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Turbulence, diffusion and the daytime mixed layer depth over a coastal city Steyn, Douw Gerbrand 1980

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til TURBULENCE, DIFFUSION AND THE DAYTIME MIXED LAYER DEPTH OVER A COASTAL CITY by DOUW GERBRAND STEYN B.Sc, University of Cape Town, 1967 B.Sc. (Hons.), University of Cape Town, 1968 M.Sc, University of Cape Town, 1970 * A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1980 © Douw Gerbrand Steyn, 1980 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The rate of dispersion of atmospheric pollutants and the volume of atmosphere available for the di lut ion of pollutants are examined in an unstable suburban atmosphere at a coastal location. Within the framework of the stat is t ica l theory of di f fusion, i t can be shown that the non-dimensional dispersion functions a y / ° v t a n c ' a z / a w t can be determined by integration of the Eulerian spectral functions multiplied by appropriately scaled sampling functions. This scaling, which arises out of the Hay-Pasquill form for the Eulerian-Lagrangian transform and the use of a non-dimensional frequency, gives rise to a dispersion scaling time t = z/o^ which is simply related to the Lagrangian integral time scale. Applying this analysis to turbulent velocity spectra measured over a selected suburban surface results in the following forms for the crosswind and vertical dispersion functions respectively. The spectra and integral turbulence stat is t ics determined in this part of the study are shown to be in general agreement with those determined over much smoother surfaces. The volume of atmosphere available for the di lut ion of pollutants is controlled primarily by the mean wind speed and mixed-layer depth. This lat ter variable can be modelled on the basis of a simple thermodynamic analysis of the mixed layer processes. The currently available models have been generalized to include advection and subsidence. The effects of S ( t * ) = (1.0 + 1.21/t*) - l i i i advection on the mixed-layer depth have been modelled by resetting the model equations in a Lagrangian frame, performing an approximate f i r s t integral in order to derive the spatial dependence of the model variables, and using these spatial forms to give a set of Eulerian equations. The effects of subsidence have been modelled by imposing a subsidence velocity on the top of the mixed layer as well as allowing subsidence-induced warming of the atmosphere above that layer. This subsidence is driven by atmospheric divergence at both synoptic- and meso-scales, the lat ter phenomenon being linked to thermally driven circulatory systems. The inclusion of these processes into the mixed-layer depth model allows i ts application to areas in which meso-scale phenomena may have a consid-erable effect on the diurnal behaviour of the mixed-layer depth. The model thus derived consists of a system of non-linear differen-t i a l equations which may be numerically solved to elucidate the temporal behaviour of the mixed-layer depth. The boundary conditions necessary for such a solution were provided by measurements made in the unstable surface layer over a coastal c i ty . The resultant mixed-layer depth behaviour is in general in good agreement with determinations of this depth made with an acoustic sounder, but can be a poor reflection of real i ty in the presence of synoptic-scale non-stationarities. The input requirements of the model are hourly values of surface sensible heat f lux, mean wind speed and upwind distance to the surface giving rise to the advected heat f lux (usually a coastline or urban-rural boundary), and estimates of the intensity of the capping inversion and horizontal divergence. The model is sensitive to al l input variables, the degree of sensit iv i ty being indicated by the dependence of the maximum mixed-layer depth on the measured boundary conditions. iv TABLE OF CONTENTS Page Abstract i i Table of Contents iv List of Tables v i i List of Figures v i i i Acknowledgements x i i 1. Preface 1 1.1 Rationale 1 1.2 Objectives 1 Part One: Turbulent Diffusion 3 . 2. Introduction 3 2.1 Specification of Diffusion Parameters 4 2.2 Gaussian Plume Model Parameters and the Stat ist ical Theory of Diffusion 6 3. An Application of the Stat ist ical Theory of Diffusion to Measured Flows 10 3.1 The Dispersion Functions from Measured Eulerian Spectra 10 3.1.1 Eulerian-Lagrangian Transformations 10 3.1.2 Transformation of the Dispersion Integral 12 3.2 Eulerian Turbulence Functions over a Suburban Surface in Unstable Conditions 13 3.2.1 Integral Statist ics 13 3.2.2 Spectra 22 3.3 Computation of the Dispersion Functions 25 3.3.1 Computational Details 25 V 3.3.2 Crosswind Spread 26 3.3.3 Vertical Spread 33 4. Conclusion 36 Part Two: The Depth of the Daytime Mixed Layer 5. Introduction 39 5.1 Specification of the Mixed Layer Depth 39 5.2 Mathematical Modelling of the Mixed Layer Depth 41 6. A Model of Mixed Layer Depth 44 6.1 Characteristics of the Observed Mixed Layer 44 6.2 Advection and Subsidence in the Mixed Layer Model 48 6.3 Subsidence 51 6.3.1 Synoptic Scale Subsidence 51 6.3.2 Meso-Scale Subsidence 56 7. Implementation of the Mixed Layer Model 59 7.1 Computational Scheme and Input Data 59 7.2 Results of Mixed Layer Modelling 62 7.3 Sensitivity Analysis 76 8. Conclusion 82 9. Summary of Conclusions 84 List of Symbols 87 References 91 Appendices 105 A. The Observational Site 106 A.l General Requirements 106 A.2 The Selected Site 107 A.3 Sectorial Roughness Length Analysis 113 A.4 Displacement Length 116 vi B. The Tower, Instrumentation and Data Logging Systems 117 B.l The Tower 117 B.2 Instrumentation and Data Logging Systems 120 B.2.1 UVW Anemometer 120 B.2.2 Yaw Sphere - Thermometer Eddy Correlation System 121 B.2.3 Differential Psychrometer System 122 B.2.4 Microvane and Cup Anemometer 123 B.2.5 Net Pyrradiometer 123 B.2.6 Theodolite-tracked Mini-Sonde System 124 B.2.7 Acoustic Sounder 125 C. Synoptic Background to the Observational Period 126 D. The Data Set 128 E. Determination of the Surface Energy Budget 129 E.I Budget Closure by Distribution of Residuals 129 E.2 Examples of Surface Energy Budgets 138 F. Spectral Analysis 140 G. Dispersion Function Program 146 H. Application of the Dispersion Functions 147 I . Theodolite-Tracked, Balloon-Borne Temperature Soundings 149 J. Comparison of Acoustic and Balloon Soundings 153 K. Subsidence Estimation from Potential Temperature Profiles 155 L. Mixed Layer Depth Program and Sample Data 158 v i i LIST OF TABLES Page, Table 3.1a vs o^/u^; i = u,v,w. 16 Table 3.1b t, vs a^/u^: i = u,v,w. 16 Table 3.2 Relation Between z/L classes and Pasquill-Gifford Classes. 17 Table 3.3 Least Squares Fitted Parameters to Equation (3.7) For Three Components. 17 Table 3.4 Comparison of Measured Values of (near) Adiabatic Non-Dimensional Wind Velocity Standard Deviations. 18 Table 3.5 Positions of Spectral Features. 25 Table 7.1 Mixed Layer Model Sensit ivi ty. 81 Table A.l Sectorial Analysis of Roughness Length. 115 v i i i LIST OF FIGURES Page Figure 3. 1 Non-dimensional Integral Alongstream Turbulence Statist ics as Functions of Surface Layer Similarity Variables 19 Figure 3. 2 Non-dimensional Integral Crosswind Turbulence Statist ics as Functions of Surface Layer Similarity Variables 20 Figure 3. 3 Non-dimensional Integral Vertical Turbulence Statist ics as Functions of Surface Layer Similarity Variables 21 Figure 3. 4 Energy Density Spectra for the Vancouver Suburban Site 24 Figure 3. 5 Crosswind Dispersion Function 27 Figure 3. 6 Crosswind Dispersion Function 32 Figure 3. 7 Vertical Dispersion Function 35 Figure 6. 1 Acoustic Sounder Trace for August 1st 45 Figure 6. 2 Potential Temperature Profiles for August 1st 46 Figure 6. 3 Potential Temperature Profiles at Various Distances from the Upwind edge of a Thermal Internal Boundary-layer 46 Figure 6. 4 Subsidence Warming 55 Figure 7. 1 Inversion Rise Modelling for July 20th 63 Figure 7. 2 Inversion Rise Modelling for July 22nd 64 Figure 7. 3 Inversion Rise Modelling for July 23rd 65 Figure 7. 4 Inversion Rise Modelling for July 28th 66 Figure 7. 5 Inversion Rise Modelling for July 29th 67 Figure 7. 6 Inversion Rise Modelling for July 30th 68 Figure 7. 7 Inversion Rise Modelling for July 31st 69 Figure 7. 8 Inversion Rise Modelling for August 1st 70 Figure 7. 9 Inversion Rise Modelling for August 2nd 71 Figure 7. 10 Inversion Rise Modelling for August 3rd 72 Figure 7. 11 Inversion Rise Modelling for August 4th 73 Figure 7. 12 Inversion Rise Modelling for August 5th 74 Figure 7. 13 Inversion Rise Modelling for August 8th 75 Figure 7. 14 Maximum Inversion Height Flux vs Maximum Surface Sensible Heat 78 Figure 7. 15 Maximum Inversion Height Layer vs Mean Wind Speed in Mixed 78 Figure 7. 16 Maximum Inversion Height vs Entrainment Parameter 79 Figure 7. 17 Maximum Inversion Height vs Inversion Intensity 79 Figure 7. 18 Maximum Inversion Height vs Horizontal Divergence 80 Figure A. 1 General Environs of Study Area, Near-Site Topography and Land-Use 108 Figure A. 2a Photographic View from the Top of the Tower to the West 109 Figure A. 2b Photographic View from the Top of the Tower to the North n o Figure A. 2c Photographic View from the Top of the Tower to the East 111 Figure A.2d Photographic View from the Top of the Tower to the South 112 Figure B. .1 The Tower and Embankments 118 Figure B. 2 Upper Sections of the Tower Showing Surface Layer Instrumentation 119 Figure E. 1 Decision Tree for Budget Determination 134 Figure E. 2 Residual e-j vs Wind Direction 135 Figure E. ,3 Frequency Distribution of e-j and 136 Figure E. .4 Suburban Surface Energy Budget 139 Figure F. .1 Spectrum for Single Stabi l i ty Class 142 Figure a,b,< F. ,2 Construction of Composite Spectra 143 X Figure 1.1 Mean Wind from Tower and Balloon Sonde 152 Figure J. l Inversion Height from Acoustic Sounder and Potential Temperature Profi le 154 Figure K.l Subsidence in Potential Temperature Profiles on August 8th 156 xi ACKNOWLEDGEMENTS I am grateful to the academic community of The University of Bri t ish Columbia and in particular my colleagues in the Geography Department of that inst i tute for providing an eclectic and stimulating intel lectual environment in which to conduct my studies. In part icular, my supervisor, Dr. T.R. Oke, served beyond the call of duty as mentor, teacher and perceptive c r i t i c of my ideas. His efforts in the bowels of City Hall provided the permission necessary for the erection of the instrumentation tower. My examining committee, Drs. J.E. Hay, S. Pond and I.S. Gartshore were always available with invaluable advice and guidance. Field assistance was provided by B i l l Broomfield in the preparatory stages, Joanne Pottier and Brian Guy during the often arduous data gather-ing phase. Their efforts with the tracking theodolites are especially appreciated. Sheila Loudon served ably as a computing assistant. Brian Kalanda provided valuable advice on the operation of the di f ferent ial psychrometer system. Research grants by the Natural Sciences and Engineering Research Council of Canada and a Scienti f ic Subvention from the Atmospheric Envir-onment Service of Environment Canada to Dr. T.R. Oke covered the consider-able funding requirements of the f ie ld work. I was personally supported by teaching assistantships, a summer research fellowship and la t te r ly by a Killam Predoctoral Fellowship, a l l from The University of Bri t ish Columbia. The Bri t ish Columbia Hydro and Power Authority gave permission to use their Mainwaring Substation as a research s i te. The Pacific Region of the Atmospheric Environment Service kindly lent their mini-sonde system and made available the services of Ron McLaren who instructed us in i ts operation. Don Faulkner of that service provided a computer program for determining the balloon position from the theodolite sightings. Dr. M. Church kindly lent an analogue magnetic tape recorder for the turbu-lence data and the Inst i tute (now Department) of Oceanography at The University of Br i t ish Columbia allowed free access to their analogue to d ig i ta l convertor and mini-computer for the digi t izat ion of those data. Richard Leslie wi l l ing ly bu i l t the bridge-amplifier and active f i l t e r s and provided much guidance on matters electronic. 1. Preface 1 1.1 Rationale The concentration of pollutants in the atmosphere and at the surface of the Earth is determined primarily (apart from source strength var iab i l i ty ) by the rate of dispersion into the atmosphere and by the volume of atmosphere available for d i lu t ion. The f i r s t determining factor is governed by the turbulent diffusion process and the second by the depth of the mixed layer and the mean wind through that layer. Under certain atmospheric conditions turbulent diffusion may be effectively absent, or there may exist no bar to vertical mixing. This study wi l l not cover those conditions, but w i l l rather concentrate on a highly turbulent mixed layer capped by an elevated inversion. Since the largest effects (in human terms) of air pollutants generally occur in urban and suburban situations, where few data are available for estimating the governing factors, this study w i l l concentrate on those factors in a suburban situation. The ci ty from which the study w i l l draw i ts data is Vancouver, Brit ish Columbia, Canada, which has a mid-latitude coastal location. The results w i l l thus be characteristic of this si tuation, but are not expected to be specific to any particular feature of the chosen c i ty . The methods used w i l l be those of micro- and meso-meteorology, and the turbulent d i f -fusion process w i l l be inferred from turbulence measurements, rather than by measuring the spread of a tracer. 1.2 Objectives The overall objectives of the study are not to present an integrated scheme for dispersion or pollutant concentration calculation, but to investigate in some depth the two determining factors already 2 mentioned in Section 1.1: Turbulent diffusion and the depth of the mixed layer. The study w i l l be approached in two quite separate parts, the f i r s t dealing with turbulent diffusion and the second with the daytime evolution of the mixed layer depth. Though treated independently, these two phenomena are in real i ty both complexes of interacting processes linked to each other and to higher order phenomena. The f i r s t part w i l l be directed towards providing estimates of turbulent diffusion parameters that can be used to determine pollutant concentrations within the suburban mixed layer via the Gaussian plume model. The s ta t is t ica l theory of diffusion w i l l be applied to turbulence velocity spectra measured within the surface layer. The historical and theoretical background of this topic w i l l be covered in the introduction to the f i r s t part of this study. The objectives of the second part wi l l be to develop a mathe-matical model for the depth of the daytime mixed layer which w i l l be applicable to situations having similar physical characteristics to the chosen s i te . The model w i l l be a generalization of existing models, and w i l l have to account for advective heat transport and meso-scale subsidence associated with thermally-driven circulation systems. The historical and theoretical background of this topic w i l l be covered in the introduction to the second part of this study. By i ts nature, this model w i l l require considerable computing power, while the diffusion scheme of the f i r s t part w i l l be easily applicable on a hand calculator. In order to reduce the clutter of secondary and peripheral analyses and background information in the body of the text , much of this material is contained in the appendices. 