloudiness and an increase in relative humidity. Table 2.1 lists the dates of measurement and gives a short account of the weather encountered. 28 Table 2.1: Dates and duration of turbulence measurements and some relevant weather variables. The numbers in brackets in the first column are the relative Julian days. Date Solar time Mean Temp. Wind- Wind-Gregorian CC) speed dir. (Julian) (m s'h (deg) 1986.08.22 (234) 13:36-20:16 23.8-21.3 2.6-1.1 229-302 1986.08.24 (236) 10:57-19:36 19.6-22.2 3.9-1.6 149-327 1986.08.25 (237) 15:11-20:02 25.7-22.8 3.2-2.0 319-213 1986.08.26 (238) 10:20-20:17 24.6-29.0 3.0-2.0 250-345 1986.08.27 (239) 05:06-09:33 19.8-22.9 2.1-0.7 180-262 1986.08.28 (240) 15:53-18:00 18.9-19.0 3.9-3.6 141-157 1986.08.30 (242) 11:03-17:55 15.6-17.9 2.0-3.5 123-142 1986.09.02 (245) 13:19-16:31 20.7-20.8 3.5-4.2 151-154 29 The cloud cover for Julian days 240 and 242 was 10/10 As and for Julian day 245 variable between 3/10-9/10 As and Ci. The raw data set consisted of about 45 hours of high frequency turbulence observations. 2.7 Data acquisition and processing Data recording and processing was carried out in 4 major phases: 1) recording with a micro data logger. 2) transfer of data from cassette tapes to magnetic tapes for further use on a main frame computer system; quality control of data set. 3) combining individual files of varying length to a standard runs length of 60 minutes. 4) applying software to perform calibrations, Reynold decomposition, spectral computations and plotting. All measurements were recorded with a Campbell Scientific (Model CR21X) data logger. The sampling frequency was 10Hz for the GTVA (VI,V2) and the SAT (w,T) components and 0.3Hz for the Gill-vane. The Gill inputs were single-ended and the Sonic inputs were double-ended (differential), the latter increasing the signal to noise ratio. The resolution of the single- and double-ended inputs are 0.666 and 0.333 mV, respectively resulting in resolution of about 0.0026 and 0.012 m s"1 for the Gill VI and V2 respectively and 0.3 mm s"1 and 0.0013°C for the sonic w and T components. The VI 30 and V2 signals were filtered at a frequency of 10 Hz by using a Rockland 452 low-pass filter to remove commutator noise. The data were subsequently transfered to cassette tapes. Transfer of the data from the cassette tapes to magnetic tapes for further use on the University's main frame computer, was accomplished by using two C20 interfaces (Campbell Scientific) connected in parallel to a Personal Computer (PC). The data were subsequently transferred in binary mode from the PC to the main frame and were converted back to ASCII. Finally a quality control program was applied to correct errors and to write the 'clean' data set to magnetic tapes. Since data acquisition during the observation periods was not always continuous (gaps of a few minutes did occur), the individual data sets were spliced together or discarded to produce 60 min continuous runs, this being the averaging period chosen. The low frequency cut-off was therefore at 0.28><10"3 Hz. In the last phase the turbulence characteristics were derived from the 'clean' 60 min time series (the names of the routines which follow correspond to those in the software provided by Roth, 1988). At first the sonic w, T and Gill VI, V2 (using the calibration procedure outlined in Appendix A.2) and vane signals were calibrated. The programs involved were P.CALD3 and S.GCORR. A second main program (P.RDEC) applied a linear regression to all time series (excepting the vane) to perform the Reynolds decomposition by subtracting the regression curve from the instantaneous signals and leaving fluctuations only with means of zero. The VI and V2 observations were just prior to the detrending converted to u and w with the derivation presented in Appendix A . l (S.GCALIB). In the same main program the variances, covariances and other statistical parameters were computed through the use of Subroutine S.VARI. The detrended time 31 series were then subjected to a test for stationarity (P.RTEST and S.RTEST) as explained in Appendix C. With the turbulent fluctuations as input the program P.SPECTRA computed the Fourier coefficients (by using the library program DFOURT), power spectral densities (S.FTSPEC) and averaged power spectral densities (S.AVLOG) which were later normalized by the total variance (S.SPECPL) to produce the final plots. Details of the Fourier coefficient computation are included in Appendix B. In addition the statistical parameters are again computed but this time in the frequency domain by adding up the Fourier coefficients (S.FVARIT) or computing the area beneath the spectra (cospectra) (S.FVARIA). The averaging procedure is an extended version of that used by Kaimal and Gaynor (1983) and resulted in equally spaced spectral densities in the log frequency domain with about 10 points per decade. To obtain good representation of the low frequency end the first 5 points were not averaged, with the result that the low frequency side is statistically very unreliable. Frequency response corrections were applied to the Gill signals in two stages using the response functions presented in Appendix B. In Subroutine S.FTSPEC all spectral densities are corrected up to a natural frequency of 2. Above this frequency noise starts to dominate the signal and makes a correction redundant. This response correction increased the standard deviations by about 3-4%. The same response functions are also used to correct the averaged spectral densities in Subroutine S.AVPLFC which has the uncorrected Fourier coefficients from S.FTSPEC as input. In addition, the high frequency end of the two velocity components are corrected in the same Subroutine with artificial -5/3 (-2/3) slopes. 32 From the 45 hours of data, 38 one-hour runs could be constructed. However, some of the data had to be discarded leaving a useful set of 26 to 33 runs (depending on the variable). The observations from Julian day 239 had to be dropped completely. Neither the GTVA or the SAT were working properly during periods with very low wind speeds and/or stable stratification characteristic of Julian day 239. The Gill measurements were not included when the mean wind speed dropped below 1.6 m s"^  for the w component and 1.1 m s"* for the u component, because in low wind speed conditions the Gill response is often non-linear and the propellers frequently stall. At the beginning of Julian day 242 the GTVA did not provide any useful results. Under near neutral conditions the temperature and the covariance signals from the SAT showed very small magnitudes and could not be interpreted properly. The runs were not used if they displayed non-stationary behaviour, however, if a run was only slightly non-stationary it was included in the final analysis. Julian days 240 and 242 were characterized by winds from the south-east sector. Due to the presence of a school building in that direction (see Section 2.4) special attention was given to the observations from these days, however, no striking differences could be observed in the spectra. 