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The structure of convection in the planetary boundary layer Davison, Douglas Stewart 1973

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.15112-THE STRUCTURE OF CONVECTION IN THE PLANETARY BOUNDARY LAYER by DOUGLAS STEWART DAVISON B.Sc, McMaster University, 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the department of Physics and the Institute of Oceanography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1973 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 i t 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PtfvS/s <; g^p QeEAfi/MRAPHj The University of British Columbia Vancouver 8, Canada Date H / f l y #21 ABSTRACT During July 1971, an intensive experimental program was undertaken in south-eastern Alberta to measure the turbulent statistics In the planetary boundary layer using a 92m tower, an array of four smaller towers and an instrumented T-33 aircraft from the National Aeronautical Establishment of the National Research Council. The height variation of the high wavenumber spectral levels of temperature and vertical velocity were found to follow the z-dependence predicted by semi-empirical theories applicable to the constant flux layer up to a height of at least 300m. However, unlike the constant flux layer, the assumption of Ri = z/L is not applicable at higher levels; -5/3 thus, at large z, the spectral levels in the k region for temperature and vertical velocity have the form, OC z and © C z°. The translation velocity of convective plumes was measured with an array of towers, at a height of 3.5m. The direction of plume motion was found to be close to that of the surface wind. The speed of plume motion was found to be close to the wind speed near the top of the surface shear zone; this result is different from previous estimates using different techniques. A dynamic model of the plume was used to show that a translation speed close to the wind speed near the top of the surface shear zone (as observed) allows plumes to exist at a lower thermal instability level then any other possible translation speed. The horizontal cross-sectional shape of the temperature field of the plumes at 3.5m was observed to be very elongated in the downwind direction with a ratio of the downstream to cross-stream diameters of about 8:1. This shape was dynamically justified. TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES ;.. vi ACKNOWLEDGEMENTS . ix Chapter 1. INTRODUCTION 1 2. SPATIAL VARIABILITIES OF TURBULENT STATISTICS CONVECTIVE CONDITIONS TO A HEIGHT OF 610 M 6 2.1 Experimental Site and Instrumentation 7 2.2 Data Sets 11 2.3 Vertical Variability of the Turbulent Statistics 14 2.3.1 Predicted Height Dependence of the Spectral Levels 15 2.3.2 Predicted Height Dependences of the Variances 19 2.3.3 Observed Height Variations of the Temperature Statistics 23 2.3.4 Observed Height Variations of the Vertical Velocity Statistics 32 2.3.5 Fluxes of Heat and Momentum 40 2.4 Degree of Statistical Uniformity at 3.5 m 53 2.4.1 Comparison of Spectra and Statistics ^ From the Array 2.4.2 Discussion of Spatial Variabilities at 3.5 m 58 2.5 Statistical Variabilities Above the Surface Layer of the PBL ... 60 2.5.1 Tower - Aircraft Comparison at 92 m ....... 60 iv Chapter Page 2.5.2 Influence of the Rolls on the Low Wavenumber Temperature Field 63 2.5.3 Influence of the Plumes on the Low Wavenumber Temperature Field 65 3. TRANSLATION VELOCITY OF CONVECTIVE PLUMES 67 3.1 Array Configuration and Significance 69 3.2 Determination of the Vertical Tilt of the Plumes 70 3.3 Determination of the Translation Velocity of Convective Plumes 75 3.4 Dynamic Consequences of the Translation Velocity 82 3.5 Dynamic Explanation for the Observed Translation Speed 94 4. HORIZONTAL CROSS-SECTIONAL SHAPE OF CONVECTIVE PLUMES 119 4.1 Experimental Determination of the Plume -Horizontal Cross-Sectional Shape 120 4.2 Dynamic Explanation for the Observed Cross-Sectional Shape 123 142 5. CONCLUSIONS BIBLIOGRAPHY 146 LIST OF SYMBOLS 149 v LIST OF TABLES Table Page I Temperature and Vertical Velocity Statistics ... 20 II Flux Statistics from Tower Measurements 22 III Temperature and Vertical Velocity Variances and Heat Flux from the Array at 3.5 m 57 v i LIST OF FIGURES Figure -I' Page 1. Instrumentation and Experimental Site at Suffield . . . . 8 2. Flight Patterns About the 92m Tower .12 3. Vertical Velocity and Temperature Traces From the Main Tower 24 4. Tower Temperature Spectra . . . . . . . . 26 5. Aircraft Temperature Spectra 27 6. Ratio of Observed to Theoretical Spectral Levels for Temperature 29 7. Variance of Temperature 31 8. Tower Vertical Velocity Spectra . . . . 33 9. Aircraft Vertical Velocity Spectra 34 10. Ratio of Observed to Theoretical Spectral Levels for Vertical Velocity 36 11. Variance of Vertical Velocity 39 12. Horizontal Velocity Spectra From the 92m Tower 41 13. Temperature, Heat Flux and Momentum Flux Time Traces to 92m 42 14. Tower Heat Flux Cospectra . 44 15. Tower Momentum Flux Cospectra 45 16. Aircraft Heat Flux Cospectra 47 vii Figure Page 17. Vertical Variation of Heat Flux 49 18. Minitower Array 54 19. Array Temperature Traces at 3.5m 55 20. Array Temperature Spectra 56 21. Array Vertical Velocity Spectra . . . . 56 22. Tower-Aircraft Comparison at 92m 61 23. Coherences Between the Temperatures on the 92m Tower. . . 71 24. Phases Between the Temperatures on the 92m Tower Versus Log Frequency ,72 25. Phases Between the Temperatures on the 92m Tower Versus Frequency 73 26. Separation Space Tower Array 76 27. Direction of Plume Motion From Coherence Analysis . . . . 78 28. Speed of Plume Motion From Phase Analysis 80 29. Upwind Edge Shape Preservation 83 30. Time Traces of Temperature and Wind Components at 3.5m 85 31. Suggested Flow Pattern in a Convective Plume 86 32. Upwind Edge Convergence Velocities 89 33. Crosswind Tilt of the Plumes 92 34. The Feeding Term: the Solution Curve of M/N = M/N (C/S) 103 35. Plume Model for Calculating Convergence Downturning 106 v i i i Figure Page 36. Suggested % and e/& Behaviour as a Function of Translation Speed 115 37. Horizontal Temperature Contours of a Plume at 3.5m . . 121 38. Plume Model with Elliptic Cross-Section 126 39. Approximation Involved in the jA -Integrations -127 40. The Assumed Form of the Pressure Term .131 c ix ACKNOWLEDGEMENTS I sincerely thank the many people who helped to make this thesis possible, especially Dr. Miyake who suggested the problem and provided guidance throughout the study. Dr. Burling, Dr. LeBlond and Dr. Pond have also given helpful comments and criticisms. The students and staff at the Institute of Oceanography here at U.B.C. were most helpful, especially Ms. Grace Kamitakahara who did much of the computer analysis and Mr. Don Hume who helped in innumerable ways especially during the field experiment at Suffield. The co-operation and support of the scientific personnel at the Defense Research Establishment Suffield and at the National Aeronautical Establishment at Ottawa are much appreciated; special thanks go to Dr. 0 . Johnson of DRES and Mr. G. Mather of NAE. The personal financial . support given by the National Research Council is gratefully acknowledged. Finally, I express my deep gratitude to a l l those people who made my years as a graduate student so satisfying: to the colleagues who were more than colleagues, to the companions on trips into the quiet of the high alpine country, and most of a l l to Susan. Chapter 1 INTRODUCTION The first few tens of meters of the atmosphere are commonly referred to as the constant flux layer. In this layer the vertical divergences of the momentum, heat and water vapour fluxes are small compared to the fluxes themselves. Above this layer is the rest of the planetary boundary layer (PBL). In the PBL the vertical shears of the mean wind and mean temperature along with the effects of the Coriolis force are significant. Much of the progress made in the understanding of the turbulent characteristics of the constant flux layer have been made in the establish-ment of empirical theories in the light of the Monin-Obukhov similarity theory. According to this theory: "... the dependence on height of any mean turbulence characteristic f in the surface layer of air which is independent of the properties of the underlying surface for not too small z may be written in the form, f (z) where f„ is a combination with the dimensions of f formulated from the parameters, g/T, , p, while f(L") i s a u n * - v e r s a l imction. Monin and Yaglom (1971, pg. 427) . II 2 The length scale, L, in the Monin-Obukhov similarity theory is defined as u3 where X is vonKarman's constant and q/(Cprf>0 ) is the heat flux, similarity theory is presumably applicable only to surface constant flux layer. In the PBL, the variation of the mean wind with height was firs t suggested to be due to a balance between pressure, Coriolis and turbulent friction forces, which leads to the well-known solution of the Ekman spiral. The classical Ekman (1905) solution assumed a constant eddy viscosity in the formulation of the stress term and so was expected and found to be only approximately obeyed. More recently, Blackadar and Tennekes (1968) showed that the mean wind structure in a neutral barotropic PBL followed Rossby-number similarity as a direct result of an asymptotic matching between the surface layer and the rest of the PBL. Faller and Kaylor (1966), Li l l y (1966) and Brown (1970) studied the characteristics of longitudinal rolls due to inertial instabilities in their secondary flow models. A l l of these theories used perturbation analysis techniques under barotropic conditions to predict the secondary flow structures in the PBL. The size and spacing of the predicted rolls agreed reasonably well, in some cases, with the observed values reported b y Lemone (1972). The added problem of a thermal instability has been considered theoretically by Asai (1970) for Couette and Poiseuille flows. Their applicability to the actual atmospheric PBL is somewhat V T' ). L is considered to be independent of height and so the 3 questionable. One of the most interesting numerical models of the unstable PBL was a. study by Deardorff (1972). Deardorff's model produced elongated eddies for both the neutral and slightly unstable cases which again agreed reasonably well with experimental results reported by Lemone (1972). However, the grid scale of the model was such that some of the convective activity in the real atmosphere would be subgrid scale in his model. In the unstable PBL, much of the heat flux is carried by organized convective plumes which are regions of hot rising air with diameters of the order of 100m. One of the first convincing measurements of the structure of convective plumes was made by Taylor (1958). Under conditions of strong surface heating, Taylor found regions of hot rising air with a significant coherence between different heights from 1.5m to 32m. His observations showed that the lower sensors lagged the higher ones. Priestley (1959) suggested that the convective elements are vertically continuous plumes as opposed to discrete masses of hot air (i.e. bubbles). Because of the observed downwind t i l t of these plumes, Priestley suggested that plumes have a translation speed close to that of the mean wind speed at plume formation height. The preferred shape would be elongated in the down-wind direction, Priestley argued, because the local "hot spot" needed to initia l l y form the plume, would exist for some time. Aircraft measurements permitted more direct measurements of plume shape. Yet there has been some discrepancy among results. Warner and Telford (1963) reported no elongation of plumes from 15m to cloudbase, but Lenschow (1970) found a distinct elongation above 100m. Among several tower-based studies of plumes one of the more interesting ones was a case study by Kaimal and Businger (1970). Simultaneous measurements at two levels (5.66m and 22.6m), were interpreted to give estimates of the motion and structure of one particular plume. However, the convective plumes are known to be fully three-dimensional structures. The objective of this thesis is to determine the structure of the unstable atmospheric boundary layer and particularly of convective plumes. Realizing the three-dimensional nature of the structure of the plume, an experiment was staged to permit spatial samplings. The experimental site used was the Defence Research Establishment Suffield in south-eastern Alberta, a site which has uninterrupted natural grass-land surrounding i t for tens of kilometers in the prevailing wind direction. This type of site is essential to avoid the effects of topography, and to ensure that the boundary layer has had sufficient distance to take on the properties appropriate to the surface. Simultaneous flux measurements were made at three levels (3.5m, 48m and 92m) on the meterological research tower at Suffield. Spatial sampling was accomplished by the use of four small towers (which formed a three-dimensional array when used with the 92m. tower) and by an instrumented T-33 aircraft from the National Aeronautical Establishment of the National Research Council, Ottawa. The experiment was undertaken in July 1971, a time of the year that ensured the existence of active convection. In this thesis, the form of the spatial variabilities of the turbulent statistics under convective conditions and the motion and structure of the convective plumes embedded in the turbulent field were determined. The height dependences of the turbulent statistics in a convective field were measured to a height of 610m and were compared to the height dependences predicted by the forms of the universal functions of z/L in similarity theory as have been determined from semi-empirical dimensional arguments in the constant flux surface layer. The effects of horizontal spatial variabilities on Eulerian measurements were determined. The speed and direction of motion of convective plumes were measured by an array of towers. The observed temperature and velocity fields associated with convective plumes were shown to be dynamically consistent and a dynamic justification was given for the observed translation speed. The shape of the horizontal cross-section of the convective plumes was determined and the observed elongated shape was dynamically justified. Chapter 2 SPATIAL VARIABILITIES OF TURBULENT STATISTICS UNDER CONVECTIVE CONDITIONS TO A HEIGHT OF 610M. Much progress has been made in the understanding of mean wind and temperature profiles and forms of the turbulent statistics under various stability conditions below the height of about 30m, (see for example, Priestley (1959), Lumley and Panofsky (1964) and Pasquill (1972)). Although above 30m experimental data are not very good, some of the characteristics of the turbulent statistics have been determined. Volkov et al (1968) examined the effects of cloudiness on the vertical scaling of the temperature and vertical velocity variances to a height of 2000m. Lenschow (1970) measured the terms in the kinetic energy balance equation (except the pressure term) and in the temperature variance balance equation to a height of 1000m under convective con-ditions. The vertical variability of turbulent fluxes over the tropical ocean were measured by Donelan and Miyake (1972), where the latent heat was found to be the dominant active scalar. In this chapter are presented the horizontal and vertical variations of the turbulent statistics as measured by an array of towers and an aircraft in strongly convective conditions over grasslands to a height of 610m. The extent of the applicability of the semi-empirical universal functions of z/L in the similarity theory (as have been determined for the constant flux surface layer) is tested. 7 2.1 EXPERIMENTAL SITE AND INSTRUMENTATION The data to be discussed come from an experiment made in July 1971 on a slightly undulating grassland with tens of kilometers of un-interrupted fetch at the Defense Research Establishment Suffield, (DRES), in south-eastern Alberta, Canada (see Fig. 1). Such a site minimized topographical effects and ensured that the boundary layer had had sufficient distance to adapt to the surface conditions. Simultaneous flux measurements were made at the heights of 3.5, 48 and 92m from a 92m tower, (Fig. IA). At 3.5m was a Kaijo-Denki sonic anemometer-thermometer (Fig. IB) and at 48m and 92m were vertical and horizontal G i l l anemometers and fast response thermistors. The details of the instrumental responses have been described elsewhere. (see Mitsuta et al (1967), Donelan (1970), McDonald (1972)). Mean temperatures at 15m, 48m and 92m were available from a temperature profile system supplied by the DRES meteorological group. The presence of spatial variabilities at 3.5m was determined by the use of an array of four small towers each with a thermistor and vertical G i l l anemometer at a height of 3.5m (Fig. IC). At higher levels, the influence of spatial variabilities were examined by an instrumented T-33 aircraft, (Fig. ID), from the National Aeronautical Establishment of the National Research Council, Ottawa. Constant level flights were made about the 92m tower at heights up to 610m. Details of the aircraft instrumentation were given by Mather (1967). In brief the instrumentation consisted of 3 accelerometers, 3 rate gyros, 2 gust vanes, a pitot tube, a static pressure sensor, and a fast response FIG. 1 INSTRUMB^ TATION AND EXPEfWOTAL SITE AT SUFFIELD c 9 c. VERTICAL GILL ANEMOMETER, CUP ANEMOMETER AND THERMISTOR AT 3.5M ON A SMALL TOWER OF THE ARRAY f < D. NOSE BOOM ON THE T-33 10 thermometer. With the aid of these sensors, aircraft motions were removed and the true environmental gust velocities and temperatures were computed digitally. The response of the instruments led to frequency limitations of 3 to 5 hz for vertical velocity and temperature. 11 2.2 DATA SETS The active experimental period was from July 3 to July 30, 1971. During this period, there was a total of 10 hours of flight time by the T-33 on 7 days. In choosing the data sets to be examined i t was decided to con-centrate on those days when wind conditions were very steady and speeds were greater than 3m/sec at 3.5m and there was very l i t t l e cloud cover so that shadows would not disturb the surface conditions. The best such days when the aircraft was present, were July 14, and July 16, 1971. On July 14, there were 0.1 Cu. clouds, a mean temperature at 3.5m of 27°C and a wind of 4.5 m/sec and 6.3 m/sec at 3.5m and 92m respectively. L-shaped patterns (see Fig. 2) were flown on July 14( with legs 18km long) roughly along and across the mean wind direction, at the heights, 610, 305, 152, 122, 92, 61 and 38m. On July 16, there were clear skies, a mean temperature at 3.5m of o 32 C, and a wind of 5.5 m/sec and 10.1 m/sec at 3.5m and 92m respectively. Right-angled triangle patterns were flown (with legs 18km, 18km, 25km) at various angles to the mean wind direction (see Fig. 2), at the heights 610, 305, 228, 152, 122, 92 and 61m. The third leg at each height allowed more accurate measurements of the size and orientation of spatial variations. As will be shown in Section 2.5, longitudinal rolls (with axes downwind) and with a lateral (crosswind) wavelength of 1.7km were found on July 16 to greatly influence the temperature statistics as measured by the Eulerian tower-based sensor at 92m. There was no evidence of rolls on July 14-The duration of the data segments used varied from 38 to 130 minutes during which conditions of wind and temperature were nearly steady; the 12 FIQ. 2 FLIGHT PATTERNS CHANGE HEIGHT SURFACE WIND DIRECTION JULY M ^^J^TOWER CHANGE HEIGHT \* 18 KM 13 times of the runs on both days were around noon, local time. The data was examined from both spectral and time domain points of view since spectral analysis alone may not give a complete picture; for example an intermittently coherent event between two sensors is d i f f i c u l t to detect with spectral analysis alone. The spectral data are presented on wavenumber plots. The wavenumber, k, i s defined by k =2trf where f is the frequency and where V is the true V + a i r speed of the aircraft (1 50 - 4 (m/sec)) for aircraft spectral plots or where V is the mean wind speed U (z) for tower spectral plots. The spectral representations in terms of wavenumber assume Taylor's "frozen f i e l d " hypothesis. 