3 Part One: TURBULENT DIFFUSION 4 2. Introduction 2.1 Specification of Diffusion Parameters Analysis of the diffusion of material in a turbulent flow has followed three dist inct l ines, each developed from a different theoretical base. The Gradient. Transfer approach is really a f i rst-order closure scheme which relates mass fluxes to mean velocity gradients by an eddy d i f fus iv i ty . The approach has a comforting feel because of the simi lar i ty i t bears to the classic Fickian (molecular) diffusion framework. The crippling flaw of this approach is that the eddy d i f fus iv i ty is a property of the flow (not the f l u i d ) , and in geophysical'flows is generally component-dependent. For these reasons the so-called "K-theory" has been largely ignored in the recent history of turbulent di f fusion, even i f i ts influence lingers strongly enough to prompt Scorer1s (1976) warnings against i ts use. In operational terms the K-theory is attractive as i t can easily be incorporated into input/output formulations of regional-scale box models of pollutant transport (Nunge, 1974), but the detailed specif i -cation of the three component K's remains a problem. A variety of more or less rea l is t ic forms for the K's have been proposed, some of which yield analytic solutions to the diffusion equations (Sutton, 1953 and Pasquill, 1974). The Similarity Theory of turbulent diffusion is based on the K-theory but uses simi lar i ty arguments to derive forms of K based on non-dimensional functions of the Monin-Obukhov length scale. These functions are invariably empirical and require extensive measurements of diffusion such as those presented by Deardorff and Wil l is (1975). An alternative view of this approach is to treat the concentration distr ibution as a 5 function of the chosen non-dimensional groups (Gifford, 1975). However, this ab i l i t y to short-circuit process is a property of the simi lar i ty theory, rather than the phenomenon. Diffusion from a continuous source may be treated from a purely s ta t is t ica l viewpoint in what is known as the Taylor (1921) Stat ist ical Theory. This approach resolves many of the d i f f i cu l t ies of the other two poss ib i l i t ies , is amenable to f a i r l y straightforward measurement and analysis and produces results which can conveniently be used to estimate pollutant concentrations using the so-called Gaussian plume model. The Gaussian Plume model estimates mean concentrations of pollutants emitted into turbulent flow with a bivarate Gaussian distr ibu-t ion. The standard deviations in the vertical and horizontal crosswind directions are used as diffusion parameters that must be specified, and w i l l be functions of the flow type and downwind distance. These parameters are usually specified as functions of downwind distance and atmospheric turbulence s tab i l i ty type. The six Pasquill turbulence types (Pasquill, 1961) form the most convenient operational scheme and can be related (Golder, 1972) to more basic s tab i l i t y measures. Under this scheme the standard deviations as functions of downwind distance are given as families of curves called the Pasqui11-Gifford curves (Pasquill, 1961; Slade, 1968; and Turner, 1969). These curves have been compiled from diffusion observations over f l a t land for distances up to 1 km. I t has been necessary to extrapolate (on the basis of solutions to the diffusion equation) these curves up to a distance of 100 km (Smith, 1972). Since the flow within the surface layer reflects very strongly the nature of the underlying surface, i t is reasonable to expect very dif ferent surfaces to be represented by dif ferent sets of Pasqui11-Gifford curves. A set of curves to represent diffusion over urban surfaces has 6 been produced by McElroy and Pooler (1968) who performed measurements of tracer spread over St. Louis. Gifford (1976) presents a set of curves derived from their data and shows them to be quite different from the . curves for much smoother surfaces. Briggs (1973) reviewed urban tracer data and their analysis up to that date and proposed sets of o- and a z curves in analytic form for diffusion over urban surfaces. Within this formulation, the need to derive diffusion parameters direct ly from observa-tions of atmospheric diffusion makes their determination tedious, time consuming and subject to large s tat is t ica l var iab i l i ty (al l these factors being inherent drawbacks of that kind of observation). The Gaussian model i t se l f remains ( i f properly used) a peerless mathematical tool for estimating diffusion because i t is simple, f lexible and in accord with most available diffusion theory. For this reason i t has remained the core of the subject while the detailed specification of the standard deviations has been the subject of much uncertainty and some research. 2.2 Gaussian Plume Model Parameters and the Stat ist ical Theory  of Diffusion Taylor (1921) in his s ta t is t ica l theory showed that an ensemble average of part icle displacement under the influence of a stationary, homogeneous turbulent flow w i l l have a variance given by: V a j = 2 a f , / / R(T)dxdt' (2.1) where R(.T) is the Lagrangian:auto-correlation of the crosswind velocity component for a lag T , and a* is the variance of this velocity component. The l im i t of the outside integral is t , the travel time. I t can be shown 7 (Pasquill, 1974) that with simple transformation equation (2.1) leads to a' V[ \ , L < " l = ^ 2 \ ' * „ i (n){stnUnt)/(Trnt)} 2dn (2.2) where $ . (n) is the Lagrangian crosswind energy spectrum, and n the V ,L frequency. Under conditions of isotropic turbulence, a similar form holds for vertical di f fusion, with the Lagrangian vertical energy spectrum replacing the crosswind function. Pasquill (1971) suggested expressing equation (2.2) as: a y / a v t 2 = s y ( t / t L ) ( 2 - 3 ) where t^, the Lagrangian integral time, scale is given by: tt= I R(x)dx. J o>._ The formulation.of equation (2.3) has the convenience of the Gaussian plume model and the theoretical backing of the stat is t ica l theory, and has met with general approval among the research community active in this f i e ld (Hanna et a l . , 1977 and Randerson, 1979). The detailed specification of S thus remains the major objective. Two l imit ing values of S are: S+1.0 as t*0 S ^ ( 2 t L / t )1 / 2 as t^» the behaviour of S for intermediate values is entirely determined by the shape of the spectrum (or equivalently, the auto-correlation function), and may be approached in three quite dist inct ways (Pasquill, 1975b). 8 Closed mathematical forms for $ L(n) or R(x) may be substituted into the integral in equation (2.2), which w i l l then yield s ( t / t L ) . This method has been i l lustrated by Pasquill (1975b) who uses a variety of forms suggested for R(x) and tabulates s ( t / t L ) . Direct observation of a y / ° v t over a range of values of t so that the large t l imi t can be used to f ind t L > This method has been used by Draxler (1976) who compiled a large body of data from tracer diffusion observations over generally f l a t land. His compilation shows wide scatter but quite dist inct trends from which he derives analytic forms for S(t / t^) for both vertical and crosswind spread. Irwin (1979) uses the same technique on vertical dispersion data under unstable conditions. His analysis uses aconvective, rather than Lagrangian integral time scale. The Lagrangian energy spectra $ ,(n) and $ (n) (crosswind and vert ical) can be estimated from measured Eulerian spectra, and the integration in equation (2.2) performed to give S ( t / t L ) . I t can be shown (Pasquill, 1974) that the integration for a particular travel time, t , is equivalent to computing variances with an averaging time equal to the travel time divided by the rat io of Lagrangian to Eulerian integral time scales. This method has been applied (Hay and Pasquill, 1959, and Haugen, 1966) in order to test the val id i ty of a particular form of the Eulerian-Lagrangian transform, u t i l i z ing tracer diffusion 9 to determine a •. Sawford (1979) (whose work was concurrent with, but independent of this study) applies the same technique for determining S y ( t / t L ) over f l a t grassland and shows that his results compare favourably with Draxler's (1976) analytic form of S ( t / t ^ ) for crosswind spread. In this study,' the crosswind and vertical dispersion functions w i l l be derived by integrating the transformed Eulerian spectral functions that were observed in an unstable-to-highly unstable suburban atmosphere. 10 3. An Application of the Stat ist ical Theory to Measured Flows 3.1 The Dispersion Functions from Measured Eulerian Spectra 3.1.1 Eulerian-Lagrangian Transformations Turbulent diffusion is a s t r i c t l y Lagrangian process, whereas v i r tua l ly a l l atmospheric measurements are Eulerian in nature. This conf l ict of viewpoint would be easily resolved i f some theoretical trans-formation existed for relating Eulerian and Lagrangian quantities. The lack of a theoretical basis for such a transformation is a reflection of our lack of understanding of the fundamental nature of turbulent flows. This rather formidable problem has been approached on the basis of a number of somewhat in tu i t ive hypotheses, each having i ts own set of (often unclear) l imitat ions. For the purposes of this study, the most convenient formulation of an Eulerian-Lagrangrian transform is one which addresses the rat io of the integral time scales from the two frames of reference. The simplest approach to the integral time scale rat io is provided by the "frozen eddy" hypothesis which suggests (Pasquill, 1974) that: t L / t E = 1/i = U/a u where t^ / t^ is the rat io of the Lagrangian to Eulerian integral time scales, i is the turbulent intensity and is equal to the rat io of the longitudinal standard deviation of wind velocity (a ) to the mean wind speed (u~). A more detailed analysis may be based on Corrsin's (1959) conjecture that after suf f ic ient ly long migration times, particles may be considered to have velocities which are unbiased samples of the turbulent velocities 11 at their positions in an Eulerian frame. This hypothesis has led Saffman (1963) and Philip (1967) to the result: t L / t E = gu/a u (3.1) where 3 = 0.80 (Saffman) g = 0.35 (Philip) (note that Hay and Pasquill (1959) use a dif ferent g, v iz ; g = t L / t £ ) . The different values for g are a result of minor differences in analytic forms chosen by the two authors. A similar treatment by Wandel and Kofoed-Hansen (1962) leads to a value of 0.44 for g. Identity of the simi lar i ty theory and s tat is t ica l theory forms of eddy d i f fus iv i ty require g = 0.44 (Pasquill, 1974). The relation (3.1) has been experimentally investigated by Angel! (1964) who performed "approximately - Lagrangian" measurements from radar-tracked tetroons. Haugen (1966) inferred Lagrangian functions from tracer diffusion experiments and so was able to test equation (3.1). Both these studies show clear inverse relationships between the scale ratios and turbulence intensity. The scatter in their data is large but the results indicate a value near 0.5 for g. A number of alternative approaches to this problem do exist and have been reviewed by Koper et a l . (1978) who derive a powerful generalized transform for the autocorrelation functions, and show how equation (3.1) is a special case of their general form. Brook (1974) uses a s tat is t ica l approach introduced by Ariel and Buttener (1966) to determine the Lagrangian velocity autocorrelation function from Eulerian wind stat is t ics gathered over an urban surface. From this function, he computes g but cannot confirm equation (3.1) 12 because his data cover only a very small range of turbulent intensity. They do, however, lead him to conclude that su/a is independent of s tab i l i t y , terrain and height. In view of the foregoing evidence, i t was decided to use equation (3.1) as the basis for an Eulerian-Lagrangian transform with e = 0.5. 3.1.2 Transformation of the Dispersion Integrals From equation (2.2) we may write: Sj ( t ) = a j / c j t 2 = / f v > L ( n ) { s i n Un t ) /Un t ) }2dn (3.2) •'0 The Lagrangian-Eulerian transform takes the form (Pasqui11,.1974) $l^(n) = r$ £ (rn) where r = t^ / t^ = u/2au Applying this transform to (3.2) gives: Sj( t ) =j $ V j E(n){sin(2Trta un/u)/(27rta un/u} 2dn (3.3) where $ F and $ . are respectively the Eulerian and Lagrangian V , t V , L forms of the transverse energy density spectra. Composite Eulerian spectra from a number of blocks of turbulence stat is t ics are computed as functions of non-dimensional frequency f = nz/u (Appendix F). Transform-ing (3.3) to an integral over f gives: f* oo S2(t) = / $ V 5 ? £ ( f ) {s in (2 1rta uf/z)/(2TTta uf/z)Fdf •'o this may be rewritten as: r oo * o; Sj(t*). =•/ $ V j [ £ ( f ) { s i n (2Trf t*) /(2^ft*)}2 df (3.4) 13 where t* = t / t = ta u /z is the non-dimensional travel time scaled by t = z/o u- This form of scaling arises naturally in the integral , and is operationally more convenient than t^ as suggested by Hanna et a l . (1977) and Pasquill (1975). The scale t can be simply related to t^ (as shown in Section 3.3.2) and hence to Draxler's (1976) empirical surrogate t . . All that remains now is to determine the form of $ F ( f ) and v, t *w E ^ a n c ' P e r f o n T I the integration to derive the form of S(t*) for cross-wind and vertical spread. Before this can be done, a brief detour w i l l be taken through the integral turbulence stat is t ics and the details of the turbulent velocity spectra that w i l l be used in the calculations. 