33 3. RESULTS 3.1 Time series representation Time series traces of the sonic (w, T, wT) and Gill (w, u, wu) signals encountered during unstable stratification are presented in Figures 3.1 to 3.4. 60 min runs from two different days are included; run 234.2 is from a period in late afternoon with clear skies and run 245.1 from an early afternoon with 8/10 Ac cloud. Stability (z'/L), mean wind speed, heat flux and friction velocity for run 234.2 were -1.4, 2.1 m s'^, 109 W m"2 and 0.26 m s"1 and for run 245.1 -1.0, 2.9 m s"1, 234 W m' 2 and 0.37 m s"1. These time series can not be regarded as representative of the whole data set. However, they show some interesting features and give a useful guide to the behaviour of u, w and T and the way they are correlated. To plot the time traces the observations were averaged over 10 data points, thereby giving a representation every second. The straight lines in Figures 3.1 to 3.4 indicate the linear regression applied to detrend the time series. Figure 3.1 is constructed from observations with the SAT. The w time series in Figure 3.1a shows the expected erratic behaviour of the fluctuations where updrafts and downdrafts follow each other randomly. In about the middle of the time series (at t = 1500-2500 seconds) some longer periods of upward and downward motions can be observed. The regression line indicates a mean different from zero. No zero point calibration was performed on the w sensor of the SAT so an offset in the signal could be responsible for the observed deviation. Nevertheless, the assumption that the vertical velocity fluctuations have a mean of zero over a long enough averaging time has to be questioned over the high roughness surface of an urban area. 34 Figure 3.1: Time series traces (60 min) of w (a), T (b) and wT (c) from the sonic (SAT) measurements for run 234.2. Straight lines an (a) and (b) indicate the linear regression applied for detrending. 35 The temperature trace in Figure 3.1b shows that a large part of the transfer occurs in up- and downdrafts of a very particular structure. A sharp drop in temperature (e.g. at t = 600, 1100, 1700, 3000 sec) is followed by a slow increase which is not continuous but rather occurs in small bursts which seem to become amplified towards the time of the next drop in temperature. The period of the entire 'structure' is between 3 and 9 minutes. It should be noted that this feature is different from the commonly observed saw-tooth structure in the temperature signal which is usually of the order of 100 seconds or less (eg. Wilczak and Businger, 1983). Coppin (1979) observed this saw-tooth behaviour in temperature time series from a suburban area (his Figure 4.5), however, the time from downdraft to downdraft was only about 2 to 3 minutes. The kinematic heat transfer (Figure 3.1c) is dominated by periods of active transfer separated by quieter periods. Since the stratification was unstable the active periods are predominately positive (upward). During the quiet periods the sense of the vertical motion is more random. Figure 3. la and c show that the active periods coincide with the upward movements of buoyant thermals (e.g. at t = 600, 900, 1500-1800 seconds). Figure 3.2 displays the observations from the GTVA. The w fluctuations (Figure 3.2a) exhibit exactly the same features as those from the SAT. This is an indication of the usefulness of the tilted-propeller as a sensor of the vertical wind. The time series traces of the along-wind component u (Figure 3.2b) show similar characteristics to those of the vertical component, however, the up and downdrafts seem to be less randomly distributed in time and are often interrupted by 'bursts' which seem to find a stronger expression in the transport downwards. Compared to the w fluctuations the magnitudes of the u velocities are slightly larger, resulting in a higher variance. 36 Figure 3.2: Time series traces (60 min) of w (a), u (b) and uw (c) from the Gill (GTVA) measurements for run 234.2. Straight lines in (a) and (b) indicate the linear regression applied for detrending. 37 The momentum flux in Figure 3.2c is characterized in a manner similar to the heat flux, by active and passive periods. The active periods occur in very short bursts of relatively large magnitude. As expected the momentum flux is predominately negative. Hence, strong updrafts are correlated with a decrease in the u component (e.g. at t = 600, 900, 1500-1700, 2000, 3000, 3200 seconds) and vice versa (e.g. at t = 1300 seconds), the latter case usually not resulting in very strong correlations. The corresponding SAT and GTVA time series traces for run 245.1 are presented in Figures 3.3 and 3.4. As before, the w fluctuations (Figure 3.3a) occur in a very random fashion and there seems to be no preferred direction of transfer. Also, similar to the previous run, the updrafts have higher values than the downdrafts. Compared to run 234.2 the overall magnitudes are larger leading to a higher variance. The Gill w time series (Figure 3.4a), again, follows the sonic observations very closely. The structure of the temperature signal described for run 234.2 (Figure 3.1b) also applies to Figure 3.3b, however, the ramp features seem to be less prominent. It is interesting to try to relate the observed behaviour to physical processes in the atmosphere. One possible explanation follows Wilczak and Businger (1983) who relate a similar feature in the temperature signal, but occurring over a very much shorter period, to large thermals which are immediately followed by cold downdrafts. Businger, 1987 (personal communication) points out that the features evident in Figures 3.1b and 3.3b show the same characteristics observed in the temperature structure of cold and warm fronts and hence relates the ramp feature to such temperature slopes of fronts (which can be found on either side of large eddies) advecting past the fixed sensor. The along-wind component (Figure 3.4b), and the kinematic fluxes of heat (Figure 3.3c) and momentum (Figure 3.4c) exhibit the same features as run 234.2. The larger 38 ire 3.3: Same as Figure 3.1 but for run 245.1. 39 Time in seconds (centered ai 13:45) b) 300 • 0 1000 2000 3000 Time in seconds (centered at 13:45) 're 3.4: Same as Figure 3.2 but for run 245.1. 40 magnitudes of the variables of interest again indicate a larger variance of the along-wind component and a larger sensible heat flux and friction velocity compared to run 234.2. 3.2 Spectral characteristics The individual and composite spectra and cospectra presented in the following section are computed and normalized as described in Chapter 2.7. Composite spectra are the average of all individual spectra of a particular variable, whereby the averaging has been performed over previously chosen non-dimensional frequency bands to give equally spaced composite spectral densities in the log frequency domain with about 10 points per decade. The spectra and cospectra represent different atmospheric stability; ranging between -2.4 < zVL < 0.06 with an arithmetic mean of z'/L = -0.75. No attempt was made to classify the spectra (cospectra) normalized by the variance (covariance) according to stability, although some stability dependence could be observed. In the following figures the spectral and cospectral densities from this study will be denoted by crosses (x). 