2.3 VERTICAL VARIABILITY OF THE TURBULENT STATISTICS According to Kolmogorov's second hypothesis, i f the Reynolds number is sufficiently high, then there will exist an inertial subrange of scales in which no production or dissipation takes place and only inertial transfer to smaller and smaller eddies is happening. Dimensional arguments then lead to the "five-thirds law" for the spectral densities of the velocity components in the inertial subrange. Similar arguments can be applied to the temperature fluctuations. Thus the spectral densities of velocity and temperature in the inertial subrange are given by (see for example, Lumley and Panofsky (1964)): where V. (k)and & (k) are the spectral densities at wave number, k, for the i component of velocity and temperature respectively, **£ and £3 are constants, and where £ is the dissipation of kinetic energy and £ & is the (thermal) dissipation of &' . (fJ£ is a non-dimensional, form of 6" . The height dependence of <£€ according to the similarity theory can be represented by a universal function of z/L , where L is the Monin-Obukhov length and is taken as a constant in the similarity theory. The forms of the universal functions of z/L have been determined by dimensional arguments and empirical results in the surface layer. In this study the observed height dependences of the spectral levels, £(k,z) in the k region, to a height of 610m are compared to those predicted from semi-empirical dimensional arguments for the surface layer, * ji 4> k' f/3 e-1) -S/3 (2.2) by calculating the ratio: 15 R(z) = Log $ (z) ^Z^observed theoretical (2.3) -5/3 -2/3 I in the k region (or k region in k <p (k) plots). For $ (z) . , = C z m$(z) I •> observed i theoretical then R = log C + m log z (2.4) Hence by plotting R versus log z, the value of the exponent m can be measured. The size of m determines to what extent the observed height dependence of the high frequency spectral levels follows the height dependence predicted from the semi-empirical dimensional arguments for the surface layer. 2.3.1 Predicted Height Dependence of the Spectral Levels -5/3 The high frequency spectral levels (in the k region) can be related to the surface fluxes and z/L (as in the similarity theory) using equations (2.1) and (2.2) and other theoretical and empirical results. In equation (2.1) the Kolmogorov constant, o( ( , was taken as equal to 0.55 following the results of McBean et al (1971) and Paquin and Pond (1971); C/ 2 and Ot^  were taken as 4/3 Of, . There is much discrepency between the observed values of |9 , in equation (2.2). Paquin and Pond (1971) found Q =0.4 based on measurements of structure functions; Wyngaard and Cote (1971) found ^ = 0.41 from an energy balance; a direct measurement of ^ by Boston (1970) gave a value of ^ = 0.8. Since 16 Boston s value was a directly measured value, g =0.8 was adopted for use in the "theoretical" spectral levels in this analysis. From the Navier-Stokes equation and the heat equation expressions can be derived for £ and £ e (see for example, Lumley and Panofsky (1964)): (2.5) = w ' T - D 9 (2.6) where primed quantities are fluctuations, and D and D & are residual terms. The residual terms include the effects of horizontal transports, pressure terms and the vertical divergences of the vertical transports. Previous energy balance studies close to the surface (Mordukhovich and Tsvang (1966), McBean et al (1971) and Wyngaard and Cote (1971) have shown that under unstable conditions close to the surface the above residual terms are not small, particularly the vertical divergence terms, ~^7T ( w' e') in (2.5) and - -r- (wT ) in (2.6), where e is the turbulent kinetic a Z 2 2 2 energy (per unit mass) given by e •= 1/2 ( u' + v' + w' ). Since detailed accurate profiles of wind and temperature were not measured, empirical expressions for iT//3 z and ~i9 / 3 z as functions of stability have been used in the calculation of the "theoretical" spectral levels to be compared to the observed spectral levels at large z . Expressions for 3U/ d z and j6/a z There are several empirical expressions for the stability , 17 dependence of the non-dimensional wind shear <J>m and the non-dimensional potential temperature profile , where <p = - s i . X z hV_ (2.7) and 3 9 (2.8) "iTr az From the definitions of the turbulent thermal diffusion coefficient, and the turbulent mechanical diffusion coefficient, ^  , Based on empirical results in the surface layer an interpolation formula for the stability dependence of cj^ between neutral stability and free convection can be written: , - (2.10) where is taken as a constant. The above equation is the so-called Keyps formula which is often written in the form, S" - ~ S 3 = / ( 2 . 1 D where S = /f) . See, for example, Lumley and Panofsky (1964, P.110). From dimensional arguments in the constant flux layer, for large ~ z /L & cc z - 1^ 3. It can be seen in (2.10) that for large -z/L (f). cc z ~ l / 3 Th n Hence from equation (2.9) K^^K^ must be a constant for large -z/L t i f equation (2.10) is correct for large -z/L. An alternate representation is the Businger-Dyer formulation 18 (see Paulson (1967) in which Ri is set equal to z/L and <Pm and 4V become: (2.12) Note, however, that for large -z/L the Businger-Dyer formula for 4>h -1/3 does not have the z dependence expected from dimensional arguments. The observed spectral levels will be used to test which of these two formulations leads to the observed height dependence at large z. The predicted spectral levels for temperature and velocity using equations (2.1) through (2.9) can be written, in,)-(2.13) and I X2 U - 1 KM (2.14) where D is the normalized form of the residual term D in equation (2.5). For the calculation of "theoretical" spectral levels to be compared with the observed levels (see equation (2.3)), the remainder terms D*and D e will be ignored and the Keyps profiles, equations (2.9) and (2.10), with = ^ » will be used. Since there is a net transport of kinetic energy and mean square temperature fluctuations out of the surface layer and into the layer near the inversion, then the divergence of the transport terms - 211 z (w'e') and -3/3z (w'T ) must change sign between the surface layer and the inversion. Hence in the middle layer of the PBL, neglecting the vertical divergences of the 19 vertical transports may not cause a large error in the "theoretical" spectral levels. It is also seen that the choice of the Keyps rather than the Businger-Dyer representation for the case of large z will be of l i t t l e consequence for the velocity spectral levels, (in equation (2.13)), since the term involving <|> will be small. 2.3.2 Predicted Height Dependences of the Variances According to similarity theory the height dependences of the variances of temperature, (T^ . and vertical velocity, 0"^ , when normalized, are representable in terms of universal functions of z/L . From dimensional arguments based on empirical results for the surface layer, (see for example, Lumley and Panofsky (1964, p. 133ff.). 0*w « Z for neutral conditions 0~- ^ * for near-neutral conditions and 01 oc Z r The observed height dependencies of [Tw and Q" will be compared to the above dependencies. A l l turbulent statistics such as variances,turbulent fluxes and spectral levels are height dependent. In the following sections each particular parameter will be examined in turn. The summary of the gross statistics of the measurements made on July 14 and July 16 are presented in Tables I and II. The statistics are evaluated by summing the observed spectral densities over wavenumbers greater than some lower limit, k^ , as shown in the Tables. Detailed discussion of the statistics are carried for free convection - z / L ^ l . ) T A B L E I A T E M P E R A T U R E A N D V E R T I C A L V E L O C I T Y S T A T I S T I C S . J U L Y 1 6 A I R C R A F T S T A T I S T I C S W'TV [o 2 a 2 ] 1 / 2 w T H E I G H T L E G [ m ] L . 4 x l 0 6 1 0 305 228 1 5 2 1 2 2 92 6 1 f l (1 (1 (1 a TOWER S T A T I S T I C S 9 2 m 4 8 m 3 . 5 m <T 2 T 32 -4 0 . 0 3 1 0 . 0 3 9 0 . 0 4 7 0 . 0 8 3 0 . 0 4 7 0 . 0 6 2 0 . 0 5 8 0 . 0 5 2 0 . 0 6 1 0 . 1 9 3 0 . 1 3 3 0 . 1 1 8 0 . 1 4 3 0 . 1 6 0 0 . 1 1 9 0 . 1 4 8 0 . 1 4 0 0 . 1 4 9 0 . 1 9 9 0 . 2 4 5 0 . 2 5 9 6 x 1 0 0 . 0 0 8 6 0 . 0 1 3 7 0 . 0 1 9 3 0 . 0 4 5 0 . 0 3 3 0 . 0 3 6 0 . 0 3 1 0 . 0 2 9 0 . 0 3 9 0 . 1 1 0 0 . 0 8 3 0 . 0 6 8 0 . 1 0 2 0 . 1 3 7 0 . 0 8 9 0 . 1 1 1 0 . 1 1 5 0 . 1 1 1 0 . 1 6 5 0 . 1 8 8 0 . 2 0 6 0 . 0 4 3 0 . 2 4 0 . 9 8 1 . 4 x 1 0 -4 3 . 2 3 . 5 4 . 0 3 . 0 3 . 1 3 . 2 2 . 2 2 . 0 2 . 0 4 7 , 5 , 5 , 4 .7 3 2 2 3 2 6 x 1 0 - 4 1 . 4 x 1 0 2 . 1 1 2 . 1 1 2 . 2 7 1 . 7 1 . 5 1 . 4 6 6 6 0 9 5 3 8 1 4 1 6 1 . 5 9 0 . 6 6 0 . 3 4 w ' T * - 4 0 . 1 5 0 . 1 5 0 . 2 6 0 . 3 5 0 . 1 8 0 . 2 1 0 . 1 8 0 . 2 1 0 . 1 8 0 . 4 6 0 . 4 0 0 . 2 8 3 2 3 6 2 8 2 4 2 6 2 6 6 x 1 0 -4 0 . 0 5 7 0 . 0 5 4 0 . 1 1 0 . 1 9 0 . 1 3 0 . 1 3 0 . 1 2 0 . 1 3 0 . 0 9 5 6 . 2 4 0 . 3 C 0 . 1 5 0 . 2 2 0 . 3 7 0 . 2 1 2 3 2 4 0 . 2 6 0 . 0 6 0 . 2 1 0 . 2 4 * w T 1.4x10 0 . 4 7 0 . 4 1 0 . 5 8 0 . 6 8 0 . 4 6 0 . 4 5 0 . 4 9 0 . 6 2 0 . 5 0 0 . 6 6 0 . 5 5 0 . 5 1 0 . 6 8 0 . 5 7 0 . 6 1 0 . 4 1 0 . 6 1 0 . 5 6 6 x 1 0 0 . 4 1 0 . 3 1 0 . 3 5 0 . 5 5 0 . 4 3 0 . 4 4 0 . 5 1 0 . 5 8 0 . 4 0 0 . 5 7 0 . 6 2 0 . 4 6 6 7 7 1 5 5 0 . 5 8 0 . 6 6 0 . 7 0 0 . 2 7 0 . 5 3 0 . 4 2 to O w h e r e K j ^ Cm"1) I s t h e l o w w a v e n u m b e r l i m i t u s e d f o r c a l c u l a t i o n o t t n e q u a n t i t y i n v o l v e d . T A B L E I B T E M P E R A T U R E A N D V E R T I C A L V E L O C I T Y S T A T I S T I C S , J U L Y 1 4 A I R C R A F T S T A T I S T I C S H E I G H T [ m ] D I R E C T I O N L E G 62 T 1 . 4 x 1 0 -4 6 x 1 0 1 . 4 x 1 0 6 x 1 0 -4 1 . 4 x 1 0 - 4 6 x 1 0 " w ' T ' w T [ O i T 2a T 2 ] l/2 1 . 4 x 1 0 6 x 1 0 - 4 6 1 0 3 0 5 1 5 2 1 2 2 9 2 6 1 3 8 / d o w n w i n d \ x - w i n d f u p w l n d \ j c - w i n d go w n w i n d 2 - w i n d 1 ( u p w i n d \ x - w i n d ( d o w n w i n d \ x - w i n d fu p w i n d x - w i n d { " d o w n w i n d \ x - w i n d 0 . 0 4 7 0 . 0 5 0 0 . 0 3 6 0 . 0 6 0 0 . 1 1 9 0 . 0 7 7 0 . 1 5 7 0 . 0 9 7 0 . 1 8 2 0 . 1 3 7 0 . 2 2 5 0 . 2 0 6 0 . 3 4 2 0 . 2 6 4 0 . 0 3 9 0 . 0 4 6 0 . 0 3 0 0 . 0 5 2 0 . 1 0 4 0 . 0 7 5 0 . 1 0 8 0 . 0 8 0 0 . 1 4 9 0 . 1 2 6 0 . 1 7 3 0 . 1 7 1 0 . 3 0 9 0 . 2 3 6 2 . 7 3 . 9 1 . 2 2 . 5 5 . 1 2 . 1 2 . 4 1 . 7 0 2 . 3 0 5 6 6 0 . 7 5 0 . 6 9 2 . 1 2 . 7 0 . 8 6 1 . 8 2 . 9 1 . 7 3 1 . 5 3 1 . 5 1 1 . 8 8 0 . 9 4 1 . 0 1 0 . 7 2 0 . 6 8 0 . 1 4 5 0 . 1 4 8 0 . 0 6 2 0 . 0 8 2 0 . 1 6 2 0 . 2 1 0 . 2 3 0 . 1 8 5 0 . 2 2 0 . 3 1 0 . 4 1 0 . 2 6 0 . 2 3 0 . 0 9 9 0 . 1 3 5 0 . 0 4 9 0 . 1 1 8 0 . 3 1 0 . 2 0 0 . 2 3 0 . 1 6 0 . 2 9 0 . 2 4 0 . 2 7 0 . 2 6 0 . 2 2 0 . 4 0 0 . 3 3 0 . 2 9 0 . 2 1 0 . 2 0 0 . 5 1 0 . 3 5 0 . 4 4 0 . 3 3 0 . 6 1 0 . 6 9 0 . 5 0 0 . 5 3 0 . 3 4 0 . 3 8 0 . 3 0 0 . 3 8 0 . 5 5 0 . 5 5 0 . 5 5 0 . 4 5 0 . 5 4 0 . 5 9 0 . 6 3 0 . 5 4 0 . 5 4 TOWER S T A T I S T I C S 9 2 m 4 8 m 3 . 5 m 0 . 1 5 0 . 2 4 1 . 4 0 . 2 8 0 . 6 6 1 . 5 2 0 . 2 0 . 2 0 . 2 7 0 . 4 0 . 5 0 . 4 3 w h e r e ( m ) i s t h e l o w w a v e n u m b e r l i m i t u s e d f o r c a l c u l a t i o n o f t h e q u a n t i t y d e v o l v e d . TABLE II - FLUX STATISTICS FROM TOWER MEASUREMENTS AT SUFFIELD July 16; 12:00-13:00 (local time) July 14; 11:50-12:20 (local time) Height [ m ] 3.5 48 92 3.5 48 92 w'T* [C° - m/sec] 0.24 0.21 0.06 (0.19) 0.27 0.2 0.2 u'w' [ m/sec]2 -0.19 -0.17 [-0.1] -0.15 [-0.1] [-0.1] T WT 0.42 0.53 0.27 (0.48) 0.43 0.5 0.4 Tuw -0.21 -0.11 [-0.04] -0.23 [-0.10] [-0.05] |rwT/ruw 2.0 5. [ 6. ] 1.7 [ 5. ] [ 7. ] O-w/u, 1.35 2. [4. ] 1.35 [ 3. ] [ 4. ] -L t m ] 26.8 26. [ 4 . X 1 0 1 ] 17. [l.xlO 1] [l.xlO 1 ] Z/L- * S -0.13 -1.8 -3.4 -0.21 -2.8 -5.4 ' 3. -> CD 6.5xl0"3 2.xl0~3 [lxlO - 3] 7.xl0~3 [3.xl0~3] [3.xl0~3] Where round-bracketed values at 92 m for July 16 are the mean values from aircraft measurements at 92 m and where square-bracketed values have very large poorly-defined low wavenumber contributions. S 3 in the respective sections in which each parameter is examined. 2.3.3 Observed Height Variations of the Temperature Statistics  Time Traces Figure 3 shows the vertical velocity and temperature traces at 3.5, 48 m and 92 m for July 16 (Fig. 3A) and July 14 (Fig. 3B). The temperature traces in Fig. 3 show a marked one-sidedness with a con-sistent base temperature as a cool limit at a l l levels on both days except at 92 m on July 16. There is a distinct intermittency at 48 m and 92 m with thermally quiescent zones separating thermally active regions. The simultaneity of tie large scale temperature and vertical velocity features at a l l levels is evident. Regions of positive (upwards) vertical velocity are seen to correspond to warm temperatures; these regions of hot rising air are convective plumes. An important feature, then, is that at 48 m and 92 m, the variance of the temperature signal is largely due to advection of temperature fluctuations from below by convective plumes. Temperature Spectral Levels The temperature spectra k^ (k) from the 92 m tower for July 16 and July 14 are shown in Fig. 4A and 4B respectively. The low frequencies are seen to have a similar shape at a l l levels for each day except for 92 m on July 16, where the low wavenumbers drop off very quickly. This f a l l off at low wavenumbers will be shown later to be due to spatial variability of the turbulent statistics at 92 m. In Fig. 5A and 5B the temperature spectra for a l l legs at a given height have been superimposed to show the effects of changes in the direction of spatial sampling on the spectral shapes. The obvious differences in the spectral shapes which were found to exist on July 16 and to a lesser extent on July 14 are discussed in Section 2.5. FIG .3A VERTICAL VELOCITY & TEMPERATURE TRACES FROM MAIN TOWER SUFFIELD JULY 16 warm . T3.5 3c- [ 4 4 ^ V ^ V \ ^ ^ T48 3*C cool T92 3C - ^ A J ^ ^ ^ / A ^ U ^ ^ ^ ^ — * J W - L _ W35 4m,s,2C W 4 8 ^ - [ V ^ w M / ^ ^ ^ w92 4 6 8 10 TIME (minutes) H h 12 K 16 18 20 LJ2= 10 M/SEC FIG. 3B VERTICAL VELOCITY & TEMPERATURE TRACES FROM MAIN TOWER SUFFIELD JULY 14 cool '3.5 r48 T92 3c° down I W 4 8 4 n,/Sec down U D ' fl -. w92 • 4 •»/.« r . v ^v v v — v ^ ^ w s ^ ^ u ^ ^ ^ ^ 4 6 8 10 12 14 16 18 20 TIME (minutes) » U 9 2 = 6.3 M/SEC (S3 26 FIG.4A Tower Temperature Spectra July 1 6 - 3 J i 1 1 1 1 1 - 4 -3 - 2 - 1 0 1 log k (m)" FIG.4B Tower Temperature Spectra July 1 4 cn p -3 J l 92 '48 ' 3 . 5 log k (m) 27 FIG.5A Aircraft Temperature Spectra July 16 FIG. 5B Aircraft Temperature Spectra July 14 - 3 . 0 J 3 0 5 2 2 8 152 Height lm) 610 Height fan) 1 2 2 o> O log k (ni1) Logk (ni1) 28 On the July 16th run, there was a large noise peak at 1 c.p.s. at a l l the heights on the temperature spectra (marked with a dot in Fig. 5A). The effects of this peak on the computed statistics have been removed. -5/3 The theoretical spectral levels in the k region as predicted by the empirically determined universal functions of z/L for the surface layer (see section 2.3.1) were drawn in for each height in Figs. 4 and 5. The Keyps form of the profiles with Kj^ /Kg = 1 and the fluxes measured at the 3.5 m level of the tower were used for the calculations of the theoretical spectral levels. In order to examine the form of the discrepancies between the observed and theoretical spectral levels of temperature the value of R was plotted against log z in Fig. 6 where as in equation (2.3), R = Log $ (z) observed $ (z) theoretical In Fig. 6 there is clearly very l i t t l e z-dependence in the values of R for the temperature spectral levels, i f the value at 610 m on July 14 is ignored. A linear regression of R and log z (omitting the 610 m value) leads to: R = 0.004 log z - 0.51 (2.15) Hence to height of 300m, the observed z-dependence of the temperature -5/3 spectral levels in the k region is equivalent to the z-dependence of the empirically derived universal function in z/L applicable to the sur-face layer (equation (2.11)). In this "theoretical" function in z/L no allowance was made for any z-dependence other than in (j>m for which the Keyps profile was adopted. The Businger-Dyer formulation would have led 29 F I G . 6 RATIO OF OBSERVED TO THEORETICAL SPECTRAL LEVELS FOR TEMPERATURE ^ TT observed ^ TT theoretical 0 . 2 1 0 0 0 - , 1— 1 0 0 -1 0 -T " • 0 . 7 0 . 3 0 . 1 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 -I 1 1 I I I L_ • • v [ ( B E S T FITTING L I N E (EXCLUDING 610M LEVEL) f \ R = 0 . 0 0 1 LOG Z - 0 . 5 1 • A I R C R A F T } V TOWER / • AIRCRAFT "> • TOWER J J U L Y 1 6 J U L Y M - 0 . 5 T 0.80 uhere p is the Rolmogorov constant for the temperature spectrum - 0 . 3 - 0 . 1 0 . 0 0 . 1 R = LOG 1 TT observed L 4> TT theoretical 30 to the coefficient of log S in equation (2.14) being about 0.17. The value of the observed constant magnitude is log~^(0.51) =0.32 of the "theoretical" magnitude. This discrepancy is equivalent to the value of j3 being 0.25 rather than the value 0.8 used for the calculation of the theoretical levels, (see equation (2.14)). For the value |3 =0.8 used in the calculations of the theoretical spectral magnitudes, the observed constant magnitude factor is as if K^ /K^  were equal to 0.31 at a l l heights. If the value 3^ =0.4 had been used for the theoretical spectral magnitudes, (following the results of Paquin and Pond (1971) and Wyngaard and Cote (1971)), then the observed constant magnitude factor is as i f ^/K^ were equal to 0.63 at a l l heights. At 3.5 m, the use of ^ = 0.4 leads to a value of K^ /Kg nearer to those observed (see Businger et al. (1971) and McBean (1970)). However, the value of K^/Y^ is not known for z/L less than around -2. and so the appropriate value of g is not clear for the higher levels. Variance of Temperature The temperature variances, (f^ Z for both days showed a marked -2/3 decrease with height at a rate very close to the z predicted by empirically based dimensional formulas for the surface layer for free convection, (see Fig. 7 and Table I). A significant horizontal spatial variation is obvious for the tower-measured value of (JJ.1 at 92 m on July 16. This effect is examined in Section 2.5. For cloudless or slightly cloudly conditions over the steppe surface, Volkov et al (1968) found a similar decrease of C^1 with height up to 700 m. -4 -4 -1 Two low wave number limits k^ = 1.4 x 10 and k^ = 6 x 10 m have been used in Table I to show the effects of the low wave numbers on 31 °f 2 (C)2 the statistics. It is seen that the difference in the C^. values between aircraft legs at a given height are roughly the same size as the differences due to the two different low wave number limits. In the estimations of for the July 16th aircraft flights, the excess variance in the noise peak at f=?l c.p.s. at a l l heights has been subtracted. The lack of a distinct low wave number fall-off for the temperature spectra in Fig. 5 at the higher levels indicates that a very large path length is needed in order to estimate the temperature variance accurately. Clearly in the presence of longitudinally oriented features, the direction of aircraft flight is important in obtaining a representative value of the variance. 