3.2. Eulerian Turbulence Functions over a Suburban Surface in  Unstable Conditions 3.2.1 Integral Statist ics In accordance with the Monin-Obukhov simi lar i ty theory, the non-dimensional velocity standard deviations in the surface layer (z<<z^) should behave as (Lumley and Panofsky, 1964): 0-../U* = (^(c), i = u,v,w (3.5) where u* is the surface f r i c t ion velocity, the are a set of non-dimensional functions and x, = z/L where z is the height, L the Monin-Obukhov length and z. is the depth of the mixed-layer. This scaling for the horizontal components appears to break down in unstable surface layers (Lumley and Panofsky, 1964) or at greater heights (z < z^), and i t has been suggested (Wyngaard and Cote, 1974 and Panofsky et a l . , 1977) that a more suitable form would be 14 (3.6a) a / u * = V v ( z i / L ) (3.6b) u,v where z- is the height of the lowest inversion (taken to be the depth of the convectively mixed layer). Considerable work has been conducted on the adiabatic l imi t o f t he ratios o^/v* and Counihan (1975) summarises the values as 2.5, 1.9 and 1.3 for u, v and w respectively. These ratios do not appear to depend on height. Binkowski (1979) develops a simple second-moment closure, Monin-Obukhov model for surface layer turbulence which predicts the form of cf^U) in equation (3.5) for -4.0<s<4.0, without expl ic i t reference to the character of the underlying surface. His func-tions compare well with the data from two independent sets of f ie ld measure-ments which show wide scatter for the horizontal components in unstable cases (as pointed out above). s tat is t ics data from different experiments exists, and as a result these analyses should be treated with caution. The problem stems from different studies having different averaging bands (in non-dimensional frequency space) for the determination of the ratios a^/u^. The most proper band is that covering the fu l l range of micro-meteorologic fluctuations ( i . e . , from the centre of the spectral gap at f ^ 6xl0~5 (Smedman-HOgstrom and Hogstrom, 1975) to the high frequency end of the iner t ia l subrange at f - 50). In practice this range is seldom achieved, and so care should be taken to compare results only i f the bands are of similar width and position. selected suburban s i te , using a Gill UVW anemometer mounted on a free-standing steel tower at an effective height of ^ 20 m into the surface A fundamental bar to compilations and comparisons of integral Turbulence measurements in this study were made over a carefully 15 layer (see Appendices A and B). The integral s tat is t ics were calculated direct ly from 62 blocks of data each containing 8192 data points sampled at 2.5 Hz (see Appendix F for details of the analysis). These data yield a non-dimensional frequency range of 3.0xl0"3<f<50. Surface layer turbulent sensible heat fluxes were determined by a variety of methods and the best estimate selected (see Appendices B.2.2, B.2.3 and E.l) so that the Monin-Obukhov s tab i l i t y length could be calculated. The depth of the mixed layer was determined by an acoustic sounder whose records were periodically veri f ied with the temperature structure measured using twin theodolite-tracked mini sondes (see Appendices B.2.6,B.2.7 and J) . From these data the ratios a^/u^. could be plotted as functions of z, = z/L and t;. = z^./L, these surface layer parameters having ranges (-0.02, -147.4) and (-0.0, -1211.8) respectively. The plots of the a.j/u* ratios against t, and £. show strong, increasing trends with increas-ing ins tab i l i t y with a great amount of scatter. This scatter is inherent in al l atmospheric measurements of this type and is in part related to the use of f i n i t e length records. In order to reduce this scatter, the data were classified into eight classes, each represented by i ts mean rat io of o^/u* and the analysis being repeated to give seven classes of This process of combining stat is t ics from blocks of data is not s t r i c t l y admissible since each block has a different mean wind (IT) and hence different range of f , a l l blocks being at the same height. However, this study the range of U was low (a mean of 2.5 m s - 1 and a standard deviation of 1.1 m s" 1 over a l l blocks), the range of f from block to block w i l l be small enough to ignore. In general, the members of a c class do not correspond to those in the same (ranking) c,^ class. These classes are shown in Tables 3.1a and 3.1b together with the cor-responding means and standard deviations of the ratios a - / u * . 16 Table 3.1a: e-vs o . / u * ; i = u,v,w Class C i a u / u ^ a v / u ^ a w / u ^ 1 -2.4±0.9 2.2±0.4 ' 1.9±0.7 1.4±0.1 2- -5.2+1.5 3.1±1.1 2.6±1.3 1.7±0.8 3 -10.0±2.2 2.1±0.5 2.4±0.8 2.0±0.3 4 -16.9+2.2 2.8±1.0 3.4+1.1 2.3±0.6 5 -39±15 2.6±0.4 2.9±0.9 2.8±0.8 6 -122±58 4.0±0.8 4.2±0.7 3.9±0.8 7 -590±350 8.6±4.1 8.4±3.7 6.9±2.5 Table 3.1b: ?vs a . / u * ; i = u,v,w. Class a u / u * V u * a , / u * w * 1 -0.2±0.1 2.1±0.5 1.7±0.5 1.3±0.2 2 -0.4±0.1 2.2±0.4 2.1±0.5 1.7+0.2 3 -0.7±0.1 2.4+1.1 2.2±0.4 2.0±0.4 4 -1.2±0.7 2.6±0.8 2.7±1.2 2.1±0.6 5 -1.9±0.3 3.0±0.7 3.4+1.1 2.2±0.7 6 -4.5±1.2 3.2±0.9 3.5±0.7 3.2±0.8 7 -11.1±3.1 6.0±3.6 5.9±2.3 4.2±1.8 8 -70±53 8.6±4.0 8.3±3.7 6.1±3.2 17 These somewhat arbitrary classes can be related to more commonly used s tab i l i ty categories using Golder's (.1972) relation between z/L and the Pasqui11-Gifford classes, the correspondences are given in Table 3.2. Table 3.2: Relation between z/L classes and Pasquill-Gifford classes.  z/L Class P-G Class  1 C 2 B 3 B 4 A 5 A 6 A 7 A 8 A Empirical functions of the form: a.j/u* = (a - bc) C (3.7) can be f i t t ed to these data (for both scaling variables) using least squares techniques. The results are shown in Figures 3.1 to 3.3, and the f i t t ed parameters given in Table 3.3. Table 3.3: Least Squares Fitted Parameters to Equation (3.7) for Three Components a b c _ a u / u * ? 5.57 17.94 0.30 3.18 0.02 0.75 a w / u * c 0.00 36.81 0.27 v ?. 5.23 0.10 0.51 a / u * ? 0.00 19.05 0.25 w * ?, 2.49 0.41 0.35 18 The adiabatic l imits for the ratios a^/u^ are not expl ic i t in the data since no t ru ly neutral conditions were experienced. Clarke et a l . (1978) investigated these ratios over an urban and suburban surface and found values in good agreement with those over homogenous terrain as reviewed by Counihan (1976). The work of Clarke et a l . (1978) uses a non-dimensional frequency band of (1.15xl0~ 3, 15.0) (Clarke, pers. comm.) which is somewhat narrower than that of this study. Given the above reservations, the ratios for the nearest neutral class (represented by C = -0.2±0.1) are in good agreement (see Table 3.4) with those presented by Clarke et a l . (1978), those summarized by Counhian (1975), and those measured over a suburban surface by Coppin (1979). Table 3.4: Comparison of Measured Values of (near) Adiabatic non-dimen- sional Wind Velocity Standard Deviations. This Study Clarke et a l . Coppin Counihan (1975) (1979) (1975) <VU* 2.H0.5 2.39 2.5 2.5 ° v / u * 1.7+0.5 1.79 - 1.88 ° w / u * 1.3±0.2. 1.26 1.1 1.25 Given the wide scatter of the data (represented by the error bars in Figures 3.1 to 3.3) i t is not possible to choose between the two scaling variables t, and 5. . , and neither can much weight be given to the actual values of the parameters in Table 3.3. The third of these parameters should, from scaling arguments (Panofsky et a l . , 1977), be equal to 0.33. This is neither supported nor contradicted by this study. 19 1-0 • ' " • -1-0 0 0 1-0 2-0 log(" z / L ] 0-0 1«0 2-0 3*0 log('Z|/L) Figure 3.1: Non-dimensional Integral Alongstream Turbulence Statist ics as Functions of Surface Layer Similarity Variables. (solid lines are equation 3.7 with parameters given in Table 3.3) 20 -1-0 0-0 1-0 2-0 log(" z / L ) l o g ( " \ ) Figure 3.2: Non-dimensional Integral Crosswind Turbulence Statist ics as Functions of Surface Layer Similarity Variables. (solid lines are equation 3.7 with parameters given in Table 3.3) 21 0 - 0 _ z 1-0 log{ \ ) 2 0 log( \ ) Figure 3.3: Non-dimensional Integral Vertical Turbulence Statist ics as Functions-of Surface Layer Similarity Variables. (solid lines are equation 3.7 with parameters qiven in Table 3.3) 22 3.2.2 Spectra Turbulent f ields are conveniently and customarily analysed according to the theory of random functions (Monin and Yaglom, 1975). The application of this body of theory generally results in the represen-tation of the fluctuations of some turbulence property (vector or scalar) in terms of i t s spectrum in frequency space via Taylor's hypothesis. The details of the derivation of the three energy density spectra are covered in Appendix F. Apart from their use in deriving the dispersion functions in this study the spectra are powerful representations of the properties of turbulence, giving a frequency breakdown of energy content (variance) of the flow f i e ld . Surface layer s imi lar i ty theory predicts that the normalized spectra w i l l be functions of a dimensionless frequency f = nz'/u (where n is frequency and z1 a length scale). Kaimal (1978) shows that the length scale is z (the height) for the vertical component and the horizontal components at high frequencies, and z^  (the depth of the mixed layer) for the horizontal components at low frequencies. His spectra are normal-ized by division by u ^ 2 ^ 3 where u* is the surface layer f r i c t ion velocity and is the non-dimensional dissipation rate, the data being collected over " f l a t featureless terra in" . The spectral properties of the atmos-phere over this type of surface have been thoroughly investigated and a review presented by Busch (1973). Similar investigations over urban or suburban surfaces have been carried out by Davenport (1967), Deland (1968), Bowne and Ball (1970), Steenbergen (1971), Brook (1974), DCichene-Marullaz (1975) and Coppin (1979), a l l of whom report their spectra as being similar in general form to those reviewed by Busch (1973). 23 The three spectra (one for each velocity component) used to represent the atmospheric fluctuations are curves extracted from composite plots of spectra for each of the eight z/L classes of Table 3.1b. Appen-dix F details the time series analysis that produced the individual spectra and shows how they collapse onto three "universal" curves. The residual scatter (between spectra) is due to inherent uncertainty in the methods of analysis. The low frequency scatter for the horizontal components is partly due to using f = nz/TJ rather than f^ = nz.j/Tr as suggested by Kaimal (1978). These more proper scalings would produce more certain spectral estimates but are not appropriate in this context as the integral in equation (3.3) would not transform to the convenient form of equation (3.4). Figure 3.4 shows the "u " , "v" and "w" spectra as functions of f , with the approximate boundaries of Kaimal's (1978) three scaling regions. In spite of the uncertainty in the measurements made over this kind of surface, the spectra bear a strong resemblance to Kaimal's (1978) curves and represent the f i r s t confirmation that urban atmospheric f luctua-tions have similar details of structure as do those over f l a t (low rough-ness length) surfaces, the previously mentioned studies having somewhat limited ranges of f . The "w" spectrum is the simplest, having one simple maximum at A = lOz where X = u/n is the characteristic wavelength, this being somewhat longer than the maximum position of x = 6z for the accepted empirical form for the unstable vertical spectrum (Kaimal, 1978). The two horizontal spectra ("u" and "v") exhibit a clear maximum at lower frequen-cies and a less prominent point of inflexion at higher frequencies shown by Kaimal (1978) to be characteristic of these components in an unstable surface layer. The positions of these two features and those found by Kaimal (.1978) are shown in Table 3.5. For rough comparative purposes, in this study Zi ranged from 3z to 30z (see Part Two for more deta i l ) . 24 0-0 -2-0 -3-0 -3-0 -1-0 -2-0 U region 3 i region 2 X=25z 1 1 1 region 1 - x=42z ^ ^ ^ ^ I i „X = 4z I I -'V' 1 region 3 i region 2 \ = U7z I I region 1 - / ^ X=72z •5z I 1 -' W ' x = 1 0z I I 1 1 --3-0 -2-0 -1-0 0-0 1-0 2-0 log(f) Figure 3.4: Energy Density Spectra for the Vancouver Suburban s i t e . 25 Table 3.5: Positions of Spectral Features. Regional Boundaries in Spectra 1-2 2-3 Maxima "u" This Study 4z 25z 42z "u" Kaimal (1978) 2z 0.67z i 1.6z. "v" This Study 0.5z 47z 72z "v" Kaimal (1978) Iz 0.25z i 1.6z. "w" This Study - - lOz "w" Kaimal (1978) - - 6z Given the uncertainty of the exact positions of the spectral features (especially when plotted in log frequency space), i t is not possible to discern any differences between spectra measured over "rough" and "smooth" surfaces. The "v" and "w" spectra of Figure 3.4 w i l l be used to complete the integrand in equation (3.4) and so provide estimates of the crosswind and vertical dispersion functions which are the object of this part of the study. 3.3 Computation of the Dispersion Function 3.3.1 Computational Details The integration of equation (3.4) with tabulated values for the spectral function is a straightforward exercise in numerical quadrature. The "v" and "w" spectra in Figure 3.4 were digit ised at 50 uniformly spaced points in log space (thus providing higher resolution at lower frequencies); multiplied by the sampling function appropriate to the 26 travel time being considered; and the integration performed with a standard numerical quadrature package available as.a l ibrary routine in The Univer-s i ty of Bri t ish Columbia's Computing Centre. This routine (called QINT4P) f i t s a fourth order polynomial to four consecutive data points and computes an analytic integral over the middle interval. This process is repeated for every interval except the f i r s t and last which are handled by forward-and backward-difference schemes (Madderom, 1978). The l i s t ing of a FORTRAN IV program which w i l l perform this analysis is given in Appendix G. The maximum non-dimensional travel time is 150, this correspond-ing to t = 54 min (the length of the turbulence data blocks) with a repre-sentative upper l imi t for a u of 0.8 m s"1 at z = 20 m. The dispersion function is computed every 3.0 units of t* . The value of this integral for zero travel time should be exactly unity for a complete, well normalized spectrum. Since normaliza-tion by the total variance is proper, departure from unity w i l l indicate an incomplete spectrum. This w i l l be referred to later. 3.3.2 Crosswind Spread The computed crosswind dispersion function is shown in Figure 3.5. The lower curve (marked a) is derived by applying the method of 3.3.1 to the "v" spectrum in Figure 3.4. The upper curve (marked b) is similarly derived but the low frequency end of the spectrum has been l inearly extended (in log-log space) to an amplitude of -1.75 at a f re-quency of -4.00. The integral at zero travel time for the extended spectrum is 1.002 while that for the spectrum as calculated is 0.950, indicating a 5% loss of total variance, probably mostly in the low f re-quency end. As in tu i t ive ly expected, the difference is most marked at long travel times. This low frequency extension of the spectrum is VO T 0-8 0-6 \-0-2 h 0-0 0-0 250 50-0 750 t* Figure 3.5: Crosswind Dispersion Function. O Draxler's (1976) function a Measured urban spectrum • b I L 100-0 125-0 150-0 Hanna et a l . (1977) values Measured urban spectrum with low frequency extension 28 somewhat arbitrary, and results derived from i t should not be accorded too much deductive weight. The form of travel time scaling used here is operationally more convenient than the theoretically more proper Lagrangian scaling, and can be related to the lat ter in a simple manner. Hunt and Weber (1979) present the relat ion: t, =0.33 z/cr L W Busch (1973) reports a Ja„ to be constant at 0.52 for a l l unstable condi-w u tions. The data of this study show this rat io to scatter widely between 0.4 and 0.8. Taking the value of 0.52 leads to t = 1.58tL (3.7) where t s = z / a u , our scaling time. From this i t can be seen that t L (and t-s) is impl ic i t ly a function of s tab i l i t y as suggested in Section 3.2. Draxler (1976) scales his travel time with t^, the time at which the dis-persion function equals 0.5. From Figure 3.5, S(47.17) = 0.5000, which means that t i = 47.17 t s (3.8) or, using (3.7), t . = 74.53 t L (3.9) Equation (3.8) was used to transform Draxler's (1976) recommended function. SU/^.) = (1.0 + 0 . 9 ( t / t i )1 / 2 ) " 1 at selected points onto Figure 3.5. The agreement between these points and curve b is excellent, with only very sl ight relative skewing. Sawford's (1979) calculated points envelop Draxler's (1976) function and so are also in agreement with this study. 