3.2.1 Vertical velocity spectra The individual dimensionless spectra (27 runs) of vertical wind plotted against reduced frequency f = nz'/U are given in Figure 3.5a. On the high frequency side relatively slight scatter is observed, whereas on the low frequency end the spectral densities spread over a large range. The scatter at the lower frequencies is partly due to variation in stability between individual runs, but also reflects the reduced degrees of freedom in the computation of the spectra. Figure 3.5b shows the corresponding Figure 3.5: Spectrum of vertical velocity normalized with the variance: a) for all 27 runs b) composite spectrum with + and - one standard deviation. composite spectrum. The end points of the vertical bars in this figure indicate plus and minus one standard deviation. As already noted the scatter increases with decreasing frequency. In general, the composite spectrum shows a fast roll-off at the low frequency end followed by a peak and the required -2/3 roll-off in the inertial sub-range. The straight line-segment in the upper right corner in this and following plots indicates a -2/3 slope. The composite w spectrum (without error bars) is again presented in Figure 3.6a and compared with the results derived from the GTVA. The agreement between the two is excellent, however, it should be noted that the Gill measurements were extended with an artificial -2/3 slope in the inertial sub-range. Nevertheless, the fact that the two composite spectra measured with two different systems, follow each other so closely is very promising and raises confidence in the use of the GTVA. At non-dimensional frequencies higher than about f = 10 the sonic measurements are contaminated by aliasing (probably caused by unfiltered signals). In Figure 3.6b the composite w spectrum is compared with that of Steyn (1980) from the same site. There are two obvious differences between the two. Firstly, Steyn's spectrum shows a faster roll-off than the required -2/3 slope in the inertial sub-range. Steyn measured the vertical wind components with a single vertical Gill anemometer which is obviously not responding to the highest frequencies (see Chapter 2.5.1). On the low frequency side the composite spectrum of Steyn shows more energy content than that from this study. However, this 'increased energy content' is only relative since the position of the vertical spectrum depends on the stratification of the atmosphere and shifts to lower frequencies with increasing instability. Steyn took observations under very unstable conditions (z'/L up to -100) which therefore probably explains this apparent increase in low frequency energy input. 43 Figure 3.6: Composite spectra of vertical velocity normalized with the variance: a) sonic compared with Gill (circles) measurements b) this study compared with measurements by Steyn, 1982 from the same site (dashed line) and a model by ffyjstrup, 1981 (solid line). In Figure 3.6b the present observations are also compared with a spectral model suggested by H^jstrup (1981), based on a model by Kaimal (1978) using the Minnesota results. According to H^jstrup (1981) the spectral density normalized by the friction velocity can be expressed by: n Sw(n)/u* 2 = (32f/(l + 17f ) 5 / 3 ) (-z7L) 2 / 3 + 2f/(l + 5.3f 5 / 3 ) (24) Using an expression for the normalized variance of the vertical component (Panofsky et al, 1977): o y u * 2 = 1.5 + 2.9(-z7L)2 /3 (25) the friction velocity in the denominator on the left-hand side in (50) can be replaced by the variance. Evaluating this new function at z'/L = -0.75 (the mean of the stabilities from all the runs included here) gives the model spectrum normalized by the variance shown in Figure 3.6b. The agreement between the observed and modelled spectrum is excellent. The minor differences on the low frequency side are probably mainly due to statistical uncertainties. Compared to the model spectrum, the peak is less well defined and at non-dimensional frequencies of 0.05 < f < 0.1 a 'flat' region with slightly higher energies than the model can be observed. This makes it difficult to decide upon one specific peak frequency. Taking the spectral density with the largest magnitude the peak frequency f m is at 0.2 with a possible range of 0.08 < f m < 0.3. Since f = nz'/U = z / l m , where l m is the wavelength, and for f m = 0.2 the dominant eddy scale computes to l m = 5z' or about 100m. Steyn (1982) observed a peak wavelength of l m = 10z', occurring at f m = 0.1, however, as discussed earlier, his results are from more unstable conditions. Visual h5 inspection of Coppin's (1979) results and those of Clarke et al. (1982) shows a peak frequency at approximately f m = 0.2. Hogstrom et al. (1982) report slightly higher peak frequencies at f m = 0.35 and 0.5 for their two sites, however, their results are from near neutral conditions. 3.2.2 Horizontal velocity spectra The individual dimensionless spectra (27 runs) of the along-wind components are shown in Figure 3.7a. The individual spectra, as well as the composite spectrum in Figure 3.7b, show some scatter at high frequencies and increased scatter at the low frequencies. Compared to the w results the scatter is slightly larger for the u component. In Figure 3.8 the present observations are compared with the composite spectrum from Steyn (1980) measured at the same site. The results are quite similar, the differences at the higher frequencies again originating in the frequency response deficiencies of the Gill anemometer used by Steyn which are not as severe as for the vertical component since the distance constant of the horizontal sensors is smaller than that for the vertical one. As mentioned earlier in this study the GTVA measurements were corrected for the frequency response, therefore exhibiting the required -2/3 slope. The main differences between the observations from this field programme and the results from Steyn (1980) are a prominent dip at about f = 0.03 and a smaller dip at f = 0.2 in the present results. In general, the observed composite along-wind spectrum exhibits a maximum at low frequencies and a weak point of inflexion at higher frequencies (small dip mentioned above). These features are shown by Kaimal (1978) to be typical for unstable spectra in 46 Figure 3.7: Same as Figure 3.5 but for the along-wind velocity component. 47 the surface layer over relatively smooth surfaces. Kaimal separates the frequency range into three regions: regions 1 (high frequency end) and 3 (low frequency end) scale with z and zj respectively, whereas region 2 provides the transition between the two and shows the inflexion. Kaimal (1978) observed the transition between region 1 and 2 at a non-dimensional frequency of about 0.5, slightly higher than the present results indicate. According to Steyn (1980) the same transition occurs at about f = 0.25. The determination of a peak frequency is made difficult because of the prominent dip at the low frequencies. The highest spectral value can be found at about f m = 0.017 ( l m = 60z' = 1320 m) with a possible range of 0.01 < f < 0.03. Steyn (1982) observed 1 0 ° Figure 3.8: Composite spectrum of the along-wind velocity component, normalized with the variance, compared with measurements by Steyn, 1982 from the same site (dashed line). 