2.3.4. Observed Height Variations of the Vertical Velocity Statistics The time traces of the vertical velocity from the 92 m tower were shown in Fig. 3. At large scales, regions of positive (upwards) vertical velocity are seen to occur simultaneously at a l l levels. The vertical velocity spectra k (k) from the 92 m tower are shown in Figs. 8A and 8B for July 16 and July 14 respectively. In contrast to the temperature, there are no obvious differences between the two days in the time traces or spectral shapes for the vertical velocities measured from the tower. The vertical velocity spectra k ^ (k) from the aircraft flights are plotted in Figs. 9A and 9B for July 16 and July 14 respectively. The plots for a l l legs at a given height have been superimposed to show the effects of changes of flight direction on the spectral forms. At each height in Figs. 8 and 9, the theoretical spectral levels -5/3 in the k region as discussed in the development of equation (2.13) 33 Log k (m) FIG.9A Aircraft Ver t i ca l Velocity Spectra July 16 FIST IK SEUCtB mratB U n g h t d n ) i i 1 1 1 1 -5 -4 -3 -2 -1 0 61 Log k (fl?) 34 FIG. 9B Aircraft Vertical Veloci ty Spectra July14 onss^ om Hunan Height fail I i 1 1 1 " •5 -4 -3 -2 -1 0 log k (cn) 35 were drawn in. In the calculation of the theoretical values (see equation (2.13)), the fluxes measured at 3.5 m on the tower were used at a l l heights, the value of the Kolmogorov constant was taken as 0.55, the remainder term D was ignored, and the Keyps profile, ^> was adopted. The values of the ratio R as defined by equation (2.3) between the observed and theoretical spectral levels of the vertical velocity in -5/3 the k region were plotted in Fig. 10. A linear regression of R and log z excluding the 3.5 m levels gave R = -0.003 log z - 0.074 (2.16) when the 3.5 m level values were included, then R = -0.042 log z + 0.012 (2.17) The coefficients of log z are sufficiently small to conclude that between 38 m and 610 m, the observed z-dependence of the vertical velocity -5/3 spectral levels in the k region is equivalent to the z-dependence of the "theoretical" spectral levels which incorporate empirical results applicable to the surface layer (equation (2.13)). For large -z/L, the theoretical spectral levels vary as z°. For the "theoretical" spectral levels, no allowance was made for a z-dependence due to vertical divergence of the vertical transport of kinetic energy. Hence, any z-dependence in the vertical divergence of the transport term must be compensated by a z-dependence of the heat flux or of the other residual terms. As will be discussed in the next section and as can be seen in Table I, the heat flux on both days shows only "a gradual decrease with height; specifically W'T' = -1.8 x 10~4 z + 0.30 for July 16 W'T' = -1.3 x 10~4 z + 0.26 for July 14 where a l l units are in the MKS system. FIG. ID RATIO O F OBSERVED TO THEORETICAL S P E C T R A L L E V E L S FOR V E R T I C A L V E L O C I T Y <0 WW observed WW theoretical 1000 icon 0.5 0.6 0.7 0.8 0.9 1.0 _ l I I I 1— • • • o 10 H B E S T F I T T I N G L I N E (EXCLUDING 3.5M LEVEL)"L^-•>', R = 0.003 LOG z - 0.074 J ! 1.5 _ L _ • A I R C R A F T " ) O TOWER J • AIRCRAFT*) • TOWER i J U L Y 1 6 J U L Y K - T H E O R E T I C A L L E V E L FOR 0^ = 0.55 where O^is the Kolmogorov constant for the longitudinal velocity spectrum -0.4 T 1 1 1 1 ' ' -0,3 -0.2 -0.1 0.0 0.1 0.2 0.3 R = LOG WW observed 0 WW theoretical 37 Since the observed spectral levels are very close to the "theoretical" levels which have a z^ dependence at large z/L, then from equation (2.10), the remainder term, D, must be either small compared to the heat flux term or else very slowly increasing with height. The mean of the R values for vertical velocity for z ^ 38 m is -0.081 (i.e. $ w u, observed = 1.21 $ w w theoretical ) . This R value leads to a value for the Kolmogorov constant, C*( , appropriate to the levels between 38 m and 610 m of 0.46. This value for d , was calculated, assuming the residual term D in equation (2.13) is negligible. The mean of the two R values at 3.5 m is +0.015 and so the decrease in the.observed spectral levels relative to the theoretical levels between 3.5 m and the higher levels is a factor of log ^  (0.096) = 1.25. This relative increase of the observed to theoretical spectral levels at 3.5 m compared to the higher levels,if not just a statistical fluctuation, may be due to a difference in the effects of the residual terms, D, due to the marked decrease of -z/L at 3.5 m compared to the higher levels. McBean et al (1971) found that for unstable conditions with -z/L ^0.2 , the dissipation is €=uJU73z t o within about 15%. If this expression is used for £ at 3.5 m, then, the "theoretical" spectral level is reduced by 11%, making the R values larger at 3.5 m. McBean et al also found that for -z/L^.0.3 dissipation exceeded total production. This effect would make the "theoretical" levels at the higher heights larger and so decrease the R values at the upper levels. Similar results to McBean's have also been reported by Wyngaard and Cote (1971). Thus, the observed change between the relative magnitudes of the observed to "theoretical" spectral levels at 3.5 m compared to those above 38 m, if real, is not resolved by the above-mentioned observed behaviour of the residual terms as functions of z/L in the surface layer. 38 Vertical Velocity Variance In spite of the lack of a low wave number fall-off at some heights, particularly above 92 m on July 14, estimates were made of the vertical velocity variances and were plotted against height in Fig. 11. The values for July 16 (Fig. 11A) and July 14 (Fig. 11B) show a definite increase with height to at least 150 m at a rate that is very close to 2/3 the rate ( z ) predicted by dimensional arguments for free convection, in the constant flux layer. Above 150 m for July 16, the (\J values appear to increase more slowly; whereas for July 14, the Glf values remain approximately constant. The differences in the values of o*^2 between the legs at a given height can be seen in Table I to be generally smaller than the differences in O*^,1 for a given leg for the two different low wave number limits used. The increase in f j ^ 1 under convective conditions at the rate of 2/3 z has been observed before by Myrup (1967) to a height of 93 m. i 2/3 The fall-off of G~w from the z behaviour was suggested by Deardorff's (1972) model to occur at about z = 0.05 to 0.1 z , where l z^ is the height of the inversion. The observed fall-off of CT^ occurs at about this level; (the height of the inversion was typically about 2 Km on clear days in July at Suffield). *. —5/3 The fact that in contrast to <TW , spectral levels in the k region followed the z-dependence predicted by the dimensional arguments for free convection in : the surface layer to a height of 610 m, may be due to the higher wavenumber spectral levels being due to turbulence advected from below by organized convection. Variance on the other hand includes large scale effects at the higher heights for which similarity 39 40 theory may not be applicable. Horizontal Velocity Components on the Tower The U and V spectra, (k), for July 16 are plotted against log k in Fig. 12. There is very l i t t l e difference between the U and the V spectra except at 3.5 m where around the wavenumbers k = 0.01 the spectral magnitude of U is roughly twice that of V. The low wave-numbers are seen to be very similar in shape and magnitude at a l l three heights except for a decreased magnitude at 3.5 m. This lack of shifting of the dominant scale with height has been noted before, (see for example Lumley and Panofsky (1964 pg. 175ff)). From the U and V spectra in Fig. 12, i t is seen that there is not a well defined low wave number fall-off and so variance estimates are not very reliable. As in the case of the vertical velocity spectra the theoretical -5/3 spectral levels in the k region were drawn in on Fig. 12. The observed spectral levels for the longitudinal velocity are seen to decrease with height between 3.5 m and 48 m faster than the theoretical levels. The lateral velocity spectral levels, however, are seen to agree quite closely with the theoretical levels (which include the 4/3 factor in the Kolmogorov constant expected under isotropic conditions.) 2.3.5. Fluxes of Heat and Momentum  Time Traces Figure 13 shows sections of the time traces of the temperatures, the heat fluxes (w'T') and the momentum fluxes (u'w1) at 3.5, 48 and 92 m on the 92 m tower for July 16. The heat flux at 3.5 m is seen to be associated with short bursts of 41 log k (m) 42 FIG. 13 TEMPERATURE, HEAT FLUX, MOMENTUM FLUX, TIME TRACES TO 92 M SUFFIELD JULY 16 '92 '48 W T 92 WT 4 8 UW 92 UVY '48 UW, '3j5 2 C° '3.5 2 L r 1 1 1 1 1 1 I 1 1 1 1 r-0 10 20 30 40 50 60 70 80 90 100 110 120 TIME (SECONDS) 43 1 or 2 second duration which are usually modulated such that the maximum heat flux is near the middle of the convective plumes, (the hot regions shown on the ^ trace). At 48 m and 92 m, the heat flux traces are more similar to the corresponding temperature traces. The momentum flux at 3.5 m is seen to resemble the negative of the heat flux at 3.5 m,both being concentrated in the regions of plumes. At 48 m and 92 m, the momentum flux is seen to be associated with much larger size scales. Flux Cospectral Plots & Statistics from the Tower The heat flux cospectra from the tower are shown in Fig. 14A and 14B for July 16 and July 14 respectively. For small wave numbers, on July 14, for the 48 m and particularly 92 m levels, the large heat flux estimates which often fluctuate widely between adjacent wave number bands are associated with large variances (not shown) of the block to block values, (each analysis block was 204.8 seconds long). The contrib-utions of these poorly defined wave numbers is very significant to the over-all heat flux and so estimates of the heat flux at these levels are very uncertain, The small wave number end of the heat flux cospectra at 92 m on July 16 is seen to drop off markedly in a fashion similar to the drop-off of the temperature signal at that level on July 16. This drop-off will be later shown to be due to spatial variation in the turbulent statistics. The cospectra of the momentum fluxes at 48 m and 92 m (see Fig. 15) have large fluctuations at the small wave number ends for both the days. Estimates of the momentum fluxes particularly at 92 m are thus very uncertain. This problem of intermittancy affecting the momentum flux estimates has been noted before at 22.6 m by Kaimal et al (1971). 44 FIG. 14A Tower Heat Flux Cospectra July 16 CO E I o> o W T 9 2 W T 4 8 WT3.5 FIG. 14 B Tower Heat Flux Cospectra July 14 o a> 10 \ E 1 0 1 o 1n 0 -1-J -1 J 1n 0--1 J W T 9 2 WT, 48 WT3 . 5 -4 -3 0 Log k (mj FIG.15A Tower Momentum Flux Cospectra July 16 3 o> q -4. -3. -2. -1. 0. Log k (mj1 45 uw92 uw, 48 uw 3.5 FIG.15B Tower Momentum Flux Cospectra July 14 to 0.0 -1.0 1.0 0.0 -1.0 1.0 -, 0.0--1.0 J 92 UW4 8 uw3.. Log k (m) 46 The height variations of the overall fluxes measured on the tower were shown in Table II along with some of the commonly measured statis-tical quantities. The huge heat flux convergence with height implied by the small value of w'l'at 92 m for July 16 will be later shown to be a spatial variability effect. The values for the momentum flux at 48 m and 92 m are very uncertain due to large poorly defined low wave number estimates. As has been found from previous experience in this lab, the values of the momentum flux even at 3.5 m are uncertain due to possible U-contamination of the vertical velocity measured due to a t i l t of the vertical velocity sensor from the vertical. The high sensitivity of the statistics to a t i l t in the W-sensor is shown by the statistics at 3.5 m for July 16 assuming a t i l t of 1° from the orientation used in Table II: <TW /u^ = 1.24; C D = 0.0078; -z/L = 0.10 The momentum flux is changed about 16% per degree of t i l t . Heat Fluxes Measured by the Aircraft The heat flux cospectra measured by the aircraft on July 16 and July 14 are shown in Figs. 16A and 16B respectively. As in the case of the vertical velocity there are more distinct differences between the legs at a given height on July 16 than on July 14. Kukharets and Tsvang (1969) suggested that at 500 m}90% of the "turbulent" heat flux is contained within the wave lengths of 150-5200 m (at 100 m, between 90 and 2600 m), and that there is a spectral gap before mesoscale and synoptic scale phenomena become important. Although the heat flux cospectra in Fig. 6 usually showed a low wave number fall-off, i t is clear that in the presence of a longitudinally oriented feature, such as a r o l l , the FIG. 16A FIG.16B 47 Aircraft Heat Flux Cospectra July 16 Aircraft Heat Flux Cospectra July 14 02 0.1 H oo 02 0.1 00 02 0.1 H ao-y Height lm) 610 305 228 152 122 92 co E b 0.0 0.1 00 00 0.1 00 00 0.0 0.1 0.0 naiwt Log k (nr1 Height (ml • 4 tl ; v J / v tt *i f \ / -, / / / / i f V / / / / / / \ \ 1 I j V V f T 1 610 305 152 122 92 61 38 -5 log Mm') 48 size of and even the existence of the lower wave number limit for representative measurement of the heat flux depends upon the direction of flight. The estimate of the heat flux for July 16 and July 14 are plotted against height on log-log plots in Figs. 17A and 17B and on linear-linear plots in Figs. 17C and 17D. Although there is much uncertainty, particularly at higher levels, due to large estimates at low wave numbers, the heat flux is seen to be nearly constant up to about 100 m and then to decrease gradually with height. If the height dependence of the heat flux is written: W ' T * ( Z ) = C l z P (2.18) where and p are constants, then the estimates of the heat flux from both tower and aircraft have a best f i t for w'T' ( Z ) =0.27 z" 0' 0 2 on July 16 w'T' (Z) =0.36 z~°' 1 3 on July 14 For the above estimates, the w'T1 value at 92 m on the tower on July 16 was omitted since its low value will be shown to be due to spatial variability effects; also the value at 305 m on July 14 was omitted since the heat flux cospectrum had shown no fall-off at low wave numbers. Equations (2.19) represent the best fitting straight lines on the log-log plots of the heat flux in Figs. 17A and 17B. However, the use of such plots emphasizes the 3.5 m to 48 m layer, where the constant flux approximation is probably best. If the height dependence of the heat flux is written w'T1 = C l z + c 2 (2.20) (2.19) 1000 _, 4 9 FIG.17A Heat Flux Suffield, July 16 100 July 16 • Ist leg • 2nd leg * 3rd legj ° tower aircraft 10 0;01 0.1 w'T' (C°- m/sec.) 1000 FIG.17B Heat Flux Suffield, July 14 100 10 -\ 0.01 • • • • July 14 • alongwind] - crosswind] a , r c r a f t a tower 0.1 wY (C°-m/sec.) 50 Fig. 17C Heat Flux Versus Height July 16 • 1 leg • 2n d leg ) aircraft A 3 r d leg • tower •u Xi 00 •H OJ W best-fitting line (omitting 92m tower) w'T' ( C°-m/sec) Fig. 17D Heat Flux Versus Height July 14 alongwind") . , t aircraft crosswind( J tower Xi 700 600 -500 -400 300 200 -100 • 0 \ best-fitting line (omitting 305m level) .3 — i — .4 — i — .6 • .7 w'T' ( C°-m/sec) 51 where and C 2 are constants and i f the 92 m tower value on July 16 and the 305 m value on July 14 are omitted as above , then the estimates of the heat flux have a best f i t for w'T' = -1.8 x 10~A 2 + 0.30 for July 16 (2.21) w'T' = -2.0 x 10~4 z + 0.26 for July 14 where the MKS-system has been used. The coefficients of z in equations (2.21) represent the best-fitting linear decreases of heat flux with height. It is noted that extrapolation of equations (2.21) implies that the heat flux becomes zero at about 1.7 km and 1.3 km on July 16 and July 14 respectively. Although the heights of the inversion (where the heat flux must reach zero),were not routinely measured at Suffield, they were typically about 2ka. Hence these data suggest that above about 100 m there is a roughly linear decrease in heat flux with height up to the inversion. An approximately linear decrease of heat flux to the inversion was observed for cloudless or slightly cloudy condition by Volkov et. a l . (1968). Lenschow (1970) (based on four points) found w'T' to decrease more rapidly between 100 m and 500 m than above this level. A linear heat flux profile above the surface layer up to near the inversion is not surprising. Suppose the local rate of heating above the surface layer is approximated by the divergence of the heat flux with height: Then, i f the whole region is heating at a uniform rate, (2.23) Thus w'T* = c 1 z + c 2 (2.24) where c . and c, are constants 53 2.4. DEGREE OF STATISTICAL UNIFORMITY AT 3.5 m. In this section the temperature and vertical velocity turbulent statistics from an array of five sensors at a height of 3.5 m are examined to determine whether there are significant differences. 2.4.1 Comparison of Spectra and Statistics From the Array The data to be discussed is from the July 16th run examined in the previous sections. The array of towers is shown in Fig. 18; the mean wind direction (looking upwind) was 236 M (as measured by the sonic anemometer at 3.5 m on the main tower), which is about 20° to the right of the main - #1 - #2 tower axis. A section of the time traces of the temperature signals from the array is shown in Fig. 19, where the signals are arranged in their cross-wind order from left to right looking upwind. There are no obvious differences in the type of shapes or magnitudes of the signals. The temperature spectra, k$ (k), are shown in Fig. 20. The scatter at the low wavenumber end of the spectrum may be simply a result of poorly defined statistics; (this particular data segment lasted 110 minutes - -4 ~1 which for U _ r =5.5 m/sec is equivalent to k = 1.7 x 10 m ). It 3.5 m M can be seen that T^ in Fig. 20 has a lower spectral magnitude than the other sensors over the wavenumber range k= 0.01 m ^  to l.Om^. However this difference is only about 15% in spectral level. The variances of the temperatures from the array are shown in Table III; (the sensors are again arranged in crosswind sequence). Due to the scatter at the low wavenumbers, the integral under the spectrum from high wavenumbers to several low wavenumber limits is shown. It is seen 214 M tower # 2 48£ m k - t ™2 4-4 m— tower # 1 tower # 24/4 m main tower (sonic) FIG. 18 Tower Array July 16, Suffield 48& m F I G . 19 J u L Y 1 5 ARRAY T E M P E R A T U R E T R A C E S AT 3-5 M warm To C ^ A / / ^ A A f c ^ cool Ti 1 A C ' TSONIC T3 C W ^ i ^ k M ^ h ^ V / T 4 [ ifHj^^ 6 8 10 12 H 16 T IME ( m i n u t e s ) • FIG. 20 Array Temperature Spectra July 16 FIG. 21 Array Vertical Velocity Spectra July 16 1.0 T 0.0--1.0--2.0--3.0-— ' — » -3 -2 -1 0 Log k (m)"' T 2 . V T 4 --1.0--2.0--3.0-A a _ -I 1 0 1 V^ , . W2 . . W 3 „ . W 4 . a Ws . o Cn O N Log k (mV 57 TABLE III TEMPERATURE AND VERTICAL VELOCITY VARIANCES AND HEAT FLUX FROM THE ARRAY AT 3.5 M. JULY 16, SUFFIELD Low wave number limit ( m ) n ., i n-3 n . .-3 n 1 0 in-2 -1 . , .. . . v ' 0.15x10 0.34x10 0.13x10 0.11x10 for calculating variance Number of estimates removed 0 1 5 10 T 2 1.16 1.14 1.04 0.85 T1 1.10 1.08 1.02 0.87 T g riCo-| 2 i.io 1.08 1.01 0.85 T 3 1.01 0.99 0.95 0.79 T. 0.82 0.82 0.78 0.63 4 w2 0.349 0.349 0.346 0.335 wx 0.330 0.329 0.327 0.318 w s [ m/sec ] 2 ° ' 3 5 0 °' 3 4 8 °- 3 4 5 °* 3 2 9 w3 0.330 0.328 0.327 0.319 0.325 0.325 0.324 0.315 W4 wT2 0.260 0.260 0.251 0.227 wT 0.232 0.235 0.234 0.219 wT 0.241 0.246 0.248 0.231 s [C° m/sec] wT3 0.229 0.234 0.232 0.210 wT. 0.219 0.219 0.216 0.199 4 Where the higher wavenumber contributions to the vertical velocity variance and heat flux for a l l the towers for k > 0.99x10 ^  m was taken to be equal to that of the sonic anemometer. 58 that has a lower value of the variance than Tg, or by about 20% to 25% (equivalent to a calibration difference of 10% to 12.5%). The value of the variance for T^ appears to be less than that of the Tg -T1 -T2 group by around 10%. The spectra for the vertical velocity sensors of the array are shown in Pig. 21. The limitations of the frequency response of the G i l l anemometers limits the comparison with W to wavenumbers less than about k= 5 x 10 ^ m \ Except for a scatter at very low wavenumbers, the spectral levels are seen to be very similar. Estimates of the variances of the vertical velocities were made (see Table III) . Because of the limitations on the frequency response of the G i l l anemometers, the integral under the spectrum for the sonic anemometer from high wavenumbers down to k = 0.99 x 10 ^ m ^ was used as the high wave number contribution for a l l the sensors. The variances are a l l the same to within 10%. Heat flux estimates for each of the five sensors in the array are also shown in Table III; the large wavenumber contributions to w'T1 for a l l the sensors for k y 0.99 x 10 * m ^ were taken to be equal to that measured by the sonic anemometer-thermometer. Agreement between the sensors was within about 15%. 