29 There remains, however, an inconsistency that must be addressed. Draxler (1976) uses the long-time l imi t of his function to relate t^ to t^ and shows that t . = 1.64tL (3.10) in conf l ic t with (3.9). A possible reason for this disagreement is the uncertainty inherent in extrapolating an empirically f i t t ed function to i ts l imit ing value at i n f i n i t y . Draxler (1976) suggests t^ = 1000s, thus implying = 610s by equation (3.10), or t L = 13s by equation (3.9). While their constancy is an unrealistic simpl i f icat ion, the lat ter value is in good agreement with the prediction of equation (3.7), using a typical value of CTu = 0.75 m s" 1 ( this corresponds to Ti" = 2.5 m s _ 1 and a turbu-lent intensity of 0.30), giving t L = 12.7 s. Fichtl and McVehil (1969) suggest that' t^ may be approximated by: t , _ 'Hnax L 2nu where x m , „ is the wavelength maximum of the Lagrangian "u" specturm. max From the spectra of this study (Table 3.4), this leads to L 2TTU * TT ' using the Eulerian - Lagrangian transform of Section'3.1.1. The/ratio of this time scale to that given by Equation (3.7) i s : = 42XL58 i 2 L L IT This rat io is unity for a turbulent intensity of 0.22, well within the range of 0.30 ± 0.15 for the present data set. Brook (1974) determines the Eulerian integral length scale over an urban surface to be 110 m at 18 m 30 height (an interpolation from his Figure 9.1), which is in agreement with the estimate of 120 m obtained from the Fichtl and McVehil (1969) formula with ^ m x = 42z. This consistency is taken as further indication of the correctness of equation (3.8), and strengthens the use of t as a time scale. Sawford (1979) has t.. ranging from 30s to 230s, depending on the run chosen (see his Table 1). His non-dimensional frequency range is 1.2xl0 _ l f<f<0.2, the.same width (in log space) and extending to 2.5 decades lower than the data in this study. The lower frequency coverage explains why Draxler's (1976) and Sawford's (1979) determinations of S(t*) agree with curve b rather than a. The foregoing leads us to the conclusion that in general t^ is a poor surrogate for t^, and w i l l depend on the non-dimensional frequency band over which the analysis is performed. External time scales such as t^ or t are preferable to t^, and in any case the averaging band is of v i ta l importance. Hanna et a l . (1979) recommended a set of values for crosswind dispersion as a function of travel distance. Bearing in mind Pasquill's (1975a) reservations about a simple Galilean transformation from distance to time, travel distance can be related to the present non-dimensional travel time. We want x = TTt. Since t* = t a u / z , we have x = y t * which for this study is equivalent to x = 66.7 t* (3.11) i f z = 20.1 m and i = 0.30. 31 Equation (3.11) is used to transform the Hanna et a l . (1977) values for f(x) onto Figure 3.5. These agree well with curve b. I t has been asserted (Pasquill 1974; Sawford, 1979) that the form of S(t*) w i l l not be sensitive to details of the autocorrelation function (and hence the spectral function). In order to test this asser-t ion , Kaimal's (1978) "v" spectrum was extended to low frequencies by linear extrapolation, renormalised and integrated to produce a form of Sy(t*) appropriate to those surfaces. The results are shown in Figure 3.6. The almost inconsequential differences in these two curves is an indication that turbulent diffusion as represented by the stat is t ica l theory is not sensitive to the differences in spectral functions used to determine the two curves. Before the dispersion function given in Figure 3.5 can be ascribed any universali ty, this analysis must be repeated on spectra determined over a range of surface types using identical methods (viz identical de-trending, smoothing, band - averaging and convolution). At present, the s imi lar i ty of the curves in Figure 3.6 is an indication that these curves may be universal ( i . e . , applicable to a range of surfaces). The high frequency t a i l of the spectra shown in Figure 3.4 ro l l off as approximately r a t h e r than the expected -2/3 behaviour of the iner t ia l subrange. This is presumably due to the inab i l i ty of the sensors to respond to high frequency fluctuations. In order to quantify this shortcoming, the "v" spectrum was extended l inearly (in log-log space) with the appropriate slope, redigitized and the integration of Section 3.3.1 performed. This correction resulted in only a 0.5 percent increase in the total variance, and no signif icant change in S^(t*). The computed values of Sy(t*) were found (by least squares methods) to f i t very closely the analytic form co ro 150-0 Crosswind Dispersion Function S(t*) for : b - Kaimal's (1978) extended Spectral function a - Extended spectral function from this study. 33 S y ( t * ) = (1.0 + 0.16 T t * ) "1 This form may then be used to determine crosswind plume width (see Appen-dix H) for input into Gaussian plume model dispersion calculations. 3.3.3 Vertical Spread The computational scheme of Section 3.3.1 can be applied to a measured vertical velocity spectrum to produce a vertical dispersion function S ( t * ) = o ^ / ^ t , analogous to the crosswind function of Section 3.3.2. The vertical dispersion function derived from the "w" spectrum of Figure 3.4 is shown in Figure 3.7. The use of this technique for determining a is much less sound than i ts use for determining a . The 3 z y major problem lies in the vertical inhomogeneity of the atmosphere which reduces the results to approximations at best. These approximations are l ike ly to be good for elevated releases in unstable atmospheres, and poor for ground-level releases in unstable atmospheres and elevated releases in stable atmospheres. A secondary problem lies in the choice of scaling time. Inherent in the method of this study is the scaling time t which can be easily related to a number of possible surface layer integral scaling times, including the Lagrangian integral time scale (see Section 3.3.2). The most rational scaling time for vertical diffusion in unstable conditions is t u = Z|/w* (Deardorff and Wil l is (1975) and Hanna et a l . (1977)) where z^  is the depth of the mixed layer, and w* = (|: w1 e 1 z^.) 1 / 3 is the mixed layer convective velocity scale where g is the acceleration due to gravity, e" the mean potential temperature of the mixed layer and w 'e 1 the surface kinematic sensible heat f lux. Irwin (1979) collects a body of data to relate cw/w* to z/z^ in the form 34 <VW*= a ( z / z i ) b where a and b are empirical coefficients varying with z/z. . Since t / t u s 1.92z./z • a w /w*, we f ind that t / t u s 1.92a (z/z.) 1-b Our data are generally in the range 0.03<z/z^<0.3 (see Part Two), where a = 0.72, b = 0.21, giving A s ta t is t ica l analysis of data from the 62 data blocks shows this rat io to be 0.12 with large scatter. In the absence of a single representative value of z/z.j for the entire data set, the time scale rat io of 0.12 was used to transfer Irwin's (1979) curve onto Figure 3.7 as a set of points. The results can be seen to agree substantially with the curve, in spite of the uncertainty about the analysis expressed before. which may be used to determine vertical plume dimensions for input into Gaussian plume model dispersion calculations. y t s = 1.38(z/ Z i ) 0.79 yielding 0 . l< t u / t s <0.5 The curve on Figure 3.7 is well represented by O-o 1 1 1 ' 1 ' • 0-0 25-0 50-0 75-0 100-0 125-0 150 t* Figure 3.7: Vertical Dispersion Function. O - Irwin's (1979) values. 36 4. Conclusion The results of this part of the study have shown that the Gil l UVW anemometer can be successfully used to measure the turbulent structure of the unstable atmosphere over a very rough surface (zQ^0.5m). I ts response shortcomings, part icularly in the vertical sensor, are partly ameliorated by the relat ively large vertical velocities encountered. In particular the non-dimensional velocity variances o./u* were shown to behave much as those over smoother surfaces. The details of their dependence on the s tab i l i ty parameters z/L and z^/L were obscured by the scatter in the data, but their general behaviour was not in contradiction with previously published empirical functions. The spectral functions over this surface were found to be of the same general form as those observed over smoother surfaces. Within the framework of the s ta t is t ica l theory of di f fusion, i t was shown that the non-dimensional dispersion functions ay/°v^ and a z / a w t can be determined by integration of the Eulerian spectral functions multiplied by an appropriately scaled sampling function. This scaling, which arises out of the Hay-Pasquill form for the Eulerian-Lagrangian transform and the use of a non-dimensional frequency, gives rise to a scaling time t . = z/a^ which is simply related to the Lagrangian integral, time scale. This treatment of diffusion is s t r i c t l y speaking only applic-able to turbulent f ields whose mean properties are uniform in both space and time, and does not take account of wind shear as a means of cross-wind dif fusion. The dispersion functions so produced agree very well with previous forms and are: 37 s y ( t * ) = (.1.0 4- O.T6/t^)" C4.1) and s z ( t * ) (1.0+ 1.21/t*) - l (4.2) The numerical coefficients being least squares f i t t ed parameters. The success of this method is due partly to the weak dependence of diffusion on the exact form of the spectral function, and hence on the details of the Eulerian-Lagrangian transform. The scaling time is shown to be a much more appropriate one than the somewhat arbitrary and unreal ist ical ly constant empirical form used previously. The dispersion functions are shown to be sensitive to the low frequency portion of the spectrum, indicating the need for careful measurement in those ranges. The two forms for the crosswind and vertical dispersion functions may be used as input to Gaussian plume model calculations of pollutant spread. The values for a y and may be obtained from direct measurement or from accepted parameterization schemes (see Appendix H). Care should be taken in the application of these dispersion formulae in the mixed layer i t se l f (z/z.>0.1 say), where the low frequency end of the spectrum may be markedly different from the ones observed at much lower alt i tudes. An alternative approach for this regime is provided by Venkatram (1980) who expresses dispersion in terms of the mixed layer variables w* and z.. 38 Part Two: THE DEPTH OF THE DAYTIME MIXED LAYER 39 5. Introduction 5.1 Specification of the Mixed Layer Depth The atmospheric boundary layer is that portion of the Earth's gaseous mantle into which the f r ic t ional and thermal effects of the under-lying surface extend. This layer is commonly in a state of highly turbulent motion which fac i l i ta tes the uniform mixing of entropy and gaseous atmos-pheric constituents throughout i t s depth, hence the commonly used term "mixed layer" (Tennekes, 1974). The structure of the mixed layer is largely determined by the exchange of turbulent energy between the layer i t se l f and the underlying surface, both of which can act as either source or sink, though the commonest configuration is for the surface to be a source of thermal and mechanical energy. In the case of a mechanically dominated layer, viscous drag at the surface provides a source of turbulent kinetic energy throughout the layer. This layer has no clear upper l imi t but can be defined as the height at which the turbulent fluxes (resulting from surface effects) have fal len to some (small) fraction of their surface values (Brost and Wyngaard, 1978), or some equivalent assumption (Niewstadt and Driedonks, 1979; Yamada, 1979). I t is often observed that an inversion of synoptic origins provides an unambiguous upper l imi t to surface driven turbulent processes. I t is the depth of this inversion-capped mixed layer in the presence of strong surface heating that is the concern of this part of the study. The depth of a thermally-driven mixed.layer generally exhibits strong diurnal variation and ranges from a few tens of metres to up to 2000 m in response to the diurnal variation of the surface sensible heat flux (Carson, 1973). This variation in depth, generally observed as a 40 monotonic increase, is principally achieved by entrainment in which the stable air above is eroded from below and mixed downward into the usually neutrally stable mixed layer. This entrainment is driven by the high levels of turbulence in the mixed layer which arise from upward heat transfer by thermal convection. The depth of the mixed layer can be measured by a variety of means, each possessing i ts own defini t ion of the exact height (Coulter, 1979). The principal methods of measurement being to sound direct ly some mixed layer parameter (usually temperature), and thereby detect the dis-continuity at the inversion base, or to remotely detect some effect of the entrainment process. Rather than ful l -scale f ie ld measurement programmes, the atmospheric boundary layer can also be modelled on a laboratory scale so as to elucidate i ts properties, including the depth (Deardorff et a l . , 1969, and Heidt, 1977). A number of schemes for the estimation of (usually hourly) mixing depths from easily available meteorologic data have been developed (Holzworth, 1967; Mi l ler , 1967; Deardorff, 1972; Benkley and Schulman, 1979) for use in air pollution modelling and as lower boundaries in general circulation models. An alternative to direct measurement or rough estimation is the mathematical modelling of the mixed layer processes so as to elucidate the depth. This modelling has been extensively developed by a number of investigators whose work has been successful enough to prompt Tennekes (.1976) to say " . . . the inversion rise problem may be regarded as solved". While this statement is in principle true, in detail there remain processes within the mixed layer and at the entrainment interface which are either poorly understood or need to be included in the models. Smith and Carson (1977) have considered the modelling of boundary layers in general and have detailed the requirements for this modelling on various scales, and in so doing have pointed out areas for further study. 41 In this study we address the short-range (see Smith and Carson, 1977) pseudo-two dimensional mathematical modelling of a dry, inversion-capped, convectively unstable boundary layer over a mid-latitude suburban surface near a large body of water. 5.2 Mathematical Modelling of the Mixed Layer Depth. The model which met with Tennekes1 (1976) approval arose from a proposal of Ball (1960) later developed by Li 11ey (1968), Tennekes (1973), Betts (1973), Carson (1973), Mahrt and Lenschow (1976) and Stull (1976a,b), among others. Crucial to the success of these models was Ball 's (1960) assumption that the downward sensible heat f lux at the inver-sion base is proportional to the upward sensible heat flux at the surface (Ball actually assumed them equal). The complete model (as presented by Tennekes (1973)) is purely thermodynamic and parameterizes the entrap-ment processes by ascribing to the interface a f i n i t e temperature step or "jump". In the presence of free convection (a condition prevalent in this study), mechanically generated turbulence has a negligible effect on the entrainment process (Tennekes, 1973). In a regime of forced convection, the vertical heat convergence in the mixed layer would be represented by a purely mechanical term derived from the surface layer f r i c t ion velocity (see Davidson et a l . , 1980). The model equations are: ( w ' e ' ) s - ( w ' e ' ) i (5.1) (5.2) '.dA dt dz. dt (5.3) - c ( w ' e ' ) s (5.4) 42 where e is the mean potential temperature of the mixed layer, w 'e ' is a kinematic sensible heat f lux where e 1 and w' are the fluctuating compon-ents of potential temperature and vertical velocity respectively, y is the potential temperature lapse rate above the inversion base, and A is the temperature "jump". The subscripts s and i refer to the surface and the inversion base respectively. Equation (5.1) is the thermal energy budget equation for the mixed layer, the rate of change of temperature being related to the (vert ical) convergence of turbulent sensible heat into that layer. Equation (5.2) relates the vertical movement of the entrainment interface to the eddy heat flux at that level and to the temperature step which serves to parameterize the entrainment process. The temporal behaviour of this step is given by equation (5.3) which is derived from the geom-etry of an idealized mixed layer potential temperature prof i le . Equation (5.4) is a parameterization of the heat f lux at the inversion base from the surface layer heat f lux , and serves to close the system of equations (5.1) to (5.3). The parameter c is the basis of Ball 's (1960) assumption, and has a range of reported values generally lying between 0.1 and 0.3 (S tu l l , 1976b). The exact value of c w i l l vary throughout a given day in response to the complex interacting processes at the inversion base (Carson, 1973; Zi1intinkevich, 1975; Tennekes, 1975; and S tu l l , 1976a). Carson (1973) shows how equations (5.1) to (5.3) can be solved analytically using a simple sinusoidal surface heat f lux. He compares his model with the results of the 1953 O'Neill boundary layer observations (Lettau and Davidson, 1957) and shows that the data imply dist inct phases in the evolu-tion of the boundary layer. Each phase is characterised by a different set of values for the four governing parameters, including c. Stull 43 (1976a) uses a constant value of c to achieve agreement between his rather more complicated model and two different sets of daily data. His values for c are in the range (0.1 - 0.2). Most recently Caughey and Palmer (1979) present a direct measurement of the vertical prof i le of turbulent sensible heat f lux that are in agreement with c = 0.2, albeit with large scatter. Mahrt and Lenschow (1976) conclude that the dynamics of the mixed layer are not very sensitive to the closure assumption. Based on the above information, a constant value of 0.20 was used for c in a l l the simulations based on real data in this study. Yamada and Berman (1979) show that this assumption provides a more than adequate f i rst-order model. The basic ideas of the model have been applied to idealized metropolitan areas by including an advected heat f lux term in equation (5.0)(Barnum and Rao, 1975) in order to simulate thermal internal boundary-layer development (Venkatram, 1977). 