48 the peak at the same wavelength. According to Coppin (1979) the peak occurs at approximately f m = 0.02 to 0.03. Agreement with the results by Clarke et al., 1982 is good. Again, Hogstrbm et al. (1982) report the peak to occur at slightly higher frequencies (fm = 0.05 and 0.1). Correspondence with u spectra from 'ideal' sites is good (e.g. Kaimal et al., 1972). Comparison is somewhat limited because the horizontal velocity components are shown to scale better with Zj and hence, most of the authors present their results within that framework. For this study only a limited number of boundary layer heights was available, hence no normalization with Zj was attempted. 3.2.3 Temperature spectra Figure 3.9a shows the individual temperature spectra (21 runs). The scatter at the higher frequencies is slightly larger than for the velocity components. The variability at the lower fequencies is similar in magnitude to that in the along-wind spectra (Figure 3.7a). Like the vertical wind component the scatter is probably due to stability variations from run to run. A surprisingly large amount of noise can be observed at the high frequency end. Recalling that the sonic signals are not filtered it is likely that noise from higher frequencies is aliased to below the Nyquist frequency. Potential sources of electrical noise are the electronics of the SAT itself and the ambient environment of the measuring site (the tower is located in a hydro-electric sub-station having high-voltage electric lines). The corresponding composite spectrum is shown in Figure 3.9b. The temperature composite spectrum in Figure 3.10 follows the -2/3 slope in the inertial subrange very closely. The T spectrum is different from the w spectrum but similar to the u spectrum in that it shows one broad peak and only a slight roll-off at the low frequency end. The peak frequency (as determined from the largest spectral density) 10" J I I I 111II I I I 11 III I I I I I I I 11 I ' ' I I 1111 10 -3 IO"2 "IO"1 10° / = nz'/U 101 102 Figure 3.9: Same as Figure 3.5 but for temperature and 21 runs. 50 is at about f m = 0.023 ( l m = 43z' = 817m). However, it is more realistic to give a range of f*m's as 0.01 < f m < 0.06. Temperature spectra from other studies utilizing measurements over rough surfaces agree quite well with the observations from this study. Clarke et al. (1982) report a broad peak at about f m = 0.02. The peak in Coppin's (1979) spectra is observed at approximately f m = 0.04. His spectra also exhibit a point of inflexion at a non-dimensional frequency of about 0.2 which is followed by a smaller higher frequency peak at about f = 0.7. Stretching the imagination a little a similar point of inflexion can be observed in the present composite spectrum (Figure 3.10) at about the same non-dimensional frequency, however, no higher frequency peak is present at f = 0.7. Hence, 1 Q -3 I I ' ' ' ' ' ' ' ' ' ' I I ' I I 1 0 " 3 10~ 2 10" 1 10° 10 1 10 2 / = nz'/U Figure 3.10: Composite spectrum of temperature, normalized by the variance, without standard deviations. 51 Coppin's spectra exhibit more energy, at and in the neighbourhood of, that non-dimensional frequency. The composite spectrum in Figure 3.10 shows relatively good agreement with spectra from smooth surfaces (e.g. Kaimal et al., 1972). The Kansas results also show the secondary peak reported by Coppin (1979), however, in the case of the 'ideal' terrain, this peak is not as prominent as suggested by Coppin. Analysis of the peak frequencies in the three turbulent components discussed so far suggests that the spectral peak for temperature is intermediate between those of the along-wind and the vertical wind components. This is again in agreement with observations from smoother surfaces. Lumley and Panofsky (1964) suggested that both velocity components contribute to fluctuations in temperature. 3.2.4 Heat flux cospectra The heat flux cospectra (27 runs) are shown in Figure 3.11a. The presentation is in area-preserving form, which means that the area beneath the/ individual cospectra is proportional to the covariance. Most obvious is the large scatter at the lower frequencies and the negative values (indicating downward heat flux) at the low frequency and high frequency ends. Negative values can be found at a non-dimensional frequency of about 0.02 and below, and at f = 10 and above. Averaging over all runs produces the composite spectrum in Figure 3.11b, also presented in area-preserving form. Now the negative values at the low frequency end disappear and only the last two points at the high frequency have small negative values. A single peak at about f m = 0.05 emerges in this kind of presentation. 52 b) 0.30 0.25 0.20 \ 1? 0.15 Co C 0.10 0.05 0.0 J I I I I I III I ' I ' I I I II | I I I I M M I I l l I I I I I I I 10" 3 lO" 2 l O - 1 10° / = nz'/U 101 102 Figure 3.11: Cospectra of sensible heat flux normalized with the covariance in area preserving form: a) for 27 runs b) composite spectrum. 53 In Figure 3.12a the composite cospectrum is plotted in the more familiar logarithmic form with indications of plus and minus one standard deviation. In the upper right corner of the same plot a -4/3 slope is drawn as required by theory for the roll-off in the inertial sub-range. Compared to the previously discussed spectra, the cospectra of heat flux definitely show more scatter in the low frequency range. In Figure 3.12b the composite cospectrum is presented without the vertical bars to facilitate the analysis of the cospectral shape. The composite cospectrum shows one broad peak which is bordered by a sharp roll-off at the low frequency end and another roll-off at the higher frequencies. At non-dimensional frquencies 2 < f < 8, hence within the inertial sub-range, the spectrum exhibits a -4/3 slope which is followed by a faster roll-off at the highest frequencies. This conforms with the theoretical expectation that at very high frequencies the cospectra should become zero because the increasing isotropy of the flow field results in a correlation approaching zero. The peak frequency, again at f m = 0.05 (L^ = 20z' = 380m), has a range of about 0.04 < f m < 0.1. Only one other author reports heat flux cospectra measured over an urban rough surface. Coppin (1979) found the peak to occur at about 0.06 < fm < 0.1 for unstable stratification which is in good agreement with the results from this study. For near neutral and stable conditions he observed a second peak at lower frequencies. A more extensive comparison with Coppin's results is not possible since he did not plot the cospectra in logarithmic co-ordinates. Observations from rural sites (Kaimal et al., 1972) show the same flat appearance in the mid-range and peaks at about the same non-dimensional frequencies. At the low frequency end, the composite cospectrum from this study falls off more rapidly than observed by Kaimal. This might be due to the increased number of negative spectral estimates observed, however, the inherent uncertainty in the derivation of the cospectral 54 Figure 3.12: Composite cospectrum of sensible heat flux normalized with the covariance: a) with + and - one standard deviation b) without standard deviations. 