2.A.2 Discussion of Spatial Variabilities at 3.5 m The data examined above indicate that over spatial separations of about 70 m in the along-wind and crosswind directions, the turbulent statistics are the same to within about 20%, and except for sensor #4, to within 10%. 59 From Figs. 18 and 20 and Table III, i t is seen that T^, the temperature sensor with the low spectral levels and variance, has a large crosswind separation from the other sensors. Hence these data do not preclude the existence of real spatial variabilities of spectral values of the order of 20% for crosswind separations of about 70 m. It will be shown in the next section that the distinct spatial variations in the turbulent statistics at 92 m are due to longitudinal rolls with transverse (crosswind) wavelengths of 1.7 km. It is possible that the lower spectral levels and variance of the temperature of tower #4 at 3.5 m, i f real, are due to effects of the rolls rather than to differences in the local surface conditions. Dyer and Hicks (1972) compared the heat and momentum fluxes between two Fluxatrons at a height of 4 m for crosswind separations of up to 150 m. The differences in heat fluxes when averaged over a period of 5 to 10 hours were generally less than about 10%. However, their data also showed that heat flux estimates averaged for half an hour varied by as much as 20%. 60 2.5 STATISTICAL VARIABILITIES ABOVE THE SURFACE LAYER OF THE PBL Although the turbulent statistics for temperature and vertical velocity were found to be similar (to 20%) at a height of 3.5 m, very pronounced horizontal variations in the statistics existed on the same day at heights of 92 m. The effect of these spatial variations on Eulerian measurements and the form of the spatial variations are discussed below. 2.5.1 Tower-Aircraft Comparison at 92 m There is considerable evidence that the time-averaged Eulerian measurement of the temperature statistics at 92 m on the main tower on July 16 was very different from the spatially-averaged measurements from the aircraft. The temperature spectra and the heat flux cospectra from 92 m on the tower on July 16 showed a fall-off towards low wavenumbers starting at a much higher wavenumber than the aircraft spectra and cospectra at the same height. This effect is clearly shown in Fig. 22 by the composite plot of the T and W spectra and WT cospectra from July 16 for the tower and leg 1 of the aircraft run at 92 m. The spectra from the other two legs at 92 m on July 16 showed some differences in details from leg 1, (see Figs. 5, 9 and 16) but none of the aircraft T spectra or WT co-spectra showed the fall-off at the low wavenumber end starting at such a high wavenumber as for the tower T spectra and WT cospectra. The markedly reduced values of the temperature variance and the heat flux measured at 92 m on the tower on July 16 were noted before in Figs. 7 and 17. FIG.22 6i Tower-A i rc ra f t Comparison at 92 m July 16 Temperature -1.-, -2. o -3. -4 . J a a D • aircraft T • tower T Vert ica l Velocity < -1. cn 5 -2 . -o ©°"i -o—O" o «?o0 o • o • aircraft W o tower W Heat Flux 0.3-. 0.2-i — 0.1 -q 0.0 -0.1 . A. A - A AA . A & A A 1 ^ A A A A A A j A aircraft WT A tower WT i 1 1— 4. -3. -2. 0. -1 1. Log k (m) 62 The mean temperatures at 15 m, 48 m and 92 m on the tower were recorded on log sheets throughout the run on July 16 and so the observed mean temperature changes could be compared with those implied by the vertical convergence of the heat flux. From the mean temperature measurements, the average rates of temperature increase at 15 m, 48 m and 92 m during the July 16 run were 2.0, 0.8, and 0.9 (C°/hr) respectively. The measured heat flux convergences between the three levels 3.5 m, 48 m and 92 m on the tower for the same run were equivalent to rates of temperature increase between 3.5 and 48 m of 2.4 (C°/hr) and between 48 and 92 m of 11.5 (C°/hr); this last value is clearly anomalous. However the equivalent heating rate between 48 and 92 m i f the mean w'T* from the aircraft is used instead of the tower value at 92 m, was 1.6 - 0.8 (C°/hr) which agrees within experimental error with the observed local rate of temperature increase. Hence the tower measurement of the heat flux at 92 m on July 16 must have been anomalous. The contrasting case of July 14 did not show this spatial variability in the temperature field, there being l i t t l e difference between 48 m and 92 m in the time traces, statistics or shapes of the spectra or cospectra. For a tower-based Eulerian measurement, time-averaged for a period of 80 minutes, to miss most of the low wavenumber contributions to the temperature field, requires a large scale feature which dominates the heat flux at 92 m; this must be either (1) stationary with respect to the ground or else (2) aligned parallel to the wind and advecting with the wind with essentially no crosswind translation, (i.e. very small phase speed). Since the surface conditions were very similar for many kilometers surrounding the site, the first possibility of a stationary feature is 63 unlikely. 2.5.2 Influence of the Rolls on the Low Wavenumber Temperature Field The large scale features on July 16, mentioned in the previous section can be identified as longitudinally stretched features by a consideration of the differences of the spectral shapes of the temperature between the three directions of flight at a given level. In the presence of a directional organization of the temperature field, then this organization must be reflected in differences in the size scales associated with the peaks of the temperature spectra obtained from aircraft flights in different directions. The 61 m level temperature spectra in Fig. 5A for July 16 show the most striking differences in spectral shape between the 3 directions of flight at a given level. Knowing (1) the directions of flight for each leg and (2) the mean wind direction at 3.5 m (236°M) and assuming (3) that from the tower results one of the large scale features must be aligned with axes downwind, (and so have a transverse wavelength), then the wavelength and orientation of this longitudinally oriented regular feature could be determined. The method of determining the wavelength and orientation is briefly as follows. A regular feature in the temperature field will have a wavelength, oriented at some angle, c< , to the direction of flight of the f i r s t leg. Hence the regular feature will be represented by a spectral peak on the first-leg temperature spectrum at a wavelength, ^ where sin <*. = (2.25) Since the second leg flown at the same height has a known orientation with respect to the first leg, then the spectral peak on the second leg 64 spectrum is } 2 sin ( S x + * ^ (2-26) where - is the known angle between the flight directions on t h e f i r s t and second legs. Equations (2. 25) and (2.26 ) are two equations in the two unknowns 2/) and . If there were only one peak on each temperature spectrum, then the problem would be solved. However, t h e presence of a second peak on at least legs 1 and 2 at 61 m (see Fig. 5A again) indicates the presence of a second regular feature. Each of the four possible pairings of peaks of legs 1 and 2 were examined to determine i f they corresponded to a longitudinally oriented feature compatible with the observed wind direction at 3.5 m (236°M) and with t h e spectral peaks on the third leg. The observed temperature spectral peaks at 61 m permitted only one solution for the transverse wavelength assuming a reasonable value of the mean wind direction at 61 m; this wavelength is 1.7-0.3 km with axes oriented at 260°M. The remaining spectral peaks can be interpreted as due to a longitudinal wavelength of 0.5 - 0.1 km, with axes oriented 94 + 12 from the transverse wavelength axes. A transverse wavelength of 1.7 km is within the range of wavelengths Lemone (1972) found in her study of rolls due to thermally modified inflectional instabilities. Lemone found that the major energy source term for these instabilities comes from the vertical shear of the transverse wind. The strong winds and the directional shear between 3.5 m (236°M) and 61 m (close to 260°M, assuming that the longitudinal features are indeed oriented longitudinally, parallel to the wind direction at 61 m) suggests that the longitudinal features are in fact rolls due to 65 inflectional instabilities. At 61 m, the effects of the rolls on temperature structure were strong but at 48 m, on the tower, (see Figs. 3A and 4A), there were no obvious ro l l effects. Hence the base of the rolls (or at least the lower level at which they noticeably affected the temperature field) was between 48 m and 61 m on July 16th. Lemone observed effects of the rolls at 18 m over the sea and Hanna (1969) argued that such rolls cause the longitudinal dunes,which have spacings around 2 km,found in large flat deserts over which strong steady winds blow. 2.5.3 Influence of the Plumes on the Low Wavenumber Temperature Field The longitudinal wavelength of 0.5 km found for the July 16 case is of plume scales. This wavelength was a distinctly longitudinal wave-length, so that peaks in the temperature spectra were shifted with changes in the directions of flight. In order to interpret the data as originating from plumes, there must have been a transverse (crosswind) ordering of the plumes with a regular downwind spacing between the rows of plumes. The crosswind spacing between the plumes was apparently not regular enough to produce a distinct spectral peak. The temperature spectra for July 14 did not show as much difference between the legs at a given height as for July 16 (see Fig. 22). The Eulerian measurements from the tower for July 14th appeared to be consistent, without any of the obvious .spatial variability effects of July 16th. However at the lowest level flown (38 m), the downwind temperature spectrum has the spectral peak at a much lower wavenumber than that for the crosswind leg. Above 38 m, the wavenumber of the spectral peaks for temperature do not show a consistent and distinct directional 66 difference. Lenschow (1970) found a horizontal asymmetry in the temperature field such that at 100 m the spacing between plumes and the size of the plume sectionings were about a factor of 2 smaller in the crosswind direction. Lenschow's data came from a run in which the mean wind speed at 100 m was 9.1 m/sec. Warner and Telford (1963) found no differences in the plume spacings under "light wind" conditions. In the data discussed for July 14, the mean wind speed at 92 m was 6.3 m/sec; on July 16, the day with rolls, the mean wind speed at 92 m was 10.1 m/sec. From a l l these results, i t appears that horizontal asymmetry in the temperature field occurs in the presence of strong wind shear effects and that these directional differences tend to disappear above the surface shear zone. CHAPTER 3 TRANSLATION VELOCITY OF CONVECTIVE PLUMES Most studies of convective plumes have been based on aircraft measurements or measurements from a single instrumented tower and have not included measured estimates of the plume translation velocity. Due to a lag in the temperature at the lower levels of an instrumented tower, Priestley (1959, pg. 72ff) suggested that such a downwind t i l t could only occur i f the translation speed of the plumes was close to that of the mean wind speed at the plume formation level; this will be shown to be a special case. Kaimal and Businger (1970) interpreted simultaneous velocity and temperature measurements at two levels (5.66 and 22.6 m) on a tower to give a translation speed close to that of the mean wind at 0.5 m. Their method required accurate absolute vertical velocity measurements in a turbulent background and assumed no change of plume t i l t between the sensors and no crosswind plume t i l t . From an analysis of the time correlation functions of the horizontal wind velocity components between 5 anemometers in a linear array, Panofsky (1962) concluded that the speed of eddy motion is close to the local mean wind speed, except for some larger scale eddies which appeared to move faster than the wind speed. Panofsky attributed this to the "centre of gravity" of the larger eddies being higher than the level of observations. Although the horizontal velocities are not good indicators of plumes, at least some of the larger scaled horizontal-wind eddies may have been associated with convective plumes. 68 The ratio of the magnitudes of the plume scale fluctuations to the smaller-scale fluctuations is much larger for temperature than for the velocity components. For this reason, temperature has been used many times previously as the best indicator for plumes (see, for example, Warner and Telford (1963), Lenschow (1970), Kaimal and Businger (1970). In the present study, a 2-dimensional array of 5 towers at a height of 3.5 m was used to measure both the speed and direction of motion of the plumes using the technique of coherence and phase analysis of the temperature signals from the array. The plumes are treated as distinct shape-preserving entities embedded in the turbulent field and translating at one speed at a l l levels. 69 3.1 ARRAY CONFIGURATION AND SIGNIFICANCE The data to be discussed is from the July 16th run discussed in the previous chapter. The array set-up used on July 16 was shown in Fig. 18. The turbulent statistics were shown in section 2.4 to be similar to within 20% throughout the array. Thus the vector separation between any two temperature sensors in the array determines the spatial dependence of the coherence and phase statistics between those two sensors. Having anticipated this statistical similarity at 3.5 m, the towers in the array were arranged along two perpendicular axes with different separations between the towers. This arrangement gave two dimensional resolution in the horizontal and avoided duplication of vector separations between the towers. The 92 m tower and the small towers together formed a three dimensional array for studying the three dimensional nature of convective plumes. \ 70 3.2 DETERMINATION OF THE VERTICAL TILT OF THE PLUMES The t i l t of the plumes in the vertical was determined from the phase between the temperature signals on the 92m tower. The coherences and phases between the temperature for both the July 16 and July 14 runs discussed in the previous chapter are shown in Figs. 23 and 24. Where the coherences are high the phases between the temperatures are consistently positive. Positive phase means the second channel lags the f i r s t . Since in Fig. 24 the lower sensor was always the second channel then this means that the lower temperature signals lagged the higher ones. Thus the convective plumes must have a vertical t i l t in the downwind direction. If there is a constant time lag,t , between the signals from two sensors over some frequency range, then thaphase shift represented by.'f is 2 TT f f , where f is frequency. Hence over that frequency range, the phase between the two sensors is a linear function of frequency. A linear frequency plot of the phases for both days over the frequency range where the coherence was greater than about 0.3 is shown in Fig. 25. The phase lines expected for various, given time delays are drawn on Fig. 25. Between 3.5m and 48m on July 16, there is constant time lag of between 7.0 and 11.5 seconds.. The roughly constant phase shift between the temperature signals at 92m and 48m on July 16 is presumably due to the longitudinal rolls interacting with the plumes. On July 14, the phases show much more scatter relative to the phase times expected for a constant t i l t ; however they are generally positive indicating that the t i l t even i f variable is in the downwind direction. In order to determine the downwind t i l t the translation speed of the 71 l.o • 0.9 -0.8 -FIG. 23A 0.7 -0.6 -CDHEFBtES BETWEEN THE TEMPERATURES B ON THE 92M TOWER i 0.5 -JULY 16 s 0.1 -0.3 -0.2 -0,1 -0.0 --3. O A o ' O O A o 0 0 ° o g g e -r~ -1. A Tg2 - % ° T « " T3.5 0. LOG FREQUENCY (HZ.) R G . 23B COHERENCES BETWEEN THE TEMPERATURES ON THE 92M TOWER JULY 11 1.0 ^ 0.9 0.8 -I 0.7 0.6 -0.5 -0.1 -0.3 -0.2 -0.1 -0.0 -1 O A o02 ° o 6" 6 -2. r - l . T92 " T18 T18 * T3.5 LOG FREQUENCY (HZ.) f F I G . 2 4 A PHASES BETWEEN THE TFiPERATURES ON T H E 92M TCWER V S . LOG FREQUENCY J U L Y 1 6 T 9 2 ~ T 4 8 T/.0 - TT • ' 1 8 " ' 3 . 5 UJ 3 1 8 0 - i 9 0 -- 9 0 -- 1 8 0 i -3. 1 - 2 . LOG FREQUENCY 0 . (HZ.) RG. 24B PHASES BETWEEN THE T B P E R A T U R E S OH T H E 92M TOWER V S . LOG FREQUENCY J U L Y W ' % - T 3 . 5 1 8 0 n 9 0 ~ - 9 0 --180 -H. T - 3 . • A A "I . ~T~ 0 . LOG FREQUENCY (HZ.) 74 plumes must be known. For a translation speed of lOm/sec. (as is determined in the next section), the t i l t from the vertical,Q , between 3.5m and 48m is given by: tan 9 = U t r / ( 4 8 - 3.5) = 2.1^0.8 Hence, Q = 64° i 10° where TJ^ is the measured translation speed and f is the measured time lag between the temperature signals at 3.5m and 48m. Since the plumes are treated as entities moving at a particular translation speed (to be discussed in the next section), then Taylor's "frozen-field" hypothesis with an advection speed determined by the local mean wind speed is not applicable to plumes. For this reason care must be taken in comparing the downwind t i l t s of the horizontal-wind eddies found by Pielke and Panofsky (1970) assuming Taylor's hypothesis, with the above-determined t i l t . 75 3.3. DETERMINATION OF THE TRANSLATION VELOCITY OF CONVECTIVE PLUMES 3.3.1 Direction of Motion by Coherence Analysis From the time traces of the temperature signals from the array shown in Fig. 19, T g and T^, are seen visually to have a high coherence, with the other pairs of sensors having less obvious coherence. Before looking in detail at the computed coherence spectra between the various temperature signals in the array their expected form will be considered. Separation Space Assuming that the temperature statistics are uniform over the horizontal field of the array, then the coherence between the signals from a pair of sensors can be represented in a separation space resolved along the directions of the axes of the array. The possible vector separations for the array used on July 16th are shown in Fig. 26. Note that since T T has the same coherence as T.T. , then the sign of the i j J i vector separation is of no consequence and there is symmetry about the origin for coherences in separation space. If the plumes advect without change between the sensors of the array, then the only separation determining fall-off in coherence is the separation perpendicular to the direction of the plume motion. Hence lines of constant coherence in separation space are expected to be straight lines parallel to the direction of motion and to the 1.0 coherence line which must pass through the origin. Lines of lesser coherence must lie symmetrically on either side of the 1.0 line. In the real case, the temperature field of the plumes will change slightly while passing through the array and so a 1.0 coherence line will not exist. But i f the changes are not very large, then the lines of constant coherence will s t i l l be nearly parallel to the motion direction. FIG. 26 SEPARATION SPACE TOWER ARRAY 76 \r3 9 TsTz Ts-T-Tz A*b JTST9 5'2 °7:r •. Tsri -TST3 -© TTT3 mT5T» w PHYSICAL SPACE TOWER ARRAY T. 7: 77 Coherence Levels The coherence between pairs of temperature signals were compared over the frequency range in which plume effects appeared to dominate (from an inspection of the time traces). The five highest coherence levels are shown in Fig. 27A. The low coherence level for T^T^ (approx-imately cross-wind) suggests that the plumes have a rather narrow lateral extent. Consider TgT^, TgT^j which form a square with one corner at the origin in separation space (see Fig. 27C). Because TgT^ has a larger coherence level than T^T^ and smaller than TgT^, then the 1.0 coherence line lies unambiguously between TgT^ and TgT^. The separation I T , is seen in Fig. 27A to have a similar coherence level to TgT^ (= T^Tg). Thus one iso-coherence line must nearly pass through T^Tg and TgT2 in separation space. Hence, tan <j> = 0.25 and <j> = 14°, where § is the angle between the direction of plume motion and the Tg-T^-T2 axis. Thus the direction of plume motion is about 228°M. If and TgT^ had been similarly paired, then (j)= 18.4°. Hence the error in the estimation of plume motion direction is probably not more than 5°. The mean wind direction is calculated from the sonic anemometer at 3.5 m was 236°M - 5°. Hence the plumes were travelling in a direction close to that of the surface (3.5 m) wind. 78 FIQ 27 DIRECTION OF PLurE ItJTION FROM COHERENCE ANALYSIS FIO 27A COIERENCE LEVELS 1.0 T 0.8-0.6-O . t -0.2 . 