44 6. A Model of the Mixed Layer Depth 6.1 Characteristics of the Observed Mixed Layer Observations of the daily course of the inversion height (see Appendices I and J) over the study area exhibit behaviour strongly at variance with the classic rise and decay modelled in previous studies over extensive homogeneous surfaces (e.g. , Carson, 1973, and S tu l l , 1976b). The inversion shown by Carson (1973, his Figure 10) rises at an i n i t i a l rate of 87 m h - 1 six hours after sunrise, and ceases rising nine hours after sunrise, after which i t stays at a constant height of 1800 m unt i l twelve hours after sunrise. Figure 6.1 is -an acoustic sounder record from August 1st, 1978, showing the typical inversion height behaviour observed in the present study. The broad features of the inversion height on this day are an approximately constant rise rate of 62 m h - 1 lasting unti l approximately eight hours after sunrise, by which time the inversion has risen to i ts maximum height of 570 m. I t then begins a rather ragged descent to nearly 50 m at sunset. The presence of intense surface-based convection is indicated in Figure 6.1 by the intermittent "plumes" within the mixed layer. The apparent gap between the top of the plumes and the inversion base is due to the inab i l i ty of the sensor to respond to signals scattered from upper parts of these "plumes" which are presumably decreasing in act iv i ty as they ascend through the mixed layer. The thickness of the entrainment interface cannot, for the same reason, be derived from the apparent thickness of the acoustic sounder representation. The sounder i s , however, able to show quite clearly (even at this compressed time scale) the contorted nature of the base of the inversion (Carson and Smith, 1974; 4^ SOLAR TIME (HOURS) Figure 6.1: Acoustic sounder trace for August 1st. 46 Z: (X 3) . C 1 Z i l x . J eix,) etxj e(x,j P o t e n t i a l T e m p e r a t u r e Figure 6.3: Potential Temperature Profiles at Various Distances Distances from the Upwind edge of a Thermal Internal Boundary-Layer. 1 0 0 0 - 0 8 0 0 - 0 h 6 0 0 - 0 h 4 0 0 - 0 h 2 0 0 - 0 h 0 - 0 2 9 1 - 0 2 9 9 - 0 Figure 6.2: Potential Temperature Profiles for August 1st. 47 S tu l l , 1976a) as i t is bombarded from below by the surface layer generated thermals. The model to be presented here is not intended to simulate this small-scale structure which is part of the entrainment process parameterized by equation (5.4). A fa i r l y common feature of the acoustic sounder returns was the apparent disappearance of the inversion base especially beyond midday which, from the temperature soundings, continued undiminished in intensity. This phenomenon, which was often associated with descending inversions, remains unexplained in this study. In interpreting these traces i t must be remembered that the acoustic pulse has a length of 11 m, thus setting a lower l imi t to vertical resolution. The exact position of the inversion base on this often obscure trace was determined by comparing the trace with temperature soundings (see Appendix H), and the daily course was digit ized at approxi-mately ten minute intervals for comparison with the model results. Potential temperature prof i les, such as those shown in Figure 6.2, were used as veri f icat ion of the inversion height from the acoustic sounder trace. They show the expected surface layer with strong lapse in the lower tens of metres and the near-adiabatic mixed layer capped by the strongly stable inversion layer, presumably associated with synoptic-scale subsidence (see Appendix C). The mean characteristics of the elevated inversions are useful parameters. The mean inversion height from the acoustic sounder traces was 490 ± 122 m at noon, somewhat lower than the value of 590 m quoted by Morgan & Bornstein (1977) for San Jose, California at the same time of the year. This is as expected, as San Jose has considerably less maritime influence and is at a lower lat i tude. The mean inversion intensity (immed-iately above the mixed layer) from al l balloon soundings was 0.019 ± 0.009 K m" 1, in good agreement with the San Jose figure of 0.012 K m - 1 . 48 The following sections describe the development of a mixed layer model based on equations (5.1) to (5.2), modified so as to simulate in a general way the inversion height behaviour observed in this study. 6.2 Advection and Subsidence in the Mixed Layer Model Growing boundary layers act as storage buffers for moisture, heat and momentum, thus implying non-zero and time-varying divergences of these quantities. Equation (5.1) expresses the thermal component of this characteristic; the two terms on the right hand side being the vertical divergence of heat (always positive in this case). In the case of f i n i t e fetch, there exists the possibi l i ty of non-zero horizontal divergence due to advected heat fluxes. Following Barnum and Rao (1975), we may rewrite equation (5.1) as: z i l j t : = ( 1 + c ) Q ( 6 - n where Q = (w'e 1) and _= + TJ^- , x being the upwind distance to S L)t o t oX the surface discontinuity causing the bounday layer adjustment. This equation may be thought of as expressing the thermal energy balance of a column of air moving with the mean wind. Similarly, equation (5.2) may be restated to include the effects of both advection and subsidence as follows: Dz. A~DT = c Q + A w ( z i ) ^6.2) where vi{z.) is a vertical velocity of as yet unspecified or igin. Figure 6.3 shows schematically the spatial growth of an idealized thermal internal boundary layer. From i t one may write: 49 A(x,t) = y z ^ x . t ) - e (x , t ) + e 0 , (6.3) where ? 0 is the early morning value of the potential temperature at what w i l l become the lower l imi t of the mixed layer. I t (~e0) is assumed inde-pendent of space and time, so that: where W(z1-,t) is the heating effect of synoptic scale processes experi-enced at the top of the mixed layer. mate f i r s t integral of equations (6.1), (6.2) and (6.4) must be found. This should produce more real is t ic results than those of Barnum and Rao (1975) who assumed a sinusoidal behaviour for both z-j and e". I t w i l l be shown in Section 6.3that the effects of subsidence are small and can, to f i r s t order, be ignored in equations (6.1), (6.2) and (6.4). Doing this and changing variables to a reduced time x and dummy distance y, where: DA Dt (6.4) In order to find the spatial behaviour of z^  and e , an approxi-x = t - x/u, then _9_ 3t _3_ 9T and _9_ 8X 1 9 _9_ ay results in _D_ Dt with u constant. The model equations therefore become: 50 9A = _ dQ_ ay Y 9y " a y (6.5) and (6.7) give: z ^ y - ^ - - fy) = ( 1 * ° ) Q 9 z i d Subtracting (6.5) gives: 1^^— - — ( A Z ^ u 1 "V YU y U replacing this in (6.6) yields V Ao = / 2 Y c % 2A° (6.7) Which upon integration yields: : i Y z2 - A Z . = ^ + f ( x ) (6.8) u i f Q is independent of y, and f ( x ) vanishes since z. = A = 0 when Q = 0. I f 2A < yZj , a zeroth approximation is : z ° = /2Qy The rat io 375 - = 2c is less than unity, just i fy ing our approximation. Y Z V ' 1 Replacing A Z ^ in (6.8) by A z^  yields: 1 YU .2cjQy_ This form is in accord with Carson's (1973) integration of the non-advec-tive equations. The quadratic spatial behaviour of z^  is supported by 51 the observations of Wiseman and Hirt (1975), Raynor et al_. (1979) and Portel l i (1979). Summer's (1965) thermodynamic model of an urban heat island (mixed layer) also has this quadratic behaviour, but is based on a stationary heat input to the mixed layer. The use of this form in the present context implies that the time scales at which the mixed layer adjusts to changes in heat input are smaller than the (diurnal) time scales at which the surface heat fluxes change. Replacing (y) in equation (6.6) yields: - Ox* 2c)u Differentiating (6.4) by y (x) and replacing the above forms for z^ and A yields: 3X T 2ux ' H + 2c -Vl + 2c J whi le ! f i - f i n 9X S 2y 2c)Q ux 8 z i 9? Using these forms for and — , equations (6.5) and (6.6) can be O A oX used to yield Eulerian time derivative for z^  and e. In addition, the Eulerian time derivative of A can be obtained from equation (6.3) to give a new set of equations which may be numerically solved to yield the temporal behaviour of z. and e under the influence of both advection and subsidence. 6.3 Subsidence 6.3.1 Synoptic-Scale Subsidence Commonly associated with the synoptic conditions encountered during this study (see Appendix C) is non-zero horizontal divergence in the momentum f ie ld . The equation of continuity has: v.(pu) = 0 52 or, sp l i t t ing the horizontal and vertical components p V H " U = " 3z ^ p W ^ where is the horizontal divergence operator. Since the synoptic condi-tions were largely stationary we may, without much fear of oversimplifica-t ion, assume the horizontal divergence to be constant over any given day, so £ (PW) = - P 3 (6.9) where 3 = v ^(y) is a constant (often erroneously called the subsidence parameter). A convenient formulation for the density of the atmosphere is (Schmidt, 1946): p(z) = e(z) ^ e " b z (6.10) °o where pQ and eQ are the density and potential temperature at some reference level and b = 10 _ 1 +m _ 1 is approximately constant. Using a two-layer thermal atmosphere, e(z) becomes e(z) = eQ 0 < z < z. e(z) = e 0 + Y (z -z . ) Z > z. (6.11) Separating variables in equation (6.9), and substituting (6.10) and (6.11) leads to: 53 after integration, manipulation and substitution of az for z^  the sub-sidence velocity is given by: w = b(eQ + " r ( i - a)z) \^ bZ-V +l M 1 - «) . b z ( l . a ) . ] (6.12) I f we confine ourselves to lower layers of the atmosphere with moderately large z. such that: bz « 1 b(l - a)z « 1 equation (6.12) is well approximated by: w - eze 0 (e o + Y ( l - a)z) (6.13) I f , in addition. and Y(1 - a)z << e be << YO - a) 0 the subsidence velocity is given by w = - ez (6.14) This approximation wi l l generally hold i f a is not greater than 0.5. Equations (6.12) to (6.14) w i l l be used to estimate the horizontal divergence from the subsidence of observed features on the upper portion of potential temperature profi les (Appendix K). The subsidence velocity at the inversion height is given from equation (6.13) as: 54 w i = - e z i with the value for g calculated from (6.12) to (6.14) and can be substi-tuted into equation (6.7) and used in the model. interface, the subsidence produces a warming of the entire column of the atmosphere, and of direct importance in this context, results in a gradual increase of the temperature immediately above the entrainment zone (Davidson, 1980). This warming wi l l affect changes in the magnitude of the temperature jump, and hence on the dynamics of the processes determining the depth and temperature of the mixed layer. Figure 6.4 shows (in idealized form) the manner in which this warming occurs. At a time, t , the "parcel" of air immediately above the inversion base (at a height of z^) has a temperature e ( t ) . This "parcel" of air started i ts subsidence at a time t Q when i t was at a height z Q . (Note that this i n i t i a l height z Q , bears no relation to the surface roughness length usually given this symbol). The inversion must steepening since the subsidence velocity increases with height. In i ts simplest form, In addition to imposing a vertical velocity at the entrainment w s integration leads to (6.15) 55 Figure 6.4: Subsidence Warming; for explanation see the text. 56 Now, e(t) = y 0 ( z 0 - z\) + e Q substituting (6.15) and differentiating leads to: de = 6 y z' e3(t - t ) dt 6 V i e 0 The value of Yq w i l l be chosen as the mean inversion intensity of the lowest 650 m of the atmosphere in the early morning temperature sounding. The figure of 650 m was chosen since this is the height zQ that would be a typical maximum for the conditions encountered in this study. (z.j = 500 m, t = 7h, 3 = 10"5 s " 1 ) . Adding this warming to the dynamics of the tempera-ture "jump" changes equation (5.3) to | | . T ^ - | | + W ^ - V (6.16, This is the complete equation for the dynamics of A and w i l l be used in the model. 6.3.2 Meso-Scale Subsidence The model as modified has no mechanism for producing the very rapid decrease in inversion height observed in the later part of most of the days studied. This rise and subsequent f a l l of the inversion has also been observed by Portel l i (1979) at a lakeshore s i te. I t is probably associated with the dynamics of a meso-scale sea-breeze circulat ion. While the detailed two-dimensional modelling of the sea-breeze circulation (e.g. , Estoque, 1961 and 1962) would be the most proper way of approaching this problem, the intention in this study is to approximate the effects of such meso-scale circulations by providing order of magnitude estimates from the results of previous numerical and observational investigations. 