55 densities makes it difficult to decide if this feature is a real difference in comparison to reference cospectra from smoother surfaces or not. 3.2.5 Momentum flux cospectra Figure 3.13a shows the 25 individual runs of the momentum flux measurements presented in area-preserving form. The magnitudes are also inverted so that the positive cospectral estimates represent negative covariance. Compared to the heat flux cospectra (Figure 3.11a) more scatter is noticed on the low frequency side even up to relatively high frequencies. Negative values are again observed, occurring frequently below f = 0.04 and at frequencies higher than f = 7. This is slightly different from Coppin (1979) who observed large negative values on the high frequency side between about f = 0.5 and f = 1.5, remaining negative up to f = 8. The composite cospectrum in variance-preserving form is shown in Figure 3.13b. Averaging over all runs resulted in positive values at all frequencies. Again, the large scatter at lower frequencies and in the mid-range is quite noticeable. The peak can be observed at about f m = 0.04. Figure 3.14 displays the composite cospectrum in logarithmic coordinates. Similar to the heat flux cospectrum (Figure 3.12b) a fast roll-off at the low frequency end is followed by a peak which is not as broad as that observed for the heat flux. The peak frequency is, based on the highest cospectral estimate, at about f m = 0.04 ( l m = 25z' = 550m) with a possible range of 0.02 < f m < 0.06. Visual inspection of the results presented by Coppin (1979) in area-preserving form suggests a peak frequency of about f m = 0.03, therefore in agreement with this study. Compared to the heat flux cospectra the peak occurs at only slightly lower non-dimensional frequencies. 56 a) 0.8, 0.6 1^  i to 0.4 i 0 . 2 0 . 0 * x» xx* * X *» f * $Jx" ,)«*# *x x x V« -Ki ^4 -• » x* _x xx ,» x. ; * x *» *« ^^^jj^t; - 0 . 2 — — i i i 11 in 1—i i i 11111 i ' i i ' ' J ' 1 0 " 1 0 " 1 0 " 1 1 0 ° / = nz'/U 1 0 1 102 b) 0 . 3 0 , 0 . 2 5 ! 0 . 2 0 0 . 1 5 to 0 . 1 0 + + + 0 . 0 5 + + + + + 0 . 0 + + + + -—' i i 11 III 1—i i M nn 1—i i 1 i m i l 1 0 ~ 3 I O " 2 1 0 " 1 1 0 ° / = nz'/U 1 0 1 I O 2 Figure 3.13: Same as Figure 3.11 but for momentum flux, 25 runs and plotted negative. 57 10° 1Q-3 I " 1 I I I I I II J I * I ier 3 io- 2 10- 1 10° 101 102 / = nz'/U Figure 3.14: Composite momentum flux cospectra, normalized with the covariance, without standard deviations. The results agree with momentum flux cospectra measured by Kaimal et al. (1972) over smooth terrain whose results show a relatively narrow peak within a range of 0.01 < f m < 0.1 for unstable conditions. Kaimal reported that the cospectral estimates at very low frequencies (f < 0.01) show a tendency to reverse sign and become negative. In particular under unstable conditions the low frequency cospectral estimates fluctuate between large positive and negative values, as observed in this study, however, the results here suggest a sign reversal up to slightly higher frequencies. In the inertial sub-range the uw cospectra are required (by theory) to follow a -4/3 slope. An examination of this behaviour is not possible in this case because the high frequency estimates are contaminated by noise. Although the cospectra were not corrected for the frequency response, the shape in the inertial sub-range is not far from 58 the expected roll-off. Noise present in the individual u or w spectra should theoretically not affect the computation of the covariance since noise is usually purely random and is not correlated between the two signals. In this case, however, the u and w components are not derived independently: the original w signal contains some of the u signal. Hence, noise contained in one signal can propagate into the other and end up being correlated (Pond, 1987; personal communication). The effect of this is higher values in the high frequency region. At non-dimensional frequencies between 0.2 < f < 1.5 the cospectra show a relatively flat region which does not show up as prominently in measurements from smoother surfaces. Inspecting the individual runs it could be noticed that at frequencies of about f = 0.7 impossibly low values appeared (which were corrected to first order). A similar observation was made by Pond et al. (1971) in flux measurements over the ocean at about f = 0.2 (their Figure 3). They related this anomaly to wave motions affecting the instrument platform. In this case, f = 0.7 corresponds to a wavelength of about 31m which is about the height of the tower (Figure 2.3). Hence, it is likely that the observed anomalies were introduced by movements of the tower. 3.3 Averaging times To obtain valid estimates of turbulent variances and covariances the averaging times have to be chosen to be long enough to enable measurement of the low frequency contributions by the variables of interest. For this site, following the recommendations of Wyngaard, 1973 (see Chapter 2.1) the averaging times for the variances of w and T were computed to be 19 minutes and for the fluxes of heat and momentum to be 56 and 59 543 minutes respectively. In practice the averaging time should be chosen so that the low frequency energy cut-off is negligibly small. Converting the averaging time for w and T to a non-dimensional frequency, by using a typical wind speed of 3 m s'^  and a height of 20 m, the frequency cut-off computes to fc = 0.006. Inspecting the low frequency ends of the vertical wind (Figure 3.6) and temperature (Figure 3.10) spectra shows that 19 minutes (or fc = 0.006) might be sufficient to obtain a reasonable estimate for the w variance, but the temperature spectrum shows a considerable amount of energy at non-dimensional frequencies below 0.006. The low frequency shape of the u spectrum (Figure 3.8) is similar to that of temperature, suggesting that a longer averaging time would be more appropriate (about 45 minutes would be required for both the u and T spectra). In the case of the T and u spectra the specification of a low frequency cut-off is made difficult since they tend to develop a mesoscale low frequency peak under certain stability conditions. Hence, it is necessary to decide which frequency can be regarded as being truly microscale turbulence. The cospectra of heat (Figure 3.12) and momentum (Figure 3.14) both show a relatively sharp roll-off on the low frequency side. For the momentum cospectra an averaging time of between 30 and 40 minutes (fc = 0.004 and 0.003) might be sufficient, particularly when considering the large statistical variation in the cospectral densities at that end. For the cospectra of heat flux Wyngaard (1973) suggests an averaging time of 60 minutes which seems to be long enough to measure all low frequency contributions. The sharp roll-off in Figure 3.12 suggests that shorter averaging