0.0 LOG FREQUENCY FIQ 27C SEPARATION SPACE MEAN PLUME MOTION DIRECTION T rT 3 -T 4 Axis 304° H 79 3.3.2 Speed of Motion From Phase Analysis Consider the phase relationship between two sensors T. and T. 1 J separated by a distance at an angle Q to the translation direction. Then for a front perpendicular to the translation direction and travelling at u"t, the time difference between the sensors for the passage of the front is *y - cosQ (3.1) U t For some frequency, f, the phase shift represented by is €ij (f) = 2vfTij =  ZirXii toS 9 (3.2) and so ^t _2T; cos e = H C.QS e = constant (3.3) 3 f Ut Hence for a given U the phase between the two sensors plotted as a function of frequency is a straight line. In Fig. 28 is plotted the phase of I T . (the pair ifith the best 1 D coherence) as a function of frequency. Also drawn in are the theoretical lines for various values of the translation velocity. The bracketed numbers beside each point are the coherence levels. For coherences less than 0.3, the phase becomes rather meaningless and so these values were not plotted. From Fig. 28 the translation velocity of the plumes was visually estimated to be 10 - 2 (m/sec). This compares with the 3.5 m mean wind speed of 5.4 (m/sec.) (which was found by a G i l l anemometer, a cup and a sonic anemometer). The translation speed is close to the mean wind speed at 48 and 92 a. (9.9 and 10.1 (m/sec) respectively). Hence the translation speed of the plumes is clearly not the mean wind speed at their formation level as proposed by Priestley (1959) and Kaimal and Businger, (1970), but rather is close to the mean wind speed near the 80 80n FIG. 28 SPEED OF PUI'E MOTION FROM PHASE ANALYSIS (PHASE BETWEEN Ts AND Tp a: ID LU 50 H 40 H 20 J 5 M/SEC / / ^ - « 0 . 5 4 • ' 0.35 8 M/SEC / 10 M/SEC J2 M/SEC 0 . 8 4 0^ PEAK OF T-SPECTRUM -ID 0.0 0.2 o.q 0.6 —I 0.8 FREQUENCY (HZ.) where the numbers beside the data points are the coherence levels 81 top of the surface shear zone. Panofsky's (1962) observation that larger-scaled horizontal-wind eddies tend to move faster than the local wind speed is perhaps due to at least some of the larger-scaled velocity eddies being associated with convective plumes. The coupling between the velocity and temperature fields associated with convective plumes is discussed in the next section This interpretation of Panofsky's result suggests that, in a study by Pielke and Panofsky (1970), the slopes of the longitudinal-wind eddies as a function of z/L might have shown less scatter i f the actual translation speeds rather than the local mean wind speeds had been used to calculate the slopes. 82 3.4 DYNAMIC CONSEQUENCES OF THE TRANSLATION VELOCITY Since the plumes have a vertical continuity and a consistent trans-lation velocity, then i t appears reasonable to assume that the plume is shape-preserving in time and, thus, that the lifetime of a plume is long compared to the time that i t takes for a parcel of air to rise from near the surface through the plume to the top of the surface layer. In this section, the observed translation speed will be shown to be compatible with shape preservation; also the coupling between the velocity and temperature fields associated with convective plumes will be discussed. Shape Preservation (i) Downwind Edge The translation speed, Ufc, is the velocity of the phase of the temperature signal. Hence individual elements do not have to travel at U^ ; rather, Ufc is the speed at which the boundary separating rising elements from non-rising elements travels. The only necessity is that the condition which allows unstable elements to begin to rise be moving at Ufc. If the passage of a region of positive vertical velocity higher up allows unstable elements below to begin to rise then the plume can have a speed greater than the mean wind at lower levels and s t i l l preserve its shape in time at the downwind edge, (i.e. leading edge). (ii) Upwind Edge Consider the upwind edge of the plume at times = t and t +T(see Fig. 29). For the plume to be shape preserving, then there can be no mean flow through the boundary. The mean flow pattern must be such that a particle on the boundary will stay on the boundary. Hence: U cos 6 = U cos & - W sind (3.4) FIG. 29 UPWIND EDGE SHAPE PRESERVATION: U t > Ub 84 where U^, W^  are the mean velocity components of the boundary particles, (W^  positive upwards) and & is the t i l t of the upwind boundary. Note that i f the air at the boundary had no vertical motion, W, =0, b then from equation (4.4) the plume boundary could only move at U = U^ . A sample of the time traces of the temperature and of the three components of the wind at 3.5 m on the main tower is shown in Fig. 30. From Fig. 30, the value of U at the upwind boundary, U^ , as defined by the sharp temperature gradient }is close to the mean U for the 3.5 m level. Since U changes with height, then must also change with height and so, must be non-zero at a l l heights where U"t for shape preservation to hold. If Wb ( z ) > 0 , then Ut < Ub (z ) If W, ( z ) < 0 , then U > IL ( z ) D C D for a downwind t i l t as in Fig. 29 When W, = 0, then U = U, and there will be a stagnation point in b t b a frame of reference moving with the plume. The height of this stagnation point will be called z g. The origin of the downward velocities W^  (z) ^  0, necessary for shape preservation at the upwind edge of the plume, can be understood with the help of Fig. 31. Hot elements which enter the bottom of the plume will have a longitudinal velocity equal to the mean wind speed at a level near the bottom of the plume. This speed will be less than U^ , the speed of motion of the plume. Thus in a frame of reference moving with the plume these hot elements will have a longitudinal component of velocity in the upstream direction. Consider now the components of velocity normal to the upwind boundary at the stagnation point in a frame of reference moving with the plume. On the plume side of the boundary, there will HO. 30 T I M TRACES Of TEMPERATURE AND WIND CQTCMENTS AT 3.5M TEMPERATURE < C ° ) - 3 J V E R T I C A L V E L O C I T Y OVSEC) LONGITUDINAL V E L O C I T Y (M/SEC) L A T E R A L V E L O C I T Y (H/SEC) TIME (SECONDS) FIG. 31 SUGGESTED FLOW PATTERN IN CONVECTIVE PLUME (iN A FRAME OF REFERENCE MOVING WITH THE PLUME) be hot elements moving towards the stagnation point. Outside of the plume, along a direction normal to the boundary, the wind w i l l also have a component directed towards the stagnation point. Hence in a direction normal to the boundary there w i l l be a convergence towards the boundary. The effect of the pressure f i e l d due to this convergence on the motion of the elements i n the plume w i l l be to force the component of motion normal to the boundary, to zero as the element approaches the upwind boundary. The remaining component of motion pa r a l l e l to the boundary w i l l cause the element to either move upwards along the boundary creating the positive W, (z) for z> z , or else downwards along the O S boundary creating the negative W (z) , for z<z . The argument for the 0 s existence of a convergence in the direction normal to the boundary is applicable to other levels i f the motions normal to the boundary are taken with respect to the motion of the air parcels on the boundary. Thus there can be a convergence at the lower levels, even though U ^ U at these levels. From observations the upwind edge at 3.5 m is clearly a region of convergence in the direction normal to the boundary. This can be seen in Fig. 30 indirectly, by the existence of a very sharp temperature gradient at the upwind edge of the plume and directly by the convergence of U towards the upwind plume boundary (as defined by the sharp temperature gradient). Priestley's (1959) argument that a plume must travel at the speed of the wind at the base of the plume is seen to pertain only to the special case when the stagnation point i s at the level of the base of the plume. o 88 Relationship of the Upwind Edge Convergence to the T i l t of the Plume Boundary For a boundary with a downwind t i l t of 6 , (see Fig. 32) the components of velocity normal to the boundary are W ' sin G - U* cosG inside the plume and -W£ sin © + U & cos & outside the plume where primed velocities are velocities measured relative to the mean boundary motions. Subscript "p" refers to inside the plume and "e" to outside. If the velocities are broken up into components normal and parallel to the boundary then the normal components must be equal and opposite in order to have the same dynamic pressure and pressure gradient on each side of the boundary. If the pressure on the. outside.of the upward edge were larger for example, then the boundary would move downwards, but these velocities are relative to the boundary and so this is not possible. It is reasonable to expect equal pressure gradients on either side of the boundary on the basis of symmetry. Hence w' sin© - U* cos 6 = - w' sin6+ u' cos© (3.5) p P e e - A U' + U' ex c\ tan © = e p_ (3.6) w + w e p If the upwind edge t i l t does not change significantly over a few meters in height, then the time traces at 3.5 m in Fig. 30 can be used to roughly estimate the velocities in equation (3.6). From a visual smoothing of the velocity time traces in Fig. 30, the vertical and longitudinal velocities were estimated at about 5 seconds before and after the passage of the large temperature shear at the plume upwind boundary: W « -0.1 m/sec u' =-3.3 m/sec e e W* == 0.5 m/sec U' •= 0.0 m/sec P P ' FIG. 32 UPWIND EDGE C O N V E R G E N C E VELOCITIES 90 where as above subscript " p" refers to inside the plume and "e " to outside. Hence from equation (3.6), tan Q = 5.5 6 = 80° The effects of a decrease in t i l t with height would be to make the t i l t estimated above too small. In section 3.2, the phase lag between the temperature signals at 3.5 m and 48 m was found to be equivalent to a time lag of between 7.0 and 11.5 seconds. For a translation speed of 10-2 m/sec, this time lag corresponded to a downwind t i l t between 3.5 m and 48 m of 64° - 10°. Hence the above estimate of the t i l t at the upwind edge at 3.5 m, as 80° seems reasonable. Compatibility of the Upwind Edge Convergence with Shape Preservation The conditions necessary for shape preservation and for the upwind edge convergence along the direction normal to the boundary, as expressed equations (3.4) and (3.6) respectively, must be compatible. This compatibility can be tested by eliminating 0 in equations (3.4) and (3.6) and estimating Ufc. If the value of Ufc so obtained is close to the value obtained from the phase analysis shown in Fig. 28, then shape preservation and the upwind edge convergence can be considered compatible. From equations (3.4) and (3.6) (3/7) The presence of large turbulent motions make the estimates of the vertical velocity at the plume boundary, W^ , from the time traces in Fig. 30 un-certain. However, from careful inspection of Fig. 30, estimates were 91 made of and which led to 8 (m/sec). This value of U agrees with the estimate of TJ"t£(10 - 2 m/sec) from the phase analysis shown in Fig. 28. Although this is a poor test due to poor estimates of W^ , i t is seen that shape preservation and the upwind edge convergence are reasonably compatible for U close to the wind speed at 50 to 90 m (i.e. the wind speed near the top of the surface shear zone). Effect of the Directional Shear of the Mean Wind . The effect of the convergence at the upwind edge is to cause a high pressure region near the upvind boundary of the plume which acts to accelerate the air in the plume to the speed and direction of motion of the mean wind. The direction of motion of the plumes was found to be a l i t t l e to the left (looking upwind) of the mean wind direction at 3.5 m; although the experimental errors were large enough to permit agreement. If the difference is real, then i t suggests, from Ekman turning, that the plumes move at 3.5 m in a direction close to the wind direction at the level of the base of the plumes, which is somewhere below 3.5 m. the presence of a crosswind t i l t (see Fig. 33A) would allow such a direction of plume motion at a l l heights while s t i l l maintaining shape preservation and allowing the air in the plume at any particular level to move in the same direction as the mean wind at that level. In such a case there must be a transverse motion of the air in the plumes relative to the plume boundary and so the air forced downwards by the upwind edge convergence would tend to curve downwards and to the left (looking upwind); see Fig. 33B. The stagnation point exists only , 92 FIG. 33 CROSSWIND TILT OF PLUME FIG. 3 3 A FIG.33B X - Z PLANE U J////// / t / / // // // // // for motion in the X-Z plane; particles reaching i t may have a transverse motion with respect to the boundary. If the plumes have a crosswind t i l t , then an individual plume might be sectioned through the middle by a sensor at one height and only through the edge or perhaps not at a l l by sensors at different heights on the same tower. The existence>of a crosswind t i l t in the presence of a directional shear of the mean wind with height means that results from two-dimensional case studies such as Kaimal and Businger (1970) are uncertain. 94 3.5 DYNAMICAL EXPLANATION FOR THE OBSERVED TRANSLATION SPEED The translation speed of the plumes measured at a height of 3.5 m was found to be significantly greater than the mean wind speed at that height. This result was shown in the previous section to be compatible with shape preservation and convergence at the upwind edge of the plume. In this section a dynamical model of the plume will be used to explain why the observed translation speed is close to the mean wind speed near the top of the surface shear zone. The hypothesis is made that the plume translation speed must be that which allows a plume to exist stably and to transport heat upwards through the surface shear zone of the wind at a lower level of thermal instability than that required for a plume moving at any other speed. The general approach to the problem will be to examine the effects on the amount of upwards heat flux in a plume, of a small increment of translation velocity. If the heat transported by the plume through the surface shear zone is increased, then this will be shown to be equivalent to the plume being able to exist stably at a lower level of thermal instability. The translation speed required by the hypothesis will be such that the heat flux through the surface shear zone is maximized. The dominant effects in determining what translation speed, U , is preferred are assumed to be the entrainment of hot elements at the base of the plume (to be called the feeding term), the downturning due to convergence at the upwind edge (see Fig. 31) and the turbulent mixing at the plume boundaries. 95 3.5.1 Effect of the Translation Speed on Plume Feeding; Model of  Plume Feeding: It is assumed that when a plume passes over a region, the hottest elements available tend to be entrained f i r s t , through the base of the plume (see again Fig. 31); the height of the base of the plume will be called the entrainment level. Suppose the elements at the entrainment level have some temperature probability density distribution, P ( T^") . Then the number of elements with T > T0' for some T e , available to the plume for entrainment in a time t , depends directly upon the total number N of elements the plume moves over in the time t , and hence on U : N = (3.8) A = Y pU tT (3.9) where pw is the areal -density of elements at the entrainment level, A is the area swept out by the plume in time T , and Y is the plume cross-stream width. P Hence as U increases there is an increasing supply of hot elements (T'> TJ ). i Te is now defined as the value of T of an element being entrained which will approach the stagnation point, (see Fig. 31) (neglecting, for now, the effects of turbulent mixing at the plume boundaries). Towards the leading or downstream edge, T* of the entrained elements is larger i than T w (by previous assumption) and so these elements rise more quickly and also have a greater distance over which to accelerate before approaching the convergence at the upwind edge. Hence those elements entrained near the downwind edge and for which T> T 0 are forced upwards by a pressure field associated with the upwind edge convergence and they are the elements which stay with the plume. Towards the trailing or upwind edge, T*^ T0' for the entrained elements and so elements from this source rise more slowly and approach the upwind boundary with a component of motion parallel to the boundary and downwards. These elements are forced downwards and are left behind by the plume. Thus, according to this model of the plume, only elements with T* ^  T,' avoid being downturned and contribute to the effective upwards heat flux of the plume. Formulation of the Feeding Term Consider two values of U : \ • \ (3.10) where £ is small and positive. From (3.8) and (3.9) A/2 • e*% \ * (3.11) and (where a l l subscripts "2" refer to U and "1" to U ). t2 1 Suppose that of elements there are with T'> Tj ; then according to the model of entrainment discussed above, in a time T , a plume moving at U will entrain M~ elements which will rise above the t 2 stagnation point. If P (T ) is the probability density distribution for the temperature of the elements at the entrainment level, then 6 = C P TM dT' (3.13) Mt \ This will be called the feeding term. There is a similar equation for 97 M^ /N^  and so we can define the ratio1. K given by R « i i i s W)*r' Vt, <3.1A> where K^ ^ is the ratio of the number of elements entrained in any time, T , which rise above the stagnation point for a plume translation speed of U compared to the number for U . t 2 . 1 If t ^ i ^ 1, then a plume moving at Ufc has a greater flux of hot elements rising above the stagnation point level, than a plume moving at U . Hence for some instability level, such a plume moving at U will 1 t 2 be able to lose some heat flux by turbulent mixing at the boundaries of the plume or by other processes and s t i l l have a flux of hot elements rising above the stagnation point level. The same turbulent mixing would cause a plume moving at U to have no flux of hot elements about the 1 stagnation point (Turbulent mixing effects will be shown in section 3.5.3 to be larger for the slower translation speed U , but this does not 1 affect the validity of the above argument) . Thus 1 is equivalent to the condition that a plume moving at TJ can exist at a lower level of 2 instability than a plume moving at TJ . 1 By a similar argument, K ^ 1 means that a plume moving at U 1 can exist at a lower instability level than a plume moving at TJ . 