57 Among the considerable l i terature on sea and land-breeze circu-lations Emslie (1968), Hoos and Packman (1974), Hay and Oke (1976) and Kalanda (1979) deal direct ly or indirect ly with those phenomena in the Vancouver/Fraser Valley region. Guy (1979) uses wind-speed and direction profi les from 23 mini-sonde f l ights from this experiment (see Appendices B and I) to characterise the structure of the meso-scale circulations over the c i ty . He finds very strong circulations on eleven of the fourteen days selected for investigation because of the absence of overriding synoptic flows. His calculations of the Biggs and Groves (1962) "Lake Breeze Index" show subcritical ( i . e . , conducive to thermally-induced meso-scale circulation) values on a l l days of the study, including the three days which showed an absence of sea-breeze circulat ion. The sea-breeze circulations occurring during this study had remarkably l i t t l e effect on observations made within the surface layer, in particular the passage of the sea-breeze front was never evident in the wind-speed and direct ion, temperature and humidity measurements made on the tower (Guy, 1979). There are, however, slow trends in both wind-speed and -direction that indicate quite clearly the existence of these circulations. The typical sequence being l ight easterly to south easterly winds in the morning freshening by about 1.0 m s - 1 by noon and gradually swinging through south to south south west by late afternoon. As dramatically i l lustrated by the tetroon f l i g h t patterns of Lyons and Olsson (1973), sea breeze circulations have regions of up l i f t and subsidence at their landward and seaward extremities respectively of between 1 and 2 m s" 1 . An examination of the two-dimensional flow f ields presented by Estoque (1961 and 1962) reveals a slow landward migration of the subsidence zone as the sea-breeze front advances. The Estoque 58 (1961) flow f ie ld for 1700 h shows that the region of maximum horizontal vor t ic i ty has migrated inland to 16 km from the coastline. An analysis of the vertical velocities at 8 km inland (the approximate distance of the present study site from the coastline) shows a horizontal divergence of 5 x 1 0 - t t s _ 1 , approximately an order of magnitude larger than that due to synoptic-scale processes (see Appendix J) . This increase in subsidence at a given inland position w i l l be gradual as the circulation matures and migrates inland. To accommodate this feature, in the model, the horizon-tal divergence was kept at i t s measured synoptic value unt i l 1130 LST, when i t was forced to increase exponentially in time so that i t reached ten times i t s original value by 1900 h, viz B(t) = 3 S t < 1130 B(t) = 6 s e ° -3 5 ( t " 1 ] - 5 0 ) t > 1130 (6.17) This form was used wherever 6 appeared in the model. Because of the approximate nature of the foregoing analysis, the modelling is expected to provide only order of magnitude estimates of the afternoon subsidence of the inversion. A major weakness of this approximation being that the time of onset of this effect w i l l in general be dependent on the upwind fetch. Whereas the form used in the model has a time of onset appropriate to a fetch of 8 km, the actual fetch does vary from 6 to 12 km depending on wind direction. 59 7. Implementation of the Mixed Layer Model 7.1 Computational Scheme Collecting the mixed layer model equations ( (6.1) , (6.2) 8 z i 3? and from (6.3)) and substituting the derived forms for ' — ' O A O X w(z .^) and W(z-,t) produces the following system of f i r s t order non-linear di f ferent ial equations: ___ a 3 3t X " 4 f c i " "5 a,z. - a c (7.2) ^ 7 3t " a 6 ^ F " 3t " a 7 z i ( 7 - 3 ) where a-j = (1 +c)Q '2 yi + 2c J a 3 = cQ 60 a 5 J O + 2c) K 2Yx a, 6 Y a 7 = gy e / o ett - t 0 ) The coefficients a-, to a 7 are a l l in general time-dependent and their values wi l l be calculated from the measured meteorologic variables. All i n i t i a l values were input as hourly averages, and the system of equations advanced in six minute steps through each hour. The surface sensible heat f lux values being l inearly interpolated for each six minute interval , and the horizontal divergence being set according to equation (6.17). The solution to the system of di f ferent ial equations was provided by a l ibrary program in The University of Bri t ish Columbia Computing Centre. This program (called DE) is based on a modified divided difference representation of the Adams predictor-corrector formulas and provides variable internal step length to control local error with special devices to control propagated round-off error (Shampine and Gordon, 1974). A l is t ing of the FORTRAN IV code to perform the simula-tion for one day and plots of the variables is given in Appendix L. The running time for a 14 h simulation on an Amdahl 470 v/6 model I I varied from 1.1 to 2.8s, depending on the "stiffness" of the equations. An example.of the input data needed to run the simulation for 14 h is provided in Appendix L. The f i r s t two lines contain sixteen hourly averaged surface sensible heat f lux values (Wm"2)(see Appendix E). The f i r s t and last of these are the pre- and post-sunrise values which are 61 used in the interpolation. The next two lines contain fourteen values for the inversion strength (K rrr 1) interpolated l inearly from the tempera-ture soundings (Appendix I ) . The next two lines contain fourteen hourly averaged mean wind speeds (m s _ 1 ) . The values used here were measured at level 4 of the tower (see Appendix B) and are taken to represent the mean wind in the mixed layer. Figure 1.1 is a plot of the mean wind at level 4 and the mean wind in the mixed layer estimated from the wind speed profi les provided by the theodolite-tracked balloons. I t shows that the tower measured wind is a good approximation for the mixed layer wind. The next two lines contain fourteen hourly averaged wind directions ( in degrees from true north) also measured at level 4 of the tower. These directions are used to calculate the distance x in coefficients a n c * a5» equation (7.1) and (.7.3). The calculation was made on the basis of an assumed e l l i p t i ca l plan of the urbanized part of Vancouver (see Figure A . l ) . The next l ine contains the i n i t i a l inversion height, mixed layer temperature (K), synoptic horizontal divergence ( s _ 1 ) , a data level, an optional model adjustment parameter which w i l l be referred to in the next section, the time of onset of meso-scale subsidence as simulation step number, the exponential parameter for this subsidence (equation (6.17)), and the early morning inversion intensity (K m - 1 ) for calculating subsidence warming. The last line shown contains up to six pairs of time (decimal hours) and mean mixed layer temperature (K) for validation of the model. Not shown in these data is a sequence of digit ized mixed layer depth and times for model validation. At in i t ia t ion the temperature step was set to 0.1 K. on a l l days. 62 7.2 Results of Mixed Layer Modelling Complete data sets for mixed layer modelling were available on thirteen days during the study period. Figures 7.1 to 7.13 show the results in graphic form. The overall performance of the model varies from poor (July 23rd) to excellent (July 31st and August 8th). The height of the observed inversion base was digit ized so as to include fluctuations with characteristic times slower than roughly 10 min, which is much faster than the characteristic times of the modelled invers-ion height. These high frequency fluctuations are presumably caused by a combination of thermal bombardment of the inversion base and breaking gravity waves at this interface, neither of which are exp l ic i t ly modelled here. There are, however, cases in which the observed inversion height deviates markedly from the modelled one at time scales larger than the aforementioned ones but shorter than the apparent response time of the model (three to four hours). These intermediate frequency fluctuations are presumably of synoptic or ig in, and are not evident in the surface layer (where the input data are measured) because of the previously mentioned buffering nature of the mixed layer. The magnitude of the potential temperature "jump" generated in the model is d i f f i c u l t to validate as i t is a mean property of the prof i le and would require much more frequent soundings than available in this study. I ts general behaviour i s , however, quite conservative and displays a gentle rise from i ts i n i t i a l value (0.1K) to a maximum of between 1.5 and 2.5 K some three to four hours after sunrise. I t remains steady at this value usually for about four hours and then begins a slow decline . 63 2 0 g 800 r 600 -E 5 0 7 0 9-0 11-0 1*0 15-0 170 19-0 Local solar time (h) Figure 7.1: Inversion Rise Modelling for July 20th. o Observed mixed layer potential temperature. A Observed (Balloon Sonde) inversion height (? = 290.5K) o The heavy line is the modelled inversion height. The l ight l ine is the inversion height from the acoustic sounder. The overall behaviour of the model is very poor on this day which was characterized by only moderate surface heating (due to a thin cover of continuous cloud). 64 20 s o I 1 1 1 1 i , I 5 0 7-0 9-0 110 130 15-0 170 19-0 Local solar time (h) Figure 7.2: Inversion Rise Modelling for July 22nd. Symbols as for Figure 7.1 (? 0 = 292.5K) The overall behaviour of the model is good with an underestimation of inversion height and temperature in the la t ter part of the day. 65 0 o © 9-0 11-0 13^ Local solar time (h) 19-0 Figure 7.3: Inversion Rise Modelling for July 23rd. Symbols as for Figure 7.1 (e = 289.7K) The model appears unable to simulate much of the inversion height variation on this day, almost certainly due to the passage of an elevated frontal system (see Appendix C). 66 Figure 7.4: Inversion Rise Modelling for July 28th. Symbols as for Figure 7.1 (? 0 = 286.7K) This day was marked by unsettled synoptic conditions, as a surface ridge developed. This non-stationarity is reflected in the relat ively poor behaviour of the model. The development of this ridge led to a sequence of days with remarkably stationary weather, reflected in the next nine simulations. 67 2 0 0 1 1 1 1 1 • I 5 0 7 0 9-0 H O 13-0 15-0 170 19-0 Local solar time (h) Figure 7.5: Inversion Rise Modelling for July 29th. Symbols as for Figure 7.1 (? 0 = 288.6K) The model behaves fa i r l y well in the f i r s t half of the day, following the mixed layer temperature well but consistently underestimating the inversion height. The model is unable to follow the sharp decrease in height observed in the la t ter half of this day. 68 2 0 g 0 I 1— 1 1 i i , | 5 0 7-0 9-0 11-0 13-0 15-0 170 19-0 Local solar timt (h) Figure 7.6: Inversion Rise Modelling for July 30th. Symbols as for Figure 7.1 (JQ = 290.3K) The simulation of both inversion height and temperature on this day is remarkably good, the sharp peak in mid-morning inversion height being an anomaly with no apparent synoptic or igin. 69 aoo f-Figure 7.7: Inversion Rise Modelling for July 31st. Symbols as for Figure 7.1 (? 0 = 284.OK) The model has near-perfect behaviour, the only disagreement being in the exact form of the inversion's descent in the late afternoon. As this phenomenon is treated by a rough approximation the dis-agreement is not fundamental. 70 600 r 0 " 1 1 1 1 1 I I 5 0 7-0 9-0 11-0 13-0 15-0 170 19-0 Local solar time (h) Figure 7.8: Inversion Rise Modelling for August 1st, Symbols as for Figure 7.1 (e"0 = 288.OK) The remarks for July 31st apply to this simulation as wel l . * 71 r 20 Local solar time (h) Figure 7.9: Inversion Rise Modelling fo r August 2nd. Symbols as fo r Figure 7.1 (? = 286.7K) In spi te of the rather complex behaviour of the inversion height, the model tracks very we l l . 72 20 0 I 1 1 1 i i i I 50 70 9-0 110 130 15-0 170 19-0 Local solar time th) Figure 7.10: Inversion Rise Modelling for August 3rd. Symbols as for Figure 7.1 (? 0 = 290.OK) As on the previous day, the inversion height is well modelled, hut the calculated temperature drops off in the afternoon. i 73 ICD iCD 800 200 600 \-i.00 70 9-0 110 130 Local solar time (h) 150 17-0 19-0 Figure 7.11: Inversion Rise Modelling for August 4th. Symbols as for Figure 7.1 (? 0 - 284.3K) The acoustic sounder record for this day was d i f f i c u l t to interpret, but the rapid early morning rise that was evident is well followed by the model. 74 20 0 I 1 1 1 _i . | 5 0 ™ 9-0 HO 130 15-0 17-0 19-0 Local solar time (h) Figure 7.12: Inversion Rise Modelling for August 5th. Symbols as for Figure 7.1 (? 0 = 283.2K) The model performs very well in the early morning when the height of the inversion is evident from the acoustic sounder trace. 75 20 Figure 7.13: Inversion Rise Modelling for August 8th. Symbols as for Figure 7.1 (? = 292.OK) The model performs part icularly well on this day, simulating both inversion height and mixed layer temperature accurately. 76 Since the input values of the inversion intensity are derived from the linearized segment of the potential temperature prof i le immedi-ately above the mixed layer, they w i l l be too large due to "contamination" by the thermal "jump". This problem is not easily resolved as the extent of the "jump" is never clear. The best solution seemed to be the use of the measured values of y reduced by a variable mult ipl icative factor (the adjustment parameter mentioned previously). By t r i a l and error i t was found that a value of 0.70 for this parameter reduced the inversion intensity to a value which gave good agreement between observed and modelled inversion heights and mixed layer temperatures. This adjustment is used for a l l days modelled. The overall sensit iv i ty of the model to a change of this magnitude can be extracted from Figure 7.17 which indicates an increase in maximum inversion height of some 120 m for a 30% reduction in y from i ts mean value in this study (0.019 K r r r 1 ) . A day-by-day discussion of the performance of the model follows. In most of the graphs the effect of the singularity in equations (7.1) and (7.3) is evidenced by sharply decreasing modelled temperatures in mid- to late afternoon. 7.3 Sensitivity Analysis In order to examine the sensi t iv i ty of the model to the magni-tude of the input variables, a synthetic data set was created based on mean values of the observed variables. The basic data set consisted of a sinusoidal surface sensible heat f lux given by: «H - « H m a x sinfliV1' with QH = 340 W r2  nmax 77 The inversion intensity (y) was set constant at 0.015 K rrr 1 , the mean wind speed (u") constant at 2.5 m s - 1 , the wind direction constant at 180°, an i n i t i a l inversion height ( z i 0 ) of 10 m and mixed layer temperature ( e Q ) of 290 K with horizontal divergence (B ) of 1.0 X 10~5 s" 1 were used. This mean wind direction implies an upwind urban fetch of 8 km. The meso-scale subsidence was set to start at 1130 h and to provide a ten-fold increase in the horizontal divergence by 1900 h. With this data set, the model produces a smoothly increasing mixed layer depth r ising to a maximum of 690 m by 1406 h (9.1 h after sunrise). The basic variables were then adjusted one at a time to investigate the behaviour of this maximum inversion height which invariably occurred at the same time. Figures 7.14 to 7.18 show the dependence of z - j m a x on QHmax» m e a n wind, inversion intensity, horizontal divergence and the entrainment parameter (c). In a l l of these analyses, the i n i t i a l rise rate of the inversion is between 113 and 190 m h - 1 and decreases monotonically from sunrise to 1400 h when i t reaches zero (the results shown on Figure 7.18 are an exception to this ) . The model can be seen to be sensitive to a l l the tested variables (which are in real i ty boundary conditions). The most sensitive being the inversion intensity which also exhibits the greatest non-l inearity, the least sensitive variable being mean wind speed, the maximum inversion height being a l l but independent of winds greater than 4.0 m s " 1 : the dependence of maximum inversion height on upwind fetch cannot be completely rat ionally investigated in this model as the present formulation is appropriate to a constant fetch of 8 km as des-cribed in Section 6.3.2. A rough indication of the model sensit iv i ty to fetch is possible in the region of 8 km. This is indicated in Table 7.1 as a mean gradient of maximum inversion height with fetch, together with 78 900 £ 7 0 0 x o E N 500 h 300 1 L — » • • • 1 0 100 200 300 400 500 QH max < W m~*> Figure 7.14: Maximum Inversion Height vs Maximum Surface Sensible Heat Flux. 79 900 E 700 L Maximum Inversion Height vs Entrainment Parameter. 