times in the order of 30 minutes (fc = 0.004) could be justified for the sensible heat flux. 60 The foregoing qualitative discussion ideally should be extended quantitatively as was done by McBean (1972) using flux measurements over relatively smooth surfaces. By computing the covariances for different low frequency cut-offs McBean concluded (for unstable conditions) that the percentage error in the flux for a cut-off at, for example, fc = 0.004 is in the order of 13%. One quantitative analysis was performed in this study. The heat flux covariances computed over 60 minutes were compared with heat flux estimates from within the same record but averaged over only 15 minutes (fc = 0.007), i.e. four 15 min periods are averaged and compared with the flux estimate of the corresponding 60 min record. According to McBean (1972) the cut-off at fc = 0.007 would result in a flux error (underestimation) of about 16%. The result of this comparison is presented in Figure 3.15. No significant under- or overestimation can be noticed over a large range of heat fluxes from about 5 to 300 W m . The few points which clearly underestimate in comparison with the 60 min averages show deviations of between 11 and 25%. However, overestimates are also present, but not to the same extent. Two remarks should be added in regard to this comparison. Firstly, in the frequency range of interest the cospectral estimates are poorly defined and show large variations leading to random departures in the fluxes. Secondly, the four 15 minute periods chosen to be compared with the 60 minute runs were usually not from exactly the same 60 minute time periods, but not off by more than 15 minutes. This might introduce some uncertainties especially during transition periods where the heat flux changes rapidly like in the early morning or late afternoon hours. 6 1 0 50 100 150 200 250 300 Sensible heat flvux averaged over 60 min Figure 3.15: Comparison of sensible heat fluxes (W m )averaged over 60 and 15 min, respectively. The solid line is 1:1. 62 4. S U M M A R Y A N D C O N C L U S I O N S The observational programme described in this thesis provides turbulence spectra of temperature, the vertical and longitudinal wind components as well as the fluxes of sensible heat and momentum measured in an unstable surface layer over a suburban surface. The results add to the sparse number of observations of this nature over rough urban surfaces and provide some insight regarding the nature of the turbulent fluxes. Not all of the data gathered were suitable for analysis, some had to be discarded. The main reasons being instrument problems (malfunction or failure), small signal inputs (mainly affecting the temperature measurements under near neutral conditions) and lack of stationarity. However, the remaining runs' (about 70% of the entire data set) are found to be useful and capable of providing some general conclusions and a basis for comparison with other studies. 4.1 Summary In Chapter 1.2 we noted that the sensor height over a rough surface has to be sufficiently large to ensure that the measurements are made above the roughness sub-layer, otherwise the turbulence quantities are known to be affected by individual roughness elements. Considering that the sensors were operated at a height of only about 20 m above zero-plane displacement, which is below the recommended height for this site (between about 20 and 75 m), the results from this study show remarkably good agreement with reference results from 'ideal' (low roughness, homogeneous fetch) sites (e.g. Kaimal et al., 1972) and with studies from other urban turbulence programmes. 63 More specifically, on the high frequency side, in the inertial sub-layer, the spectra of w and T follow the required (by theory) -2/3 slope very closely. No conclusions can be made concerning the high frequency shape of the u spectra, since the Gill signals were corrected for frequency response on the basis of adjusting the roll-off to a -2/3 slope. In the case of the wT cospectra the required -4/3 slope in the inertial subrange is matched very well and is followed by an increasingly steeper roll-off which is in agreement with theory and results from smoother surfaces. Because of the noise problem decribed earlier, the uw cospectral densities at the high frequency end cannot be analysed. Compared to the spectra, the inertial sub-range of the cospectra start at higher non-dimensional frequencies in the case of the heat flux (> 2 vs. about 1). On the low frequency side the spectra from this study show the same behaviour as the observations from smoother surfaces. The w spectra exhibit a rapid roll-off whereas the the spectra of u and T are characterized by considerable scatter and only drop slightly with decreasing frequency. A large amount of scatter with an additional frequent sign reversal is noticeable at the low frequency end of the two cospectra investigated. Compared to the reference cospectra from smoother sites they seem to roll-off slighly faster. The analysis of the peak frequency, as an indication of the dominant eddy scale involved in the energy transfer, deserves special attention. Table 4.1 summarizes the results reported from this and other studies in terms of the statistics of the spectral peaks. Comparing the f m values from this study with the reference values from Kaimal et al., 1972, no differences can be observed. The peak frequencies of all components either coincide with the reference value or are within the range given. The agreement with other urban studies is also good. The only difference is in the temperature spectrum where Coppin (1979) reports a slightly higher value than observed in this study. The 6k peak frequencies from Hbgstrom et al., (1982) are in general slightly higher, however, their results are from near neutral conditions. Table 4.1 also gives the dimensions of the predominant eddy sizes expressed in terms of the height of measurement and, for this study, in real units. Minor deviations from reference spectra observed over smoother surfaces include the flatter appearance of the peak regions in the w and u spectra, the slightly faster roll-off Table 4.1: Summary, of spectral peak frequencies (f m ) and corresponding peak length scales ( l m ) from different studies. The first four studies are from urban sites and the last is from 'ideal' rural terrain. The numbers in brackets give possible ranges. The first three studies are from unstable conditions, the fourth one from near neutral stability. The last study is considered a reference from relatively smooth surface in unstable conditions. w u T wT uw This study f m 0.2 (0.08-0.3) 0.017 (0.01-0.03) 0.023 (0.01-0.06) 0.05 (0.04-0.1) 0.04 (0.02-0.06) 5z' 95m 60z' 1320m 43z' 817m 20z' 380m 25z' 550m Steyn (1980) fm 0.1 10z' 0.017 60z' Coppin (1979) fm 0.2 0.025 0.04 0.1 0.03 lm 5z' 40z' 25z' 10z' 33z' Hogstrom et al. (1982) f m J m 0.4 2.5z' 0.05 20z' Kaimal et al. (1972) f m 0.2 (0.01-0.02) 0.02 (0.02-0.2) (0.01-0.1) 65 at the low frequency end of the cospectra and a relatively flat region in the momentum cospectra between the peak and the inertial subrange. However, more careful investigation is needed before it can be concluded that these differences are true urban anomalies. Inspecting the low frequency end of the spectra and cospectra regarding the appropriate averaging time to be used, the recommendations by Wyngaard (1973) seem to apply for the vertical wind component (20 min), but not for the temperature spectra where considerable energy can still be observed at a frequency corresponding to 20 minutes and higher. This suggests that the averaging time should be longer, in the order of 45 minutes. The same holds for the longitudinal wind component. In the case of the cospectra Wyngaard recommends about 60 min. This is shown to be clearly long enough and in fact it could be reduced to about 30 minutes for both the heat and momentum flux. For the heat flux an averaging time of even 15 minutes seems to be appropriate (Figure 3.15). 