2 The condition = 1 means that the increment in translation velocity between U and U has no effect on the number of hot elements t 2 t x rising above the stagnation point level (ignoring for now the differences in turbulent mixing). So, = 1 coincides with a maximum for the heat flux (or flux of hot elements) at a given instability level. Thus 98 = 1 is the condition which determines the translation speed of the plumes as required by the hypothesis. With the use of equations (3.13) and (3.14), the feeding term, ~ , can be parameterized. We define N T0 = Ta' ( t r € ) (3.15) where T is the value of T, corresponding to U and Ta is the value Z C2 1 corresponding to U . Thus € is the increment for TJ corresponding to 1 the increment 5 in U (see equation (3.10)). For several assumed forms of the temperature probability density distribution, Py (T*), then equation (3.13) will be shown to reduce to a form, $ - 7, U',<rrJ (3.16) where QJ.1 is the temperature variance. With the use of (3.16) for a given temperature distribution, equation (3.14) will then be shown to be reduced to the form The translation speed required by hypothesis is the one defined by the condition = 1 and so (3.17) for = 1 is of the form, § • g / U ) (3.18) where £ is an arbitrary increment to U . It will be shown that for «S 1 , then equation (3.18) can be written Tr* n u ,v (3.19) Equation (3.19) is the required relationship between ~ , (the fraction of elements that are entrained and rise above the level of the stagnation point) and (where €l\ is the change in temperature required for an element to rise above the level of the stagnation point for a change &U in the translation speed). 99 The Feeding Term for Specific Temperature Distributions Equation ("3.19) was derived for several simple but very different assumed forms of the temperature probability density distributions of curve of equation (3.19) turned out to be quite insensitive to the form of the assumed temperature distribution. (i) Triangular Temperature Distribution: Consider a temperature probability density distribution of the triangular form: the hot elements at the entrainment level. For — ^0.3, the solution (3.20) for | T1! < L ; zero, elsewhere. Since for any distribution / (3.21) and (3.22) then i t is easy to show that P. = i*/4 and so (3.23) From equation (3.13) (3.24) 100 This is a simple quadratic in T0 and so Equation (3.14) for K^, with the use of (3.25) and then (3.10) and (3.15) becomes for a triangular distribution (equations (3.23)), u ftr <rT - KY vt, _ /<re<rr -T.'. (i+i) (3.26) For K 2 1 = 1, and again using (3.25) this becomes, to first order in % and € , an equation of the form of (3.19), TJ - z(T£T) <3-27> (ii) Rectangular Distribution A similar analysis can be used for a rectangular distribution f^tr') = P, for |T'j < L; zero elsewhere Applying the conditions of equations (3.21) and (3.22) and then i t is easy to show 3 - «-28) ( i i i ) Normal Distribution - r'1 p f r ' j = -f=r C (3.29) This is the known form of a normal probability density distribution and so the equation for can be solved for immediately. Since tables for the normal function are usually written in the form: 101 then we can write N r * J - e * 4 (3. 30) where a. * Equation (3.14) becomes: <J-o© J27 e "V* -co (3.31) For a given value of M../N then Te can be found using (3.30) and the X X | tables of values for the normal function. For K^ ^ = 1, and for given values of M^ /N^  and & , U^/ll^ can be found from (3.31) and ^solved for, using (3.30); hence € can be solved for (since &x /a, * / f €) M An equation for — of the form of (3.19) involving only €/X N can be developed for &4 1. For K^ ^ = 1, (3.31) becomes i <»<> and so 1 -f where PM was written for •—, e To f i r s t order in £* and £ .a, ci+e) (3.32) for the sake of brevity. (3.33) For fal and thus £<< 1, then p can be taken as a constant between N 0., and a , (l +ej , 102 1 3x Thus equation (3.33) can be written r •/ M But Q, =yflLj: and is uniquely defined for a given value of — 1 regardless of the value of €.fs. Hence equation (3.34) is of the form f = 0.35) Thus only the ratio comes into the formulation of the feeding term for the normal distribution. Note, however, that the specific form of the normal distribution was never used in the development of (3.35). Hence this equation is valid for any physically reasonable temperature distribution. Graphic Form of the Feeding Term: The equations of the form *— = for the condition of = 1 (condition for the required translation velocity) were plotted in Fig. 34 for the temperature distributions examined. It can be seen the very different distribution shapes did not make a large difference in the form of the curves. M Consider physically what this graph shows. Recall that — is the ratio of the number of elements, M, that are entrained by the plume and rise above the stagnation point to the total number of elements, N, that the plume passes over. £ is an arbitrary increment added to the FJQ. 3H 104 translation speed of the plume and £ is the corresponding change in i T e , the temperature an entrained element must have to approach the stagnation point. = 1 is the condition for no change in the number of elements entrained which can rise above the stagnation point level, for a change in translation speed of ^ TJ . Thus i f the change-in T^ , for a given change $ U in the translation velocity is such that (-li lies on the curves for K 2 i = 1> t n e n there is no change in the number of elements rising above the stagnation point level for that change S u"t . If (-Ii lies to the left of the curve = 1, then the increase in T„ necessary for an element to rise above the stagnation point level, z , is less than that required to keep the number of elements s constant; that i s , the number of elements rising above z is increased s (thus > 1). If fi/£<0, then an increase in TJ^ of & TJ leads to a greater supply of hot elements and also a lessening of the temperature excess needed by the elements to rise above the stagnation / M point. Thus for £ / f<0 , regardless of the value of — , a larger translation speed is required by the hypothesis i f such a larger speed is possible for the plume. 3.5.3 Effect of Translation Speed on Convergence Downturning In this section, an expression for €'/$ is derived from the dynamic constraints at the upwind edge convergence discussed in section 3.4, relating the value of e/S to the translation speed, TJ . This result combined with the results from the feeding term analysis of the previous section and from a qualitative analysis of the turbulence effects in the next section allows determination of the required translation speed, . U • This section is considered to be more speculative than the previous 105 sections but a relationship between €/£ and U is necessary to close the problem and to determine the required translation speed. In section 3.4 i t was shown that the components of velocity normal to the boundary particle motions at the upwind edge convergence must be equal and opposite on either side of the boundary. See again Fig. 32. From that requirement equation (3.6) was developed, tan G - + Up (3.6) where subscript "p" refers to velocities inside the plume and "e" to velocities outside the plume. Consider now these velocities at a distance 3. normal to the boundary at the stagnation point, where the depth of the upwind edge convergence zone is taken as A , (see Fig. 35). The height inside the plume at the edge of the convergence zone at a distance yl from the stagnation point will be called z^. The subscript "c" will be used for values pertaining to that level. Thus Wp' (zt) = Wc (3.35A) - Up (Zc) = l/ c = Ut- U, (3.35B) where U is the downwind speed of the entrained elements. The negative i sign appears in -TJ (z ) because TJ is taken as positive in the p c c upstream direction. Since is taken at the edge of the convergence zone then the entrained elements up to this point have experienced no longitudinal acceleration; hence (3.35B) follows immediately. At the edge of the convergence zone outside the plume, the vertical velocity is assumed to be very small compared to the other velocities in (3.6) and so H =r O (3.35C) i • FIG. 35 : PLUME MODEL FOR CALCULATING CONVERGENCE DOWNTURNING / /normal to the boundary '\~) coinciding with the direction 7 of element motion at z z (level of stagnation point) s / / / / / s edge of upwind edge convergence zone o O N 107 The longitudinal velocity Uc is approximated as simply the mean wind speed at the height z + (z - z ) minus the speed of the . o S C boundary particle at the stagnation point, U, (z ) (i.e. the translation b s speed, Uj.)* Hence If we write l/t = Ub(zs) 5 U(ZS) - C,(ZS) (3.36) where C0 (z ) is defined by this equation, then s <4' +  X' in9fzj + (3.35D) C0 (z ) is the difference between the longitudinal velocity s of the parcel on the boundary at the stagnation point,U, (z ), and the b s mean wind at that level. Since the. pressure field associated with the upwind convergence accelerates the air in the plume, then the change of C» (z ) for a change in the translation speed U (and hence a change s t in z ) is assumed to be small compared to the change in TJ itself, s t For an element to approach the stagnation point, its direction of motion at the edge of the convergence zone at Z = ZQ must have no component parallel to the boundary at the point where the vector of the element motion intersects the boundary, (see Fig. 35) i.e. fc = 0(7.,) (3.37) where £c is the angle between the horizontal and the direction of element motion at z = zc i n a frame of rererence moving with the plume. Hence fan H = ~ <3-38> 14 (Recall that U was taken as positive in the upstream direction), c 108 The substitutions of the velocities (3.35) into (3.6) leads to with the use of (3.37) Un .Sfc, - hnft - **L ZL, + (3.39) (3.40) sin £ We shall now consider two translation speeds U and U 2 1 where as in (2.10) = ^ O + S) (3.41) where £ is small and positive. If subscript "2 " refers to values corresponding to U and "1" to TJ , then equation (3.40) leads to i ,J?I rZ tl Vc> -tan ht + Ifr. - UL - 1 ^ ZH L sin h. Writing - ^ 4 1 o-j where Ot is small, then equation (3.42) reduces to (3.42) (3.43) (3.44) Hence from the dynamic constraints at the upwind edge convergence, the velocities at the edge of the convergence zone inside.the plume (at z = z ) normal to the boundary from the stagnation point can be 109 related, for translation speeds differing by a small increment. Parameterization of W: and TJ in Terms of & and 6 ~~ • xzi c 2  The velocities W and U shall now be written in terms of C2 C2 W and U and of & and £ where as previously defined c c where T^  is the temperature an entrained element must have to approach the stagnation point, (i) U : From the expression for Uc in (3.35), i t is seen that UCi = tic, + (3.45) where (ii) W : c2 W t « WL + y - T* T (3.46) where is the i n i t i a l vertical velocity of the element at the level of entrainment at the base of the plume, z^,and f is the time for the element to rise from the entrainment level,z^ , to the edge of the convergence at z^. Hence: c 2 T In order to express W in terms of W expressions for W. and z are c2 C l 2 c2 needed. The effect of W. on W is very small and so rough estimates will i c be used to evaluate , is assumed to have form similar to (3.47). A temperature excess of 2C° and a distance to rise to the entrainment level of 1.5 m leads to W £ 2 * ' W t , l l * * * * ) (3.48) An expression for z will now be developed. Since C0(z ) 2^ s has been assumed to vary slowly compared to U (z) then which becomes Z * 2 - " ( 3 - 4 9 ) This is the change in stagnation point level. From Fig. 35. ZC t « Z<r, - Zst-2$< " ^V* , d l"^» 5 i" *J ( 3* 5 0 ) The value of sin© does not change much for a change in the translation speed, U . If a change in U leads to a change in W and t t c hence in (tan Q (z )) of say 10%, then the change in (sin Q (z )) s s is only about 1% for 6(zJ= 6 4 ° , (the measured average t i l t , & , between 3.5 m and 48 m) If an increase in and hence the stagnation point level zg , leads to an increase in ^  as shown in Fig. 35, then we shall assume that the change in 2- can be written. * z - a, = cL ( z S t - z S l ) ( 3 5 1 ) where is a constant. Thus equation (3.50) becomes ZCt- Z', • (2^'Zs,)( C L  s l «  6 ( Z S > ) (3.52) I l l Assuming z also increases for an increase in z then c s CLSii>e(Zs) <l. With the use of equation (3.49) and (3.52), Using expressions (3.47), (3.48) and (3.53), then an expression for W can be written, C2 where Wc 2 « U/C| + X2£ * (3.54) #2 - 0 2 W t + 7 - (3.55) and f t l^Q- ^ Sin OCZs)) Xj = 1 ^ -i± (3.56) Note that X3 represents the increase in W due to the element having C2 further to rise and so more time for the buoyant acceleration to act. X2 represents the increase in W due to the temperature of the c2 element being larger (if £ positive) causing a larger i n i t i a l vertical velocity and a larger buoyancy term. Note also that ) f = Wc, so that a relative change of € in T0( causes a relative change in W^  of €/g Development of the Equation in The expressions for W and U can now be substituted into equation c 2 c 2 (3.44) which was derived from the dynamic constraints at the upwind edge convergence. To first order in S, € and c( , equation (3.44) becomes: 112 (3.57) For "typical values" W± = 0.4 m/sec , U = 2m/sec , T,' - 2 C° , U « 10 m/sec , t l then equation (3.57) can be simplified by some first order expansions using: We* « Vc] f Wt « J  To, Equation (3.57) after some algebra becomes: (3.58) + 2 « where the subscript " 1" has been dropped, for clarity. Consider now the form of o( . 0< was defined by equation (3.43) as: „ 317/ -j 1U For a logarithmic wind profile: 22.1 iSi fe 121^ [ Z5,j (3.59) ,4/3T For the free convection case, the last factor would be (z /z ) With S2 S l the use of equations (3.49) and (3.51) for the change in /t and equation (3.59), then (3.43) becomes: A ml . .(,- S-M± \ O.60, Thus 113 >*!z (3.61) where C is the constant defined by (3.51) and where CL sin 6(zs) < ' la With the use of (3.61), (3.58) becomes: / - ~T (3.62) where the square-bracketed factors are the expansion factors, and hence have a value close to 1. From Fig. 35, i t can be seen that Sin 9 CZS) a Z< - Zc (3.63) Thus C sin 9 (z ) is always less than (C z ) / ^ in equation (3.62). jLi S ' XJ S * Consider now the relative sizes of the terms in equation (3.62) for £/$% Assuming a logarithmic wind profile such that and neglecting the expansion terms, equation (3.62) becomes after some algebra (3.64) If (i) the thickness of the convergence zone,^, changes slowly with . height so that C t< 1 where from (3.51) (ii) A is not small compared to z , and (i i i ) TJt is large enough so that ^ f then the positive 114 terms ( l S t and 3 r ) in (3.64) are both less than unity; whereas the negative 2nc* term is close to z / (z - z ). s c p Thus i f conditions (i), (ii) and ( i i i ) are satisfied and i f (iv) z £ 2(z -z ) s r c p then £/£ AO. Recall from the discussion of the feeding term, that €/$ A 0 means that a faster translation speed leads to a greater heat flux above the stagnation point. We shall now consider these conditions more closely. It can be seen that i f condition (iv) is satisfied then so is condition ( i i ) ; hence condition (Li) can be ignored. Condition ( i i i ) , assuming a logarithmic wind profile becomes (i i i ) z ^ 5.5 z s p where as before z g is the stagnation point level and z^ is the level of entrainment of elements into the base of the plume. Physically, the above conditions mean that i f the convergence 2one at the upwind edge of the plume at the stagnation point is reasonably thick (as shown in Fig. 35, so that condition (iv) is satisfied), then a plume moving at a speed close to the mean wind speed at a height about 5 times greater than the level of the base of the plume (satisfying condition ( i i i ) ) , always has a greater heat flux above the stagnation point i f i t moves faster, (&/$ < o). The form of the dependence of % on U based on the above considerations is shown in Fig. 36. This result doss not yet include effects of turbulent mixing which are discussed in the following section. For the use of very light winds, the neglect of the expansion terms in going from equation (3.62) to (3.64) must be reconsidered. In 2 particular, the approximation (W /U ) «1 may not be valid since , FIG. 36 S U G G E S T E D AND *A BEHAVIOR AS A FUNCTION OF TRANSLATION S P E E D T u u p p e r 116 U < U £ TJ . The effect of the expansion terms is to make c t s upper G/& more positive. The effects of turbulent mixing at the plume boundaries w i l l be considered i n the next section. Then the effects of the translation speed on plume feeding, convergence downturning and turbulent mixing w i l l be considered together to determine the required translation speed. 3.5.3 The Effect of the Translation Speed on the Amount of Heat Flux Lost by a Plume Due to Turbulent Mixing. Thus f a r , no consideration has been given to the losses of the plume heat flux due to turbulent mixing and how these are affected by U^ .. It i s reasonable to suppose that most of the heat flux loss of a plume due to turbulent mixing occurs at the upwind edge convergence where the gradient of temperature at the plume boundary i s very sharp. (This w i l l be shown to be the case i n the next chapter.) Since the a i r at the upwind boundary below zg is not going to stay with the plume, turbulent losses below z are of l i t t l e consequence. For a smaller U , z is lower and s n t s so there i s a greater height within the shear zone where turbulent losses at the upwind edge convergence affect the amount of heat flux i n a plume, reaching the top of the surface shear zone. Suppose that including turbulent losses, the temperature excess needed by an element at the entrainment level to rise above the shear zone i s : T / - - T.' t T' '(3.65) where Te* as defined previously is the temperature excess needed by an entrained element to approach the stagnation point and where Tt is the extra temperature needed by the element so that i t remains far enough from the upwind boundary to avoid being mixed with the non-plume a i r at the boundary. Writing T± = 7^ ' (l-fj where J is small and positive since losses are less for the larger U , then, Thus the results from the analyses of the effects of the translation speed, , on the plume feeding and on the convergence downturning which led to relationships shown graphically in Figs. 34 and 36 can now be expanded to include turbulent effects to the top of the shear zone. This is done by replacing £ by € where: €' * € - - i - i ; (3.67) For a plume translation speed, U , close to the speed of the wind near the top of the shear zone,(U ), T < T and € r € . upper t i For a small U /U , T* becomes large and T* / T ' . approaches 1. t upper t ° t A . Thus for small U /U , e's= f and so £'/S < 0 , The suggested ^ upper ' 0 0 form of the behaviour of &'/S as a function of U is shown in Fig. 36. Since for small u t/ u Upp e r> then €'/& is negative even in the case of very light winds for small ^^upper" 3.5.4 Required Translation Speed. The i n i t i a l hypothesis was that the plume translation speed must be that which allows a plume to exist stably and to transport heat upwards through the surface shear zone of the wind at a lower level of thermal instability than that required for a plume moving at any other speed. From Figs. 34 and 36, i t is seen that the analysis based on the model of the plume shown in Fig. 31 leads to the result that a larger value of the translation speed always leads to a greater heat flux and 118 hence to a plume being able to exist at a lower instability level. An upper limit on the permitted value of the translation speed is presumably the speed at which a region- of upwards moving air can advect at the top of the shear zonej this restriction is needed for shape preservation (see section 3.4) In the case of very light winds in which the vertical velocities are larger than the mean wind at the top of the shear zone, then the translation speed may be less than this upper wind speed, but is s t i l l greater than the speed of the entrained elements. Thus, except for very light wind conditions, the translation speed of convective plumes is predicted to be close to the mean wind speed at the top of the surface shear zone. This is what was observed on July 16, 1971 at Suffield. Chapter 4 HORIZONTAL CROSS-SECTIONAL SHAPE OF CONVECTIVE PLUMES In previous studies of the structure of convective plumes in the atmospheric boundary layer, several attempts have been made to determine the shape of the horizontal cross-section of the plumes. Warner and Telford (1963) suggested circular cross-sections on the basis of an examination of the time traces of the temperature signals from aircraft flights at various directions to the mean wind and at heights from about 15 m up to near the cloudbase. However, Lenschow (1970) found a distinct elongation of the plumes in the downwind direction above 100 m again using aircraft based results, over land. Frisch (1973) using aircraft measurements, found l i t t l e evidence os elongation of plumes over grasslands and suggested a stability dependence might be involved with circular plumes occurring only for the condition of free convection. Priestley (1959) suggested that the local "hot spot" needed to form a plume initiall y would exist for some time and so the preferred type of plume would be drawn out in the downwind direction as i t advected along with the wind. Clearly the shape of the horizontal cross-section of the plume has not been established. In this study, the horizontal cross-sectional shape of the temperature field associated with convective plumes at a height of 3.5 m was determined using the temperature signals from an array of sensors. A dynamic explanation is given for the observed elongated form. The data to be discussed is from the same run on July 16, 1971 as was discussed in the previous chapter. 