1100 V 900 I 3 700 500 h 300 .00 .01 .02 .03 Y (Km"1) .04 .05 Figure 7.17: Maximum Inversion Height vs Inversion Intensity. 80 Figure 7.18: Maximum Inversion Height vs Horizontal Divergence. 81 the mean gradients for the other parameters investigated. The gradients are a l l determined at the basic values of the parameters. Table 7.1: Mixed Layer Model Sensitivity Parameter Basic Value Gradient Qu 340 W rn'2 1.13 m3W_1 H u 2.0 m s- 1 158 s Y 0.010 K m-1 2.82 x 10^ m2 K - 1 c 0.020 250 m 6 1.0 x IO" 5 S " 1 9.3 x 103 m s x 8.00 km 1.26 x 10"2 82 8. Conclusion This part of the study has shown that the accepted forms of inversion rise models ( typif ied by that of Tennekes (1973)) can be success-fu l l y generalized to include the effects of advection and subsidence. The effects of advection have been modelled by including an advected heat f lux term into the thermal budget equation for the mixed layer. The magnitude of this f lux is determined from observed forms of the spatial structure of growing thermal internal boundary layers. The effects of subsidence have been taken into account in the model by allowing subsidence-induced warming of the atmosphere above the growing layer as well as imposing a subsidence velocity on the entrainment interface. This sub-sidence is driven by atmospheric divergence on both synoptic- and meso-scales. The magnitude of the synoptic-scale divergence has been estimated from observations of subsidence in potential temperature prof i les, while the meso-scale effect has been approximated from modelled results of thermally induced meso-scale circulations. The inclusion of these processes in the model allow i ts application to areas in which meso-scale phenomena may have a considerable effect on the diurnal behaviour of the mixed layer depth (e.g. , coastal regions). The model has been applied to observations of mixed-layer depth and surface-layer variables made over a mid-latitude coastal c i ty . These observations show the diurnal behaviour of the daytime mixed layer depth to be quite different from the behaviour expected over wide stretches of homogeneously featureless terrain. In general the maximum mixed layer observed in the present study was approximately half that observed over f l a t ter ra in, and showed a decline in mid- to late afternoon that is absent in the contrasted environment. 83 The results of the modelling are generally in reasonable agreement with the observed mixed layer depth and mean temperature. This success is an indication that the generalizations are necessary and at least par t ia l ly suff icient to account for the mixed layer properties in this type of environment. I t is probable that the model w i l l be able to estimate quite reasonably the daytime inversion height from parameterized or climatologic input variables. As none of the model properties are exp l ic i t ly urban or in any way related to the character of the underlying surface or surface-layer i t should have general appl icabi l i ty in a l l aspects excepting the details of meso-scale subsidence which have been approximately treated. The most obvious extensions of the model would be in the detailed modelling of meso-scale, thermally driven circulations so as to exp l ic i t ly compute the imposed subsidence f i e l d . This sort of extension would need spatially-resolved mixing heights for proper validation. The entrainment processes at the inversion base are stochastic in nature and i t is unlikely that the high frequency fluctuations of the inversion height w i l l y ield to simple modelling of this kind. A surprising feature of this part of the study was the apparent absence of any effect related to the passage of the sea-breeze front. This may be a regional characteristic due to the complexity of the coastline, and not generally true. I t must be emphasized that the model performs well in the restricted and very "simple" synoptic conditions encountered during this study but produces very poor results in the presence of synoptic scale non-stationarit ies, as shown by Figure 7.3 which depicts a day during which a very weak front passed over the study area. 84 9. Summary of Conclusions The two major themes of this study (turbulent diffusion and mixed layer depth) have been developed using a body of data gathered over a coastally situated suburban surface under conditions approaching free convection. In developing the f i r s t theme, the following conclusions have been drawn: The non-dimensional integral turbulence stat ist ics a - j /u* over very rough surfaces (z = 0.5 m) have adiabatic l imits that agree with those measured over much smoother surfaces. The behaviour of these stat is t ics with increasing ins tab i l i ty is consistent with previous results but somewhat obscured by large scatter. The integral s tat is t ics are a l l related to the mixed layer variable ~ z i /L and show strong increasing trends with this variable. Turbulent velocity spectra can be successfully measured in these highly turbulent flows with an orthogonal array of helicoid pro-peller anemometers. The velocity spectra thus produced are remarkably consistent with unstable spectra measured over much smoother surfaces. In particular the horizontal components show the three spectral regions defined by Kaimal (.1978). The stat is t ica l theory of diffusion may be used as a basis for a convenient form of dispersion function whose determination reduces to the integration of the appropriate energy spectrum, multiplied by an averaging function. The form presented here has an internal scaling time that is relat ively easily available and can be related to the more proper Lagrangian integral time scale. 85 The dispersion function thus derived is in good agreement with previous estimates made from measurements of tracer spread and from turbulence measurements. The crosswind and vertical dispersion functions are presented as empirical forms which may be used to perform diffusion calculations i f the conditions of diffusion are consistent with the assumptions underlying the s tat is t ica l theory of dif fusion. In developing the second theme, the following conclusions have been drawn: In situations such as the one presently being studied, the behaviour of the daytime mixed layer depth may be quite different from that observed over homogeneous terrain. In part icular, the mixed layer depths are notably lower than expected and show downward trends in mid- to late-afternoon. The behaviour of the mixed layer depth may be successfully modelled by including the effects of advection and subsidence (at both synoptic-and meso-scales) in currently available mathematical models. Under non-stationary synoptic conditions the model results can be only poor reflections of the actual mixed layer depth. The generalized model is sensitive to a l l input variables, the sensi-t i v i t y being roughly 100 m for the expected range of values of surface sensible heat f lux , mean wind, inversion strength and subsidence parameter. In addition to these conclusions, during the course of the data analysis the following techniques have been ut i l ized. 86 The closure of an open energy budget was achieved by distr ibuting the budget residual among the turbulent flux terms, thus providing a more certain estimate of the fluxes. The horizontal divergence parameter was determined by applying a simple model of a compressible atmosphere to observed rates of sub-sidence of thermal features from temperature soundings. The form of the subsidence velocity is a simple function of height which, under successive approximations, can be shown to reduce to the incompressible form. 87 LIST OF SYMBOLS Symbols are defined on f i r s t introduction in the text, and for ease of reference are summarized here. In a few cases the symbolism is not unique; this is indicated by a multiple defini t ion in the l i s t below, and wi l l be obvious from the context within the text. Subscripting is used for axis (x ,y ,z ) , velocity component (u,v,w), level (s , i for surface and inversion base respectively) and frame of reference (L,E for Lagrangian and Eulerian respectively). An overbar represents a mean 0 value and primes represent departures from a mean. Symbol Meaning S. I . Unit a Constant in mixed layer scaling ( - ) a-| Coefficient in mixed layer model (K m s"^) a£ Coefficient in mixed layer model (K s) a^ Coefficient in mixed layer model (K m s"^) a^ Coefficient in mixed layer model (s - ^) a^ Coefficient in mixed layer model (m s"^) ag Coefficient in mixed layer model (K m~^ ) a 7 Coefficient in mixed layer.model (Km'^s- ^) b i ) Constant in mixed layer scaling ( - ) i i ) Constant in approximate formulation for height dependence of atmospheric density and temperature (nH) c Entrainment closure parameter ( - ) 88 d Displacement height (m) f Non-dimensional frequency ( - ) h* Height of roughness elements (m) i Longitudinal turbulent intensity ( - ) k von Karman's constant ( - _ L Monin-Obukhov s tab i l i ty length (m) n Frequency (s - ^) Q Kinematic eddy heat f lux (K m:s _ 1) Turbulent latent heat f lux (W m~2) Qu Turbulent sensible heat f lux (W m~2) hi Q* Net all-wave radiation (W m~2) r i ) Error rat io ( - ) i i ) Ratio of Lagrangian to Eulerian time scales ( - ) R Velocity autocorrelation function ( - ) S Non-dimensional dispersion function ( - ) s Silhouette area of roughness elements (m ) t Time (s) t^ Eulerian integral time scale (s) t.j Empirical scaling time for dispersion function (s) t^ Lagrangian integral time scale (s) t Surface layer scaling time for dispersion function (s) t Scaling time for vertical diffusion in u 3 unstable conditions (s) t* Non-dimensional diffusion time ( - ) 89 u Longitudinal component of wind velocity (m s" ] ) Tj Mean wind speed (m s" T ) u* Surface layer f r i c t ion velocity (ro s~ V Cross-stream component of wind velocity (m s" W Warming due to subsidence (K s" ]) w Vertical component of wind velocity (m s" ] ) w s Subsidence velocity (m s" ] ) w* Convective velocity scale (m s" ] ) w' e 1 Kinematic heat f lux (subscripted s for surface layer, i for inversion base) (K m S" X Upwind distance or fetch (m) y Dummy distance variable Cm) z Height ( m ) z. 1 Inversion height (m) z o Surface roughness length (m) a Scale height ( - ) (3 i ) Constant in Hay-Pasquill form of Lagrangian-Eulerian transform ( - ) i i ) Bowen's rat io ( - ) Y Inversion intensity (lapse rate), subscripted o for some i n i t i a l state (K nr ! ) <5 Depth of adjusted layer (m) A Potential temperature "step" at inversion base (K) AQS Heat storage in urban canopy layer (W m" 2 ) e Residual in energy budget closure (W m" 2 ) 90 Non-dimensional wind velocity variance Non-dimensional turbulent energy density spectrum (subscripted for component (u, v or w) and frame of reference (Lagrangian or Eulerian)) Wavelength Density (of air) subscripted 0 for a reference state Crosswind and vertical RMS plume dimensions respectively Alongwind crosswind and vertical standard deviations of wind velocity respectively Potential temperature Mean potential temperature of mixed layer Potential temperature of some reference state Reduced time Lag in auto correlation function Monin-Obukhov s tab i l i ty parameter (z/L) Mixed layer s tab i l i t y parameter (z^./L) 91 REFERENCES Angell, J.K., 1964: Measurement of Lagrangian and Eulerian Properties of Turbulence at a Height of 2300 f t . 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The Observational Site A.l General Requirements Observational studies in micrometeorology are plagued by the need to assume that surfaces surrounding the site of observation are homogeneous in a l l properties which may affect the atmospheric surface layer. While this may not present any problems in studies of the marine surface layer, i t has led to the selection of environmentally extreme sites for ter rest r ia l studies. This is part icularly true for sites at which pioneering studies have been carried out. A prime cr i ter ion on which site selection must be based is the fetch required for the surface layer to adjust to a change in surface characteristics. This adjustment must be completely propagated through the layer of atmosphere being studied, so that no flux divergences exist due to upwind changes in surface properties. The process of adjustment has been investigated both theoretically and experimentally, and reviewed by Munro and Oke (1975) who present the relation « ' (x) = O . l x 4 7 5 z 01 / 5 (A.l) for the depth of complete adjustment <5'(x) as a function of fetch (x) and surface aerodynamic roughness length ( z Q ) . This relation describes the adjustment of an adiabatic turbulent surface layer in transit ion from a smooth-to-rough surface. The depth of adjustment w i l l be greater for increasing ins tab i l i t y , and smaller for a transit ion from rough-to-smooth (Peterson, 1969) or increasing s tab i l i t y . 107 The relation (A.l) gives the required fetch which must be homo-geneous for a given roughness length and tower height (assumed equal to <5 '(x)). The present study was directed at surfaces with a roughness length of ^ 0.5 m, and ut i l ized a tower 30 m in height, thus requiring a fetch of 1485 m for smooth to rough transitions in an adiabatic atmosphere. In addition to this theoretical requirement, for logist ical reasons i t was necessary to f ind a site with easily accessible electr ic power, reasonable security, and one at which the erection of a 30 m steel tower would not be in violation of c i ty zoning laws. A.2 The Selected Site The requirement of 1.5 km of homogeneous fetch makes i t a l l but impossible to f ind a site t ru ly representative of urban meteorology, but i t is relat ively easy to find a site surrounded by homogeneous sub-urban surfaces, part icularly in North American c i t ies . Nevertheless because of the restr ic t ive nature of the ideal requirements, i t was inevitable that some compromise would have to be made in selecting the s i te. The site f ina l l y chosen is a transformer station (operated by the Bri t ish Columbia Hydro and Power Authority) known as the Mainwaring Substation. The substation is situated in suburban South Vancouver, in the 6400 block of Inverness Street (Kalanda, 1979). The setting of the ci ty of Vancouver has been described by Hay and Oke (1975) from a meteoro-logic point of view, while the general environs of the study area, the near site topographic details and land use are shown in Figure A . l . As can be seen from these figures, the site is surrounded by suburbia in al l directions for well over the requisite 1.5 km, and these surrounds are essentially f l a t except for a gradient of 1:16 to the southwest, starting 1.0 km away. The visual impression of the surrounding topography 108 Figure A.1: General Environs of the Study Area, near-site Topography and Land-use. Figure A.2a: Photographic view from the top of the tower to the west. Figure A.2b: Photographic view from the top of the tower to the north. Figure A.2c: Photographic view from the top of the tower to the east. Figure A.2d: Photographic view from the top of the tower to the south. 