4.2 Conclusions In relation to the objectives of this study, the following conclusions can be drawn: • Spectra and cospectra measured in this study in an unstable turbulent surface layer over a suburban surface do not show significant differences from reference spectra measured over smoother surfaces. Comparison between these results and those from other urban studies gives good agreement. 66 » To get an useful estimate of variances and covariances the averaging time recommendations suggested by Wyngaard (1973) seem to be appropriate for the w component, but should be longer in the case of the u and T components and can be relaxed for the heat and momentum fluxes. • The Sunset site which has been used in several previous micrometeorological studies seems to be suitable in regard to turbulence and energy balance measurements representing suburban terrain. • In particular, this study provides the first evidence that the heat flux cospectra agree well with observations from smoother surfaces. • The fact that measurements over a rough surface, at a height that is less than is normally assumed to be required exhibit more concurrences than differences with reference data is somewhat puzzling. It raises the possibilities that either the measurement height constraints over a rough urban surface can be relaxed or (assuming the sensors are placed in the roughness sub-layer) that the effects of individual roughness elements is not strong enough to be observed in spectral analyses. Having shown the appropriateness of this site for turbulence measurements a more extensive study of suburban turbulence characteristics with presentation of the results in the proper MOST framework is warranted and would provide primary evidence concerning the validity of the similarity principle over rough urban surfaces. In particular more data concerning the temperature spectrum and the momentum flux would be desirable, since they are the two quantities that have been least defined. 67 REFERENCES Bendat, J.S., and Piersol, A.G., 1986. Random Data; Analysis and Measurement  Procedures. Wiley-Interscience, New York, 407 pp. Biltoft, C.A., and Gaynor, J.E. , 1987. 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Oke, T.R., Kalanda, B.D., and Steyn, D.G., 1981. Parameterization of heat storage in urban areas. Urban Ecol. 5, 45-54. Panofsky, H.A., Tennekes, H., Lenschow, D.H., and Wyngaard, J . C , 1977. The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary Layer Meteorol. 11, 355-361. Pasquill, F., 1974. Atmospheric Diffusion. Ellis Horwood, Chichester, 429 pp. Pond, S., and Large, W.G., 1978. A system for remote measurements of air-sea fluxes of momentum, heat and moisture during moderate to strong winds. Manuscript Rep. No. 32, Inst. Oceanogr., The University of British Columbia, 55 pp. Pond, S., Phelps, G.T., Paquin, G.A., McBean, G.A., and Stewart, R.W., 1971. Measurements of the turbulent fluxes of momentum, moisture and sensible heat over the ocean. J. Atmos. Sci. 28, 901-917. Pond, S., Large, W.G., Miyake, M., and Burling, R.W., 1979. A Gill twin propeller-vane anemometer for flux measurements during moderate and strong winds. 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Soc. 98, 590-603. 71 A P P E N D I X A : M O D I F I E D G I L L A N E M O M E T E R A . 1 D e r i v a t i o n o f u a n d w f r o m the G i l l V I a n d V2 s i g n a l s The Gill twin propeller-vane anemometer is fully described in Pond and Large (1978). The following outline of how the velocity components are resolved from the VI and V2 measurements is based on Large (1979). For the definition of angles and the notation refer to Figure A. 1. Where symbols used in this Appendix are different from those used elsewhere in the text they are defined immediately following their first appearance. A problem arises when the wind vector makes an angle 9 (angle of attack) greater than about 20 degrees to the propeller axis. In this case the apparent axial velocity component is less than the expected cos(6) times the magnitude of the wind, by a factor B(8). Pond and Large (1978) approximate this non-cosinus behaviour for angles of attack between 35 and 75 degrees with: Geometrical considerations lead to the following equations for the Gill-w (taking the non-cosine behaviour at 9 = « into its calibration) and Gill-u signals which supply the velocities: B(9) = 1.103 - 0.279 (8 in radians) (A.l) where 9 is defined by: 6 = « + 6 - tan' 1 (w/Q) (A.2) Gill-u = VI = Qcos6 + wsino (A.3) 72 Gill-w/B(«) = V2 = (Qcos(« + 6) + wsin(oc+6)) (1 - ((0.27/B(«))(£ - tancT^w/Q))) (A.4) where VI and V2 are the instantaneous measured velocity components of the 'horizontal' and tilted sensor respectively; Q and w are the derived instantaneous horizontal and vertical wind components and o is an angle to account for the tilt of the VI sensor from the horizontal plane (Figure A.l). The tilt angle 8 needs to be evaluated before Q can be removed from the Gill-w signal and w calculated. A good estimate of 6 is the angle of rotation needed to make the calculated mean of the w over the averaging period exactly zero, since the average vertical velocity is assumed to eventually go to zero. 6 is derived from the ratio Y = <V2>/<V1>, where brackets denote time averages. Assuming tan_1(w/Q) ^ w/Q and <w> = 0, it follows from (A.3) and (A.4): <V1> = <Q>cos6 (A.5) <V2> = <Q>(cos(« + 6)(l - 0.27 /B(«)) + (<w 2>/<Q>" 2) sin(*+6)0.27/B(ef)) (A.6) which is correct to second order. The term in <w^>/<Q> is small compared to the previous term, leaving: Y = (1 - 0.2767B(eO)(cos*- sin«tan6) (A.7) which, assuming 6 — tan6, gives a quadratic in 6. Hence 6 is evaluated from: 73 6 = b - (b2 + (B(«)/0.27)(Y - cosor)/sin«) 1 / 2 (A.8) with b = 0.5(cot« + B(«)/0.27) (A.9) Straight forward algebra yields: V2 - (<V2>/<V1>) <V1> = Bw + (A/Vl)w 2 (A.10) with A = (0.27/B(e<))sin(*+6)cos6' (A. 11) B = cos(« + <S)0.27/B(cx) + (1 - 0.2767B(«))sin«/cos6 (A.12) where Q has been replaced by Vl/cos6 and a term in <w 2 >/<Q> 2 and terms of the o order (w/Q)° have been neglected. The quadratic in (A. 10) is solved to give an estimate of w. The instantaneous horizontal velocity component is found from: Q = Vl/cos6 - wtan6 (A. 13) The mean horizontal wind speed, the horizontal and vertical velocity components