120 4.1 EXPERIMENTAL DETERMINATION OF THE PLUME HORIZONTAL CROSS-SECTIONAL SHAPE: Time-to-Space Transformation: In the previous chapter the statistical values of the coherences and phases between pairs of the five temperature signals of the array at 3.5 m were used to determine the average translation velocity of the convective plumes. For the case of July 16, the translation velocity was found to be 10t2 m/sec from the direction 229° - 6°M. Knowing this translation velocity and the physical separations of the sensors (see Fig. 18), then the time traces of the temperature signals can be appropriately spaced and delayed to effect a time-to-space transformation, with the assumption that the plumes do not change shape significantly in their passage through the array. The average plume passage (see Fig. 30) took about 15 to 20 seconds corresponding, for a translation speed of lOm/sec, to plume lengths between 150 and 200 m. Since the total length of the array used on July 16 was only 73 m, then the above assumption seems reasonable. On this assumption, temperature contours were visually interpolated between appropriately time-shifted simultaneously recorded time traces of the temperature signals. In this way a "snapshot"-like picture of the horizontal cross-section of the temperature field at a height of 3.5 m was made in a frame of reference moving with the plumes, (see Fig. 37). For this procedure to produce meaningful results, the magnitude of the fluctuations due to plumes must be significantly larger than those due to random turbulent effects which may not advect at the same speed as the plume and which do not form a shape-preserving feature in time. As can be seen from Fig. 30, the temperature signal satisfies FIG. 37 100 M ( for U = 10 m/sec ) < U H 122 this criterion much better than the vertical velocity signal. This is why the temperature rather than the vertical velocity was used to outline the plume boundaries. Only four sensor lines are shown in Fig. 37. Sensor #4 was sufficiently displaced crosswind from the other sensors for its temperature signal to be so poorly coherent with the other signals that temperature contours involving i t could not be reliably drawn. Note that information about the temperature field exists only along the sensor lines in Fig. 37; so a l l the contours are visual interpolations. There are uncertainties in the details of the temperature contours in Fig. 37 due mostly to changes in the temperature field during its passage through the array, to deviations of particular plume motions from the average plume motion used for the time-space transformation, and to simplifications in the process of visual interpolation. However, the general features of the plume cross-section are clear. The most obvious feature of the temperature contours is the marked elongation of the plumes in the downwind direction. This had been suspected from the very sharp crosswind fall-off of the coherences between the temperature signals which was found in the previous chapter, If the plume cross-sectional shape is approximated by an ellipse with the major axis in the downstream direction, then the ratio of the major to minor axes (or the downstream to cross-stream diameters) in Fig. 37 is about 8-4. The large uncertainty is due mostly to uncertainty in the cross-stream width (minor axis). For the ratio of the axes to be close to 1, the translation speed of the plume would have to be about 1 or 2 m/sec. compared to the measured average translation velocity of 10-2 m/sec. 123 4.2 DYNAMIC EXPLANATION FOR THE OBSERVED CROSS-SECTIONAL SHAPE: Approach to Problem In the previous section the horizontal cross-sectional shape of convective plumes was found to be very elongated in the direction of the mean wind. In this section a dynamic explanation for this elongated shape is given. Analogously to the hypothesis used in the explanation for the observed translation speed in section 3.5, a hypothesis is made that the plume cross-sectional shape must be that which allows a plume to exist stably and to transport heat upwards through the surface shear zone of the wind at a lower level of thermal instability than that required for a plume of any other shape. A plume scale heat flux equation is developed and applied to a model of a plume having an elliptic horizontal cross-section. The terms which contribute to the heat flux lost by a plume due to turbulent mixing at the plume boundaries are formulated in terms of measurable statistics, and the ratio of the semi-major to semi-minor axes of the plume horizontal cross-section. The ratio required by the hypothesis can be shown to be that ratio which minimizes the heat flux lost by the plume through turbulent mixing. Thus the expression for the heat flux loss is minimized to give the required ratio. The numerical value of the required ratio is estimated by substituting temperature and velocity statistics from the July 16th run and is compared with the ratio measured from the temperature contours in the previous section. Although there are some rather gross simplifications used in the following analysis, this study shows that the observed shape can be explained in terms of a dynamic balance and indicates the parameters controlling the plume shape. 124 Plume Scale Heat Flux Equation Following Lumley and Panofsky (1964, pg. 59 ff.) but using three scales of motion (as was done by Lemone (1972)),rather than two scales, then a plume-scale equation of motion and a heat equation can be derived. These are combined to give a plume-scale heat flux equation which when averaged over plume scales (to be defined below) becomes: where superscript "p" refers to plume scales, "s" to synoptic scales, and "t" to turbulent scales. Overbars denote plume-scale averaging (to be defined below) except when followed by a " t " which denotes turbulent-scale averaging. The x, y, z, components are represented by subscripts 1, 2, 3, respectively. X is t n e thermal diffusivity; yj is the kinematic viscosity. Differentiation is denoted by a comma, L~e.g. = ^^/"bx^ J In the above equation, i t was assumed that the three scales of motion could be separated by appropriate averaging. Plume-scale averaging is taken in a frame of reference moving with the plumes over a period of time (of say, several minutes) short compared to the plume lifetime but long enough for random turbulent effects to average to zero. In their influence on the amount of heat flux lost (or gained) by a plume due to shape changes, the transport terms (the first two terms on the right hand side of equation (4.1)) are considered unimportant; the 125 molecular diffusion terms are negligible compared to the turbulent terms, 2~ and the two source terms Jg/xJ (the buoyancy term from the equation I of motion) and w (a source term i f the mean temperature profile is unstable) are assumed to have no direct dependence on plume shape. This leaves the pressure term and the turbulent interaction terms as those terms in equation (4.1) which contribute to the heat flux lost by a plume and which have a possible dependence on the shape of the plume; these terms will be examined below. Plume Model A model of the plume will be adopted with an elliptic cross-section in the X-Y plane, (see Fig. 38). The ratio of the semi-major to semi-minor axes (or the downstream to cross-stream diameters) of the cross-section is defined as Y and is given by: tf - f . (4.2) where "a" and "b" are the semi-major and semi-minor axes of the horizontal plume cross-section, respectively. The dependence of the amount of heat flux lost by a plume due to interactions represented by the pressure and turbulent terms, on the plume shape, can be found by taking a volume integral of the terms over plume scales. The pressure and turbulent terms are formulated such that only measurable quantities and the axes a and b appear. Minimization of the heat flux lost by the plume, then leads to an equation for V • FIG. 38 PLUME MODEL WITH ELLIPTIC C R O S S - S E C T I O N 127 Form of the Volume Integrals Because of the assumed elliptic cross-section of the plumes in the X-Y plane, the volume integrals are made using an elliptic cylindrical co-ordinate system. Following the notation of Morse and Feshback (1959 pg. 997 ff.) the volume element is h ^ h g d j f d£ dj ? = -(cosh /4 - cos (f) dft d(f dz (4.3) In the X-Y plane this co-ordinate system consists of ellipses and hyper-bolas, ft = constant is an ellipse; the fJ -vector points outwards from.the ellipse. (f = constant is a hyperbola whose asymptote forms an angle with the line between the two foci of the ellipse, (i.e. the major axis or the X-axis in Fig. 38). is a length scale given by 2 ,2 where a and b are the downstream and cross-stream radii defined earlier. The plume boundary is taken to be, - f*^ The turbulent and pressure terms in equation (4.1) will be shown to be of the form F ( ^ c ) G {(f) H (z) where F, G and H are functions of jt, Cf and z respectively. The vertical variation H (z ), will be considered only qualitatively in this study. A l l the terms will be shown to involve JT'/j^' or ^W^/ifx' as the f*- dependence, where f*' is the ft -vector in the X-Y plane along the normal to the ellipse perimeter, d^ t' is the line element h^d£ in (4.3). If the region where plume scale gradients in temperature and vertical velocity are large, is confined to a sufficiently small region near the 2 2 boundary of the plume so that JI cosh JA can be taken as a constant where the gradients are large, see Fig. 39, then (for example) 128 The FIG. 39 APPROXIMATION INVOLVED IN THE LI-INTEGRATIONS Form of .A cosh fA and /2/u' a t t n e Plume Boundary tf*.~fb (boundary of the plume) 129 I 5 fff Ftp) G((f)HCz) A(Mj dft dft Jf3 A) where iJT is the total temperature difference between the inside and the outside of the plume and where fi = fJ^ is the boundary of the plume. Some of the terms have no (f - dependence, ( G((f) = 1 )• For these terms (4.4) reduces to an elliptic integral of the second kind: I « - f HCz)*k A T JX (toskXfJb - COS1?) V ? n—p Since the perimeter of the ellipse is given by la + b 2jr , then V 2 the ^-integration for terms with no tf- dependence is just the distance around the perimeter of the plume cross-section. Several terms have an angular dependence of the form, cos of , where -*t is the angle between the perimeter normal in the X-Y plane and the X-axis, (see again Fig. 38). The relationship between d and Cf can be shown to be {in of - y* fan tf (4. 6 ) where as defined above }f is the ratio of the semi-axes (Y = a/b). Since the terms having the OJ-dependence will be shown to have this dependence on the upwind edge (from £f =-jto + then equation (4.4) with G ( tf) equal to cos«l can be shown to reduce to: I * - (H(z) Mz A T j 1 cos* ((*skxyb - cos1?)'* Jcf = ( MzMz AT 2f (4.7) 130 2 2 In arriving at (4.7), the approximation was made that (b /a ) <*< 1; for the observed b/a ratio, this is a valid approximation. . With the use of equation (4.5) and (4.7) the volume integrals of the pressure and turbulent mixing terms can be solved. Formulation of the Pressure Term From the observations and the analysis discussed in the previous chapter in section 3.4 a region of convergence was found to exist at the upwind edge of the plume due to the vertical shear of the mean wind. In this section the dynamic pressure due to this convergence is estimated. By integrating across the boundary at the upwind edge the heat flux lost by the plume due to the pressure-temperature interaction can be estimated. The pressure gradient at the upwind edge convergence is assumed to be normal to the boundary along r in the ^i' -z plane. The assumed form of the term is shown in Fig. 40. Hence, p P sin 0 « where fx is the f*-vector in the X-Y plane along the normal to the ellipse perimeter; yu'is associated with the line element, h^d ^( , in (4.3). 0 is the downwind t i l t . (In the presence of a crosswind t i l t , then, & represents only the downwind component of t i l t , where "downwind" at any height is defined with reference to the wind direction at that height.) Contributions to the pressure term are assumed important only at the up-wind edge convergence (i.e. for \(f\ < TT /2 ). It is assumed that the dynamic pressure at the upwind convergence can be represented by P? s £ UZ <4-9> 2 Sin  p f (4.8) v FIG. 40 THE ASSUMED FORM OF THE P R E S S U R E T E R M : T p P^ DEFINING THE r-DIRECTION z 132 where from the t i l t relationships in equation (3.6): TJ = (u*ce - U^ ) cos© outside the plume at the upwind edgef(o(so) and TJ = . (W - w ' ) sin© inside the plume, c p where \ ] ^ is the speed of the wind outside the plume beyond convergence effects, and TJ^ ' are as defined previously in conjunction with equation (3.6) and Wc is the vertical velocity inside the plume at the edge of the convergence zone. The component of (U - U*) cos0 that is ce e normal to the surface for of+O is (U - U /)cos 9 cos o(. ce e Thus, i*3 2 Cos 0 outside the plume >3 2 P/K inside the plume The volume integral of the pressure term on the inside of the plume from f*'f*x to f* * fty (where f*x is a value of jt inside the plume, where the pressure term is small, and p is the plume boundary) becomes: n {A. 12) If from equation (3.6) the t i l t at the upwind edge is approximated by Un 9m Wpm where subscript "m" refers to values at the upwind edge, and i f the effect of the -dependence of the t i l t over the range (- ~ to ) 133 is limited to the change in the component of Ue normal to the boundary, then tan 0 » i a n Om cos « (4.13) Since most of the contribution to the integral will be in the region where the convergence is strong, and hence where the gradient P is large, then the value of T for the integration inside the plume will be taken as a constant of AT/2 , where 4T is the total change of temperature from inside to outside the plume. Assuming the dynamic pressure is symmetric across the boundary, then the contribution from outside the plume is similar to that from inside the plume. (If the division of T^of 4T/2 inside the plume and - 4 T/2 outside the plume is incorrect, then this just changes the distribution of the pressure term between inside and outside the plume, but not the total magnitude). With these considerations the volume integral of the pressure term over a layer of thickness h after some algebra becomes: f Z Y ^ v o W ) A h J^fl AT(Awfbl ( 4.I 4 )  111 f '3 cos Bm a where A W is the difference between the maximum value of yP inside the plume and the value outside the plume, and where i t has been assumed (in the evaluation of a sin l d factor) that ^,^45° The effect of this approximation is to make the pressure term too large, the effect increasing as the pressure term becomes small. For the B value of 80° as measured on July 16 at 3.5 m, the error in this approximation is negligible. It is noted that the pressure term has a stronger dependence on 134 the semi-minor (cross-stream) axis b, than the semi-major (downstream) axis, a. Thus a narrower plume diminishes the amount of heat flux lost to the plume through the pressure-temperature interaction at the upwind edge. Formulation of the Turbulent Terms: The heat flux lost by a plume due to pressure-temperature 2 interactions was found to directly depend upon b /a. Some of the heat flux losses due to the turbulent terms have a similar dependence; other parts of the turbulent interactions do not have an angular dependence. The ratio = ^  obtained by minimization of the heat flux b lost by the plume will involve a balance between these two sets of terms. Consider first the term describing the turbulent interaction with the plume temperature, % j For the case of incompressibility (as used in the development leading to the equation (4.1)), this can be written: where the overbar for plume scale averaging has been omitted for clarity. This term is now clearly seen as the interaction between the vertical component of the Reynolds stress and the plume temperature gradient. As for the case of the pressure term, the gradient is assumed to be normal to the boundary along "r" in the ^t! -z plane. Hence: 135 s-w? ^ - ' J v ( 4-1 5 ) where 0 is the downwind t i l t of the surface. Before a volume integral can be taken, the turbulent scales have to be estimated. The term t/* can be evaluated from the integral under the vertical velocity spectrum from high frequencies to some lower frequency limit which is s t i l l above the plume scales, i.e. (4.16) turbulent scales l~1 T fc| The term ^  J can be divided into a mean wind shear contribution (call i t ^ TJ ^  ^ ), and a contribution due to the W*5 gradient across the boundary (call i t (u c u fc t ) ) \ j * 3 3 ^ e w^ n <^ s n e a r effect will be important only on the surfaces normal to the wind (since the VW correlation is very small) and hence will depend roughly on cospf as does the pressure /— rt\ term; rough evaluations show that I U~ n / is over an order of magnitude smaller than the pressure term and so will be neglected. However, since from the time traces, the angular dependence of the gradient at the boundary around the plume appears small, then this term will be assumed to have no ^-dependence. Thus its dependence on a and b will be different than the pressure term and so this term cannot be neglected even though i t is of the same order of magnitude as \ U ^  J \U ^  U_ can be roughly estimated as follows. The TJ t J 3 p average value due to the W gradient across the plume boundary is roughly the turbulent mixing length (approximately equal to the height, z ) multiplied by the gradient; the average value of u fc can be estimated from the integral of the velocity spectrum for turbulent scales. Thus * - ( £ • « ) • ! / ^ 3 *• turbulent J scales The expression for the required ratio of the semi-major to semi-minor axes will turn out to be rather insensitive to errors in the above expression or its evaluation. With the above considerations, the volume integral over a layer of thickness, h, of the turbulent term involving becomes (to the same approximations as used earlier): -i T? Uj* tf,* d (volume) J (4.18) '3 The turbulent term involving the plume scale vertical velocity: L a  r 3 v 2 W^  T^ *" TJ ' can be treated in an analogous way. The turbulent parts can be considered to have contributions from the overall turbulent correlations and from the effects of the plume scale gradient in . The effects when roughly evaluated are a l l very much smaller than previously considered effects except for one term. This is the term analagous to the one in equation (4.17), and is of a similar form for \(f J > •?