are hence derived by implementing equations (A.5), (A.13) and (A.10). The mean angle-of-attack (non-cosine response) was corrected for at the beginning by dividing V2 by B(«) in (A.4). Initial testing of the programs gave mean values for the vertical wind component which were considerably off zero. Pond (1987; personal communication) suggests it is necessary to account for the second order term in (A.6) which had been neglected in the initial derivation of (A. 7). Because of the relatively high turbulence intensities over urban 9 9 surfaces the <w >/<Q> term is significant and hence is introduced by subtracting the 9 9 <w>/<Q> part in (A. 6) on both sides. Operationally this results in subtracting this part from Y in (A. 7). By estimating a typical turbulence intensity for the site, and by assuming a o-angle, the results were considerably improved. A.2 Correction for non-linear response As pointed out in Chapter 2.5.1 the Gill anemometer stalls below a certain threshold velocity and its response is non-linear between this threshold and about 1 m s"1. This behaviour was accounted for in this study by using a non-linear calibration curve in the form of an hyperbola between 0.3 and 1 m s"1. A lower limit of 0.3 m s"1 was chosen as the mean of reported start-up speeds (usually higher than 0.3 m s"1) and stall-speeds (usually lower than 0.3 m s"1). The corresponding calibration functions for the non-linear range for the VI and the V2 components are derived from the basic equation of an hyperbola: V l c / a 2 - V l / b 2 = 1 (A.14) and are: V l c = ( a ^ C V l 2 ^ 2 ) 1 / 0 , 9 + 1)) 1 / 2 (A. 15) V2„ = (a 2 ( (V2 2 /b 9 2 ) 1 7 0 - 9 + 1)) 1 / 2 (A. 16) 75 Figure A . l : Diagram of propeller system to define angles (in the GTVA). 6 is tilt of VI propeller (tilt down taken to be positive), « is angle between propeller axes, nominally 60* , 0 is angle of attack (from Pond and Large, 1978). GILL V1/V2 NON-LINEAR RANGE CALIBRATION 0.0 0.2 0.4 0.6 0.8 1.0 1.2 wind speed (m/sec) Figure A.2: Correction for the Gill non-linear response region (0.3 - 1.0 m s ). Dashed lines: Calibration functions according to equations A. 15 and A. 16. Solid lines: Standard linear calibration based on the calibration of the propellers used. 76 where V l c and V2 C are the calibrated VI and V2 wind speeds in m s and the coefficients of the hyperbola are a = 0.3, b-^  = 91.6 and b2 = 19.7. In Figure A.2 the non-linear calibration of the VI and V2 components are compared with the standard linear calibration. (A. 15) and (A. 16) where used in all computations even though comparisons between linear and non-linear calibrations in the region of interest did not show significant differences in computed variances and covariances. 77 APPENDIX B: SPECTRAL ANALYSIS To clarify the exact methods used in Subroutine S.FTSPEC (see Roth, 1988) the computation of the spectra and cospectra are further described here following Garrett (1970). The complex Fourier coefficient for the k-th harmonic is defined as: where A and B are the real and imaginary parts. The harmonic index ranges from 0 to M/2, where M is the number of samples. The power spectral densit}' (SD) at a frequency n=k/Mdt (where dt is the sampling interval and Mdt is the averaging length) is estimated by dividing the product of the Fourier coefficient and its complex conjugate (denoted by a *) with the bandwidth (dn=l/Mdt) and including a factor of 1/2 which gives: F C k = A k + i B k (B.l) SD(n) = (Mdt) (FC k F C k / 2) (B.2) = 1/2 (Mdt) ( A k 2 + B k 2 ) (B.3) The cospectral density (Co) is denned by: Co(n) = (Mdt) Real part(FC l k F C 2 k * / 2) (B.4) = 1/2 (Mdt) ( A l k A 2 k + B l k B 2 k ) (B.5) where the subscripts 1 and 2 refer to two different variables 1 and 2. 78 The discrete Fourier transform used in the UBC library program DFOURT is defined by: F C k = 2 x-e'2™^ k = 0,1, , M- l (B.6) where Xj is a set of real numbers. To compute the proper magnitudes of the Fourier coefficients the real and imaginary parts in the returned arrays of S.FTSPEC were divided by M. Since only one side (positive frequencies) of the total (positive and negative) spectrum is returned by DFOURT the modulus of the Fourier coefficients were multiplied by 2 to account for the missing negative frequencies. Summing these Fourier coefficients resulted in the variances and covariances of the respective variables. By dividing with the bandwidth the spectral densities were derived. It should be pointed out that the spectral densities were not divided by 2 as suggested in (B.2) and (B.4) by Garratt (1970). By including the factor 1/2 the variances would have been too small. The discrepancy between the two approaches is explained by differences in the definitions of the Fourier transforms (equation B.6) used. To account for the high frequency losses in the Gill measurements the spectral densities were divided in S.FTSPEC and S.AVPLFC by the following response function (Garratt, 1975): RF(n) = 1/(1 + (4 ^ D 2 ^ 2 ) ) (B.7) where D is the distance constant for the respective velocity sensor. 79 APPENDIX C: TEST OF STATIONARITY Each of the four variables in all of the 60 minute runs were tested for stationarity. Stationarity implies that the properties of a variable do not change significantly over time. The test applied was a non-parametric run test (Bendat and Piersol, 1986). The time series of length 60 minutes were divided into 40 equal time intervals, where the data in each interval may be considered independent. A sample standard deviation was computed for each interval and then compared with the median standard deviation. The hypothesis of stationarity was then accepted at the 0.05 level of significance if the number of runs observed in the sequence of standard deviations, relative to the median of all standard deviations in one 60 minute period was at least 14 but no more than 27. Table C l summarizes the results from the run test. In the final analysis the respective variables of runs 237.3, 237.4 and 238.8 which showed a strong non-stationary behaviour were not included. Julian day 239 (characterized by a stable stratification of the atmosphere) and run 242.1 also showed non-stationarity at times but were dropped on the basis of other reasons as well. 80 Table C l : Summary of results from run test of stationarity for all turbulence runs. The number of the run-ID also refers to the Julian day. The hypothesis of stationarity is accepted at the 0.05 level if the number of runs lies between 14 and 27. Number of Runs Turbulence Gill Sonic run-ID w u w T 234.1 18 21 20 18 2 21 19 23 17 3 13 23 13 19 4 15 19 15 16 236.1 22 21 19 19 2 23 20 22 17 3 18 15 16 19 4 19 16 19 21 5 19 13 18 22 6 20 21 18 18 7 23 11 18 13 237.1 18 16 16 16 2 18 13 20 21 3 13 13 12 13 4 6 16 8 19 238.1 20 16 16 18 2 17 19 17 22 3 18 24 20 21 4 28 19 24 17 5 17 20 23 20 6 21 20 22 15 7 16 16 16 19 8 8 12 14 21 239.1 15 14 8 13 2 14 14 6 16 3 18 16 18 16 4 9 18 14 12 5 17 17 19 16 240.1 15 15 18 16 2 20 17 20 18 242.1 11 9 18 18 2 18 15 16 22 4 21 22 18 22 5 20 22 18 17 6 19 22 17 15 245.1 20 20 17 16 2 13 18 13 11 3 18 21 17 25 


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