• • JI 137 (4.19) turbulent scales where the subscript q refers to the part of the term arising from the existence of the temperature gradient at the plume boundary / — - t ) analagously to the term ^ J discussed above (4.17), except that (4.19) is applicable only for \C?\>f • Hence the turbulent term has a volume integral, expression with the same form of fl"!" W f T,f U. 1 V(volume) £ dependence on a and b as (4.14) but much smaller Determination of the Required Ratio of Diameters of the Plume Cross- Section In the previous two sections the plume scale heat flux losses due to the pressure and turbulent mixing terms were formulated in terms of measurable quantities and of "a" and "b", the downstream and cross-stream semi-axes of the plume cross-section. These terms were assumed to be the only, terms in the plume scale heat flux equation which contributed to the heat flux lost by a plume and which had a dependence on the shape of the plume. The value of the ratio, (given byV = a/b), which minimizes the heat flux lost by the plume due to turbulent mixing at the plume boundaries will be the ratio that will permit a.plume to exist at the lowest possible level of instability. A plume with a different ratio, V , will have a greater heat flux loss. So there will be an instability level at which a plume with the optimum ratio can exist stably; 138 whereas a plume with any other ratio of diameters cannot exist stably. Thus the hypothesis at the start of this section is equivalent to the requirement for a value of which minimizes the heat flux lost by the plume. The sum of the volume integrals of the pressure and turbulent terms can be written, Iff (j i * T / Y n \ * * *" u*') * ( v o l u m e ) - h Sin3 a cos e. m If estimates are made for each term from the data of July 16 then the right hand side (R.H.S.) of equation (4.21) reduces to 2 .2 I c R.H.S. = h |j.0.2 - +7.0- + 0.44 J a2+ b 2 ' + 0.17 J a 2 + b2^J (4,22) where the terms are in the same order as in equation (4.21). Hence i t is seen that the pressure term and the mixing of temperature through the upwind edge are very large and depend more strongly upon the width of the plume, (i.e. on the cross-section of the plume exposed to the wind), rather than upon its length. The 139 turbulent mixing terms without angular dependence coming from the plume scale gradients around the whole perimeter of the plume are roughly an order of magnitude smaller. Defining Sin'0m A T (AW) Cz = 2 A T Un em Uj' CH = A W ir (T* llf )j cos em (4.23) Z A T and since A = "77 ab where A is the cross-sectional area of an ellipse with semi-axes a and b Then writing fe - fl (4.24) 4' Y * then the R .H .S . of equation (4.21) becomes R.H.S. of (4.21) = (4.25) The above expression represents the heat flux lost by a convective plume due to the pressure and turbulent terms parameterized in terms of V, To minimize the heat flux loss, expression (4.25) was differentiated with respect to tf and set to zero, After some algebra, the following equation for V results ) X n 2 (4.26) c3 / For the July 16 data, the R.H.S. of (4.26) was estimated to be about J60' , which leads to a value for V of 8. This value for V , the ratio of the semi-major to seni-minor axes of the plume horizontal cross-section-, agrees very well with the observed ratio, 8 - 4 in section 4.1. Discussion of the Plume Cross-Sectional Shape Although there are many rather gross assumptions and approximations in the development of equation (4.26), i t is seen that a minimization of the heat flux lost by a plume due to turbulent and.pressure-induced mixing at the plume boundaries leads to a predicted value of the ratio of the semi-axes of the plume cross-section very close to the observed ratio, The upwind edge mixing due to the presence of the vertical shear of the mean wind is much larger than that due to the part of the turbulent mixing that has l i t t l e angular dependence. This is the physical basis for the elongated form of the plume cross-section. At higher levels where the vertical shear of the mean wind is much less, the shape which would minimize heat flux loss would be much less elongated in the downwind direction than closer to the surface where the wind shear is larger. However, if the plume cross-section, rising up to a level where the wind shear is smaller, already has an elongated form due to the wind shear at lower levels, then this shape would be 141 expected to persist for some time. In the cases of very light winds, the elongation would be decreased due to the decreased wind shear. The fact that Lenschow's (1970) observations of elongated plumes were taken under conditions of strong winds (U at 100 m was 9.1 m/sec) and that Warner and Telford's (1963) observations over the grasslands of no plume elongation were made under conditions of "light winds", might explain the apparent discrepancey between their observations. CONCLUSIONS The horizontal and vertical variability of the turbulent statistics of the temperature and vertical velocity from 3.5 m to 610 m under convective conditions were measured with an array of towers and an aircraft. The overall heat flux was found to be roughly constant to about 100 m. and then to decrease gradually with height to 600 m. Extrapolations beyond 600 m of the best-fitting linear decreases of heat flux with height for the two days examined implied that the heat flux was zero at heights (1.7 km and 1.3 km) that were not dissimilar to typical inversion level heights (about 2 km). A linear decrease of heat flux with height is a much slower rate of decrease than that found by Lenschow (1970) but agrees with results by Volkov et al (1968). The temperature variance was found to have a height dependence to -2/3 600 m very close to the z predicted by dimensional arguments for free convection in the surface constant-flux layer. 2 The vertical velocity variance (0"w ) n a s a z-dependence to at 2/3 least 150 m at the z rate predicted by dimensional arguments for free convection in the surface constant-flux layer. From 150 m to 600 m, the vertical velocity variance increased more slowly or remained 2 approximately constant. This height at which ( j ^ falls off from the 2/3 z rate of increase agrees with the height of the falloff predicted by Deardorff's (1970) model of the PBL. The momentum flux u'w'is very hard to measure reliably, particularly above the very near-surface layer. By 48 m, there was a large degree of intermittency which produced very large poorly-defined 143 low wavenumber estimates of u'w'. According to the Monin-Obukhov similarity theory, the height and stability dependence of any mean turbulent characteristic is determined by a universal function of z/L, where L is the Moniri-Obukhov length, an indicator of stability. In this study the validity of this theory as to its height dependence was tested to a height of 610 m. The "theoretical" universal function of z/L adopted for the -5/3 temperature spectral levels in the k region, neglected the effects of the pressure term and of the vertical divergences of the vertical transports, and used the Keyps formulation of the wind temperature profiles with K^ /K^  = 1.0. The observed z-dependence of the temperature -5/3 spectral levels in the k region agreed with this theoretical function of z/L to a height of 300 m. Thus, unlike the constant flux layer, the assumption Ri = z/L used in the Businger-Dyer formulation of the wind and temperature profiles is not applicable at higher levels. At large x -4/3 z, the observed height dependence is ( J^^z The value of the Kolmogorov constant, |3 , which gave the best f i t between the observed temperature spectral levels up to 300 m and those predicted by the universal function of z/L was ^(K^/K^) = 0.25. For Boston's (1970) value of (3 = 0.8 the value for ^ / I ^ is 0.31. For P = 0.4, -(following the results of Paquin and Pond (1971) and Wyngaard and Cote (1971)), ^ / l ^ =0.63. The "theoretical" universal function of z/L adopted for the vertical -5/3 velocity spectral levels in the k region also neglected the effects of the pressure term and of the vertical divergence of the vertical transport of kinetic energy and used the Keyps formulation for the wind profile. The observed z-dependence of the high frequency spectral 144 -5/3 levels (in the k region) of the vertical velocity spectra from 38 m to 610 m was also found to be very close to the z-dependence predicted by the empirically-based universal function of z/L adopted. The value of the Kolmogorov constant Ofj , ( = 3/4 CX )^* which gave the best f i t between the observed vertical velocity spectral levels from 38 m to 610 m and the levels predicted by the adopted universal function of z/L was Of^  = 0.46. At a height of 3.5 m, the turbulent statistics for vertical velocity and temperature were the same to within 20% for spatial separations of 70 m both along and across the wind. Longitudinal rolls with axes downwind and with a cross-wind wave-length of 1.7 km were found for the case of July 16th, from the directional differences in the spectral peaks of the temperature at 61 m. For this case, the heat flux estimated from an Eulerian measurement at 92 m on the tower time-averaged for 80 minutes was found to be a factor of 3 less than the three spatially-averaged heat flux estimates from aircraft flights in different directions at the same height. The translation velocity of convective plumes was found from the low wavenumber statistics of the temperature field measured by a 2-dimensional array of sensors at 3.5 m. The direction of plume motion was found to be close to the wind direction at the surface. The translation speed on the July 16 case was found to be 10 - 2 m/sec which is very close to the speed of the wind near the top of the surface shear zone (U / 0 = 9.9(m/sec): U n o = 10.1 (m/sec)); the observed translation speed 4om yzm is distinctly larger than the mean wind speed at the array level (U„ c = 5.5 m/sec). This result is different from some previous estimates 3.->m of the translation speed found with different techniques. The observed temperature and velocity fields associated with convective plumes were shown to be dynamically consistent. A plume translation speed close to the mean wind speed near the top of the surface shear zone (as was observed) was shown to allow plumes to exist at a lower thermal instability level than any other possible translation speed. The shape of the horizontal cross-section of the temperature field associated with a plume was observed to be elongated in the downwind direction, at a height of 3.5m, with a ratio of downstream to cross-stream diameters for the July 16 case of 8:1. This elongated shape was shown to allow the plume to exist at a lower level bf thermal instability than any other shape. 146 BIBLIOGRAPHY Asai, T. (1970). Stability of a plane parallel flow with variable shear and unstable stratification. J.M.S. of Japan, 48, p. 129. Blackadar, A.K. and H. Tennekes, (1968). Asymptotic similarity in neutral barotropic planetary boundary layers. J.A.S., 25, p. 1015. Boston, N.E.J. (1970). An investigation of high wavenumber temperature and velocity spectra in air. Ph.D. Thesis, University of British Columbia. Brown, R.A. (1970). A secondary flow model for the planetary boundary layer. J.A.S., 27, p. 742. Businger, J.A., J.C. Wyngaard, Y. Izumi and E.F. Bradley (1971). Flux profile relationships in the atmospheric surface layer. J.A.S. 28, p. 181. Deardorff, J.W. (1972). Numerical investigation of neutral and unstable planetary boundary layers. J.A.S. 29, p. 91. Donelan, M.A. (1970). An airborne investigation of the structure of the atmospheric boundary layer over the tropical ocean. Ph.D. Thesis, University of British Columbia. Donelan, M. and M. Miyake (1972). Spectra and fluxes in the boundary layer of the trade wind zone. J.A.S. (submitted) Dyer, A.J. and B.B. Hicks, (1972). The spatial variability of eddy fluxes in the constant flux layer. Q.J.R.M.S., 98, p. 206. Ekman, V.W. (1905). On the influence of the earth's rotation on ocean currents. Kongliga Svenska Venlenskaps Akademien Arkiv for Matematik, Astron, och fysik, 2/11). Ellison, T.H. (1957). Turbulent transport of heat and momentum from an infinite rough plane. J.F.M. , 2^, p. 456. Faller, A.J. and R.E. Kaylor (1966). A numerical study of the instability of the laminar Ekman boundary layer. J.A.S., 23^  p. 466. Frisch, A.S. (1970). A study of convective elements in the atmospheric boundary layer. Ph.D. Thesis, University of Washington. Hanna, S.R. (1969). The formation of longitudinal sand dunes by large helical eddies in the atmosphere. J.A.M., J3, p. 874. 147 Haugen, D.A., J.C. Kaimal and E.F. Bradley, (1971). An experimental study of Reynolds stress and heat flux in the atmosphere surface layer. Q.J.R.M.S., 97, p. 168. Kaimal, J.C. and J.A. Businger (1970). Case studies of a convective plume and a dust devil. J.A.M., j), p. 612. Kaimal, J.C, J.C. Wyngaard, Y. Izumi and O.R. Cote. (1972). Spectral characteristics of surface-layer turbulence. Q.J.R.M.S., 98, p. 563. . . Kukharets, V.P., and L.R. Tsvang (1968). ^Spectra of the turbulent heat flux in the atmospheric boundary layer. Isv. Atm. and Oceanic. -. ••; Phys. Vol. 5, No. 11, p. 1132. Lemone, M.A. (1972). The structure and dynamics of horizontal r o l l vortices in the planetary boundary layer. Ph.D. Thesis, University of Washington. Lenschow, D.H. (1970). Airplane measurements of planetary boundary layer structure. J.A.M. 9_, p. 874. Li l l y , D. (1966). On the instability of Ekman boundary flow. J.A.S., 23, p. 481. Lumley, J.L. and H.A. Panofsky (1964). The Structure of Atmospheric  Turbulence. Interscience Publishers, New York. Mather, G.K., (1967). The NAE T-33 turbulence research aircraft. Report LR-474, National Aeronautical Establishment, National Research Council, Ottawa, Canada. McBean, G.A., R.W. Stewart and M. Miyake (1971). The turbulent energy of budget near the surface. J.G.R., 76, p. 6540. McDonald, J.W., (1972). Fluxatron and sonic anemometer measurements of momentum flux at a height of 4 meters in the atmospheric boundary layer. M.Sc. Thesis, University of British Columbia. Mitsuta, Y., M. Miyake and Y. Kobori (1967). Three dimensional sonic anemometer-thermometer for atmospheric turbulence measurement. Kyoto University Internal Report; 10 pp. Miyake, M. (1961). Transformation of the atmospheric boundary layer over inhomogeneous surfaces. M.Sc. Thesis, Department of Atmospheric Sciences, University of Washington. Monin, A.S. and A.M. Obukhov (1954). Basic regularity in turbulent mixing in the surface layer of the atmosphere. Trudy Geophys. Inst. ANSSSR, No. 24, p. 163. Monin, A.S. and A.M. Yaglom (1971). Statistical Fluid Mechanics: Mechanics of Turbulence. MIT Press, Cambridge, Mass. 148 Mordukhovich, M.I. and L.R. Tsvang (1966). Direct Measurement of turbulent flows at two heights in the atmospheric ground layer. Isv. Atm. and  Oceanic Phys. , Vol.. 2, p. 786. Morse, P.M. and H. Feshbach (1953). Methods of Theoretical Physics, • McGraw-Hill, New York. Myrup, L.O., (1967). Temperature and vertical velocity fluctuations in strong convection. Q.J.R.M.S. , 93_, p. 350. Panofsky, H. (1961). An alternative derivation of the diabetic wind profile. Q.J.R.M.S., 87, p. 109. Panofsky, H.A. (1962). Scale analysis of atmospheric turbulence at 2 m. Q.J.R.M.S., 88, p. 57. Paquin, J.E. and S. Pond (1971). The determination of the Kolmogoroff constants for velocity, temperature and humidity fluctuations from second- and third- order structure functions. J.F.M., 50, p. 257. Pasquill, F. (1972). Some aspects of boundary layer description. Q.J.R.M.S., 98, p. 469. Paulson, CA. , (1967). Profiles of wind speed, temperature and humidity over the sea. Ph.D. Thesis, University of Washington. Pielke, R.A. and H.A. Panofsky, (1970). Turbulence characteristics along several towers. B.L.M., 1, p. 115. Priestley, C.H.B., (1959). Turbulent Transfer in the Lower Atmosphere. University of Chicago Press, Chicago. Taylor, R.J., (1958). Thermal structures in the lowest layers of the atmosphere. Austral. J. of Phys. 11, p. 168. Volkov, YU. A., V.P. Kukharets and L.R. Tsvang (1968). Turbulence in the atmospheric boundary layer above steppe and sea surfaces. Isv. Atm. and Oceanic Phys. Vol. 4, No. 10, p. 1026. Warner, J. and J.W. Telford (1963). Some patterns of convection in the lower atmosphere. J.A.S., 20, p. 313. Wyngaard, J.C. and O.R. Cote (1971). The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer. J.A.S. 28, p. 190. Yamamoto, G. (1959). Theory of turbulent transfer in non-neutral conditions. J.M.S. Jap. , 37, p. 60. LIST OF SYMBOLS 149 semi-major axis of plume horizontal cross-section (Fig. 38). semi-minor axis of plume horizontal cross-section (Fig. 38). drag coefficient, C D = (u^ /U) specific heat of air at constant pressure remainder term in the expression for the dissipation of Kinetic energy (2.5) normalized form of D (2.13) remainder term in expression for thermal dissipation (2.6) frequency in hz Earth's gravity wavenumber k = 2TT f/fj turbulent thermal diffusion coefficient turbulent diffusion coefficient of kinetic energy ratio of M value corresponding to U to that for U t2 t l Monin-Obukhov length (defined in Introduction) number of elements entrained through the plume base in time X which have T' > T E ' direction in degrees magnetic number of elements that the plume passes over in time T probability density distribution for the temperature fluctuations of the elements at the entrainment level, z 150 heat flux, q = c^p W ' T ' log of the ratio of the observed to theoretical spectral levels (2.3) correlation coefficient between longitudinal and vertical velocities (Table II) correlation coefficient between vertical velocity and temperature (Table I) temperature temperature excess of the elements at the plume entrainment level, z P value of T' of an entrained element which will approach the plume stagnation point mean wind at the height z translation speed of the plumes 2 — friction velocity, u^ = - u'w' height inside the plume at the edge of the convergence zone opposite the stagnation point (Fig. 35) entrainment height, the height at which elements are entrained into the base of the plume (Fig. 35) stagnation point level in a plume (Fig. 35) Chapter 3: increment in 2. ?U/3z corresponding to S (3.43) Chapter 4: angle between normal to perimeter of plume horizontal cross-section and the major axis (Fig. 38) i*"* 1 component of Kolmogorov constant for velocity spectra (2.1) ! 1 5 1 p Kolmogorov constant for temperature (2.2) tf ratio of the semi-major to semi-minor axis of the plume cross-section (4.2) $ arbitrary increment in the plume translation speed (3.10) £ Chapter 2: dissipation of kinetic energy (2.1) Chapter 3: increment in corresponding to & (3.15) £* Chapter 3: value of £ including turbulent effects (3.67) i f 9 Chapter 2: thermal dissipation (of ) Q Chapter 2: potential temperature Chapter 3 and 4: downwind t i l t of the upwind edge of the plume. X von Karman's constant =0.4 X Chapter 3: depth of convergence zone at the stagnation point (Fig. 35) Chapter 4: length scale in the elliptic cylindrical co-ordinate system (Fig. 38) po average density <|>m non-dimensional wind shear (2.7) <^  non-dimensional potential temperature shear (2.8) subscript "b" values pertaining to the upwind boundary of the plume subscript "c" values pertaining to z: .• subscript "e" velocities outside the plume measured relative to particles on the upwind edge of the plume subscript " i " i n i t i a l velocities of the entrained elements subscript "p" velocities inside the plume measured relative to particles on the upwind edge of the plume subscript "3.5" experimental values at 3.5m on the main tower subscript "48" experimental values at 48m on the main tower subscript "92" experimental values at 92m on the main tower 

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