113 is "gently ro l l ing" with a vertical length scale of about ten metres. This is borne out by Figure A.2 (a,b,c, and d), four photographic views from the top of the tower. The large building in Figure A.2c is a three storey school which is situated 200 m to the east of the tower. As the mean wind was seldom from this direct ion, i t was easy to ensure that wake effects from this building did not contaminate the results of the turbu-lence measurements. In order to quantify the degree of horizontal homogeneity of the roughness length a land-use analysis was performed using a 1:10,000 photomosaic of the c i ty . (Figure A.l is based on this analysis). The surface roughness length is most properly determined from wind prof i le measurements, a task of considerable complexity in this environment, so a surrogate approach was taken. A.3 Sectorial Roughness Length Analysis For irregular arrays of reasonably homogeneous roughness elements, the surface roughness length can be estimated (Lettau, 1969) by z = 0.5h*-f- (A.2) o S v ' where h* is the height, s is the silhouette area, and S the total area of the roughness elements. Similar estimators using different formulae have been presented by Kutzbach (1961) and Counihan (1971) and the tech-nique has been applied to urban surfaces by Nicholas (1974) and Clarke et a l . (1978). In order to apply this method of analysis to Vancouver, a land-use classif ication scheme was designed which served to differentiate between the different types of roughness elements in the area. Figure 114 A.l shows the results of this classif ication which differentiates between purely residential use (mostly single family dwellings and garages, the mean frontal dimensions being 10.5 m and 6.0 m respectively and the mean heights being 8.5 m and 3.5 m respectively), commercial and multi-family dwelling use (having a mean frontal dimension of 41.5 m and a mean height of 15.0 m), and open areas (mainly parkland, parking lots and playing f ie lds) . The mean number density of roughness elements in the f i r s t two land-use types was found by counting those elements on representative sample areas on the map, and was used to determine the total number of roughness elements in each of sixteen 22.5° sectors centred on the tower. Because of the diversity of roughness elements, the terms in equation (A.2) were replaced by composite values as follows: ™ n. 1=1 m s = E i f s i i = l m where m is the number of roughness element types, n. is the number of elements of the i t n type in the sector being considered, N is the total * t h number of elements in that sector, h^  is the height of the i- " element type, s-j is the silhouette area of that element type, and A is the area of the sector. The results of this analysis are shown in Table A . l . While the absolute value of z 0 thus produced is not expected to be much more 115 than an order of magnitude estimate, the homogeneity of the parameter among the 16 sectors is a powerful indication that the assumption of surface homogeneity is at least a fa i r one. The mean roughness length of 0.52 m is somewhat lower than the values reported by Counihan (1975) and Clarke et a l . (1978) for this kind of surface (0.7 to 1.7 m) . The standard deviation of 0. 09 m indicates the degree of homogeneity • Table A . l : Sectorial Analysis of Roughness Length Sector No. of No. of No. of Percentage z o houses garages larger open space (m) buiIdings S 760 388 42 64 0.49 SSW 941 500 16 64 0.50 SW 1148 543 10 42 0.57 WSW 1149 790 23 54 0.61 W 1219 745 75 0 0.70 WNW 867 350 45 46 0.48 NNW 1073 553 7 27 0.53 N 1045 459 15 0 0.53 NNE 1117 406 19 0 0.56 NE 880 445 60 48 0.52 ENE 796 449 31 0 0.43 E 1100 677 32 55 0.59 ESE 972 572 9 60 0.49 SE 811 530 14 57 0.43 SSE 1005 680 13 64 0.54 Overall roughness length 0.52 m ± 0.09 m (mean and standard deviation) 116 A.4 Displacement Length Outside the laminar sublayer which surrounds a l l surfaces exposed to the atmosphere, there exists a highly turbulent wake layer which contains constantly fluctuating horizontal inhomogeneities. This layer, called the urban canopy layer by Oke (1976), provides a lower boundary for the surface layer in which our measurements are made, and must be accounted for in any calculations involving height. In effect, the top of this layer must serve as a zero for a l l height measurements when using surface layer theory. The momentum surface layer is taken to be based a distance d (the aerodynamic displacement height) from the ground's surface. This height is usually determined from measurements of mean wind prof i les, but can be estimated from land-use analyses. Estimation formulae have been presented by Kutzbach (1961), Counihan (1971) and Nicholson (1975). This sort of analysis has been applied by Clarke et a l . (1978) to suburban surfaces, and w i l l be used here. The Kutzbach (1961) form gives d/h as a function of the fraction area covered by roughness elements of height h. Using a weighted mean for the different types of roughness elements yields a displacement height of 3.7 m. The Nicholson (1975) form is derived from Lettau's results and is a somewhat more complicated function of roughness length and building height. This yields a value of 3.2 m for d, in close agreement with the f i r s t estimation. A displacement height of 3.5 m was used in a l l analyses to adjust a l l height measurements on the tower to a more real is t ic datum for the surface layer. 117 B. The Tower, Instrumentation and Data Logging Systems B.1 The Tower The instruments probing the surface layer in this study were mounted on a triangular section, steel la t t ice free-standing tower con-structed by the LeBlanc and Royle company to their LR324 series S/S specifications., The tower consisted of six la t t ice sections (A,A,B,C,D and E of their specif ications), having a base of 2.03 m, tapering to 0.5 m at an elevation of 18.40 m, and thence being parallel-sided to 27.45 m. A tubular extension of 1.60 m was added to give a total elevation of 29.05 m. The tower was f i t t ed with an external climbing ladder on the tapered section, while horizontal rungs were incorporated into the upper sections. The instruments were mounted on booms fixed to these upper sections (Figure B.l and B.2). The upper sections have a shadow fraction of 0.14, and the booms are a l l at least two tower diameters in length, thus ensuring that tower influences on the measurements w i l l be at an acceptable minimum (Moses and Daubeck, 1961). The tower was erected in the south-east corner of the Main-waring substation, some three metres from the embankments and hedges of the south and east boundaries, the base being 6.0 m below the base of the hedges. A t ra i l e r at the base of the tower housed the recording and logging equipment (Figure B. l ) . The surrounding houses are bu i l t at the same level as the base of the hedges, and allowing 3.5 m for the displacement length (see Appendix A), 9.0 m must be subtracted from a l l tower station elevations to obtain heights in the surface layer (Figure B. l) . 118 Level 5 Level U 20-1 m 18-5 m Level 3 Level 2 Level 1 13-4m -11-6 m -9-4 m -Surface layer datum 0-0 m 85m a.m.s.I. 9-0m Zo+d W//////////  5' 0m - 0-0m 1:200 Figure B . 1 : The Tower and Embankments 119 Figure B.2: Upper sections of the tower showing surface layer instrumentation. 1. U.V.W. Anemometer. 2. Microvane and cup anemometer. 3. Differential psychrometer, upper sensor. 4. Net pyrradiometer. 5. Yaw sphere-thermometer. 6. Differential psychrometer, lower sensor. 120 B.2 Instrumentation B.2.1 U.V.W. Anemometer This instrument consists of an orthogonal t r i p l e t of helicoid propel lor anemometers, each driving a miniature DC tachometer ( G i l l , 1974b). The instrument has been intensively studied (Drinkrow, 1972; Fichtl and Kumar, 1974; Hicks, 1973; Horst, 1973; McBean, 1972) and used in studies of the urban atmosphere (Brook, 1974; Coppin, 1979; Clarke et a l . , 1978). The instrument used in this study was the stock model manufactured by R.M. Young Co. with 0.3 m pi tch, four-blade polystyrene propellors. The response length (MacReady and Jex, 1964) has been found to be a function of the angle of attack (Raupach, 1977), and for the RMS angles of attack (approximated by t a n _ 1(a /u)) encountered in the study w should be 1.3 m for the horizontal sensors and 1.5 m for the ver t ica l . The starting speed of the sensors is in the region of 0.15 m s _ 1 (McBean, 1972). Different configurations of these three sensors have been considered ( G i l l , 1975; Christiansen, 1971; Pond et a l . , 1979) in order to minimise the effective response length, which can be unacceptably long in conditions of high horizontal wind velocities and small vertical velocity variances. This problem was not encountered in this study so the more standard and simpler orthogonal t r i p l e t was ut i l i zed. This instrument was used as a sensor of turbulent wind velocity fluctuations and was mounted at level 5 (Figures B.l and B.2) of the tower. The instrument was mounted on top of the tubular extension to the tower and was levelled as accurately as possible with a sp i r i t level. The levell ing required an operator at the top of the tower, and the flexing of the tower would certainly al ter the level l ing; i t i s , however, expected that the plane of the horizontal sensors is within at most 1° of true 121 horizontal, thus ensuring a t i l t error of less than 14% in the measured velocity covariance (Dyer and Hicks, 1972). The three signals were led down the tower to the t ra i le r where, after passing through an active low-pass f i l t e r with a signal reduction of 3.2 dB at 25 Hz, with ten-fold amplif ication, they were recorded on an FM analogue instrumentation tape recorder (Hewlett-Packard model 3960A). The data tapes were sub-sequently played back into an analogue to d ig i ta l converter (having 12 b i t resolution) linked to a minicomputer (PDP Model 10) which wrote the sampled data onto a computer-compatible 9-track magnetic tape. The analysis of the data is described in Appendix F. B.2.2 Yaw Sphere-Thermometer Eddy Correlation System Turbulent fluxes of sensible heat can be direct ly measured by determining the correlation of temperature and vertical velocity f luc-tuations. Any of a wide range of velocity and temperature sensors can be used to achieve this end. A part icularly convenient combination is a vane mounted pressure-sphere anemometer and platinum resistance thermometer (called a Yaw Sphere-Thermometer or YST) as described by Tanner and Thurtell (1970) and Yap et a l . (1974). A YST system has been used to estimate turbulent sensible heat fluxes over an urban (really suburban) surface (Yap and Oke, 1974; Oke, 1978), and was ut i l ised for that purpose in this study. The sensor assembly was mounted at level 2 on the tower and the signals were led down the tower where they were transformed and conditioned to produce an hourly averaged value of the turbulent sensible heat f lux. In atmospheric environments, such as the one presently under study, a compromise must be reached between the need for long averaging 122 times (to achieve s ta t is t ica l stationarity in signals with large variances), and the need for short averaging times (to satisfy the assumption of temp-oral stat ionarity in the signals). The problem of s ta t is t ica l stationarity has been approached by Wyngaard (1973) whose results, when applied to the conditions of this study indicate a desired averaging time of one hour for a 10% accuracy in flux estimates at a mean wind speed of between 2.0 and 3.0 m s _ 1 . As this is one order of magnitude less than the diurnal cycle, i t should be within the dominant temporal variations, and so w i l l be used as the basic averaging time. This choice is consistent with a number of previous urban meteorology studies (Brook, 1974; Clarke et a l . , 1978; Coppin, 1979; and Yap and Oke, 1974). The estimation of a detailed energy budget is described in Appendix E. The errors in fluxes determ-ined with this instrument amount to 5% to 15%, dependent on mean wind and the range settings used. B.2.3 Differential Psychrometer System The rat io of turbulent sensible- to turbulent latent-heat flux in the surface layer may be estimated by measuring simultaneously the vert ical gradients of atmospheric temperature and humidity. The method, as implemented in this study ut i l ises a pair of vert ical ly separ-ated wet-bulb/dry-bulb temperature sensors, as described by Black and McNaughton (1971). The system used in this study is described by Kalanda (1979), and Kalanda et a l . (1980), and was used to give hourly averaged estimates of the Bowen rat io. The system was fixed to the tower so that the sensor positions were at levels 1 and 3 (Figures B.l and B.2). These data were used in conjunction with other measurements to provide estimates of the surface energy budget as described in Appendix E. 123 A detailed error analysis of fluxes determined by this system has been presented by Kalanda (1979), who finds errors in the range 10% to 20%. B.2.4 Microvane and Cup Anemometer Wind speed and direction at level 4 (Figure B.l and B.2) were sensed by a three-cup anemometer and microvane manufactured by the R.M. Young Co. (Model 12101 cup and 12301 vane). The cup has a distance constant of 3.0 m, and the vane a delay distance of 1.0 m and a damping rat io of 0.44. The analogue signals from these sensors were conditioned and integrated by a Campbell Scienti f ic data logging system (Model CR5) to produce hourly averaged values of mean wind speed and direction. B.2.5 Pyrradiometer The net all-wave radiant f lux density of the surface was measured with a net pyrradiometer (manufactured by Swissteco Pty. Ltd. , Model SI) mounted at level 3 (Figures B.l and B.2) and 1.8 m from the tower. The polyethylene domes were kept.inflated and free of internal condensation by a stream of dry commercial-grade nitrogen piped up the tower. The signal from the sensor was led down the tower where i t was integrated and logged on the CR5 data logger to produce hourly average values of the net radiant flux density. Appendix E details how these data were used (in conjunction with others) to estimate surface energy budgets. At this height (29.05 m) the gravel-coated transformer site has a view factor of approximately 0.24 (the site is rectangular with dimensions 140 m x 110 m). This represents a considerable f ract ion, and though there are no installations in the site with temperatures s ign i f i -cantly different from those of the surrounding suburbia, the net radiation 124 may not be entirely representative of a suburban surface. A radiation budget study based on the data gathered during this study (Steyn and Oke, 1980) shows this site and the surrounding suburbia to be represented by an albedo of between 0.12 and 0.14, in good agreement with albedos typical of urbanized surfaces (Oke, 1974). B.2.6 Theodolite tracked Mini-Sonde System The thermal structure of the planetary boundary layer was probed intermittently with miniature radio transmitting temperature sensors. The sensors were of the "mini-T-sonde" variety (manufactured by Sangamo Co.), and provided an accuracy of ± 0.1GC and a time constant of 2.5 to 3.5 s using a miniature thermistor as a temperature transducer. The sondes were carried aloft on Helium-filled p i lo t balloons inflated to provide an ascent rate of ^ 3m s - 1 . The temperature was transmitted as an FM analogue radio signal centred on 403 MHz. The receiver demodu-lator (Beukers Model 4700B) was f i t t ed with an output lineariser and chart recorder which provided a temperature-time plot. In order to trans-form to a temperature-height p lot , the position of the balloon was tracked with two tracking-theodolites (Askania Model 5700), having a vernier least count of 0.1 and a high power telescope with graticule circles at 0.5° and 0.1°)\ They were set up on a 301.4 m baseline just to the west of the substation. The baseline was aligned in a N-S orientation to accommodate the expected preponderance of E-W flows. Each f l i gh t was tracked for 15 min, with azimuth and zenith sightings being taken every 30 s. The cueing for these readings was provided by a controlling operator with a portable radio transmitter, each of the trackers having a receiver. The angles were read into portable tape recorders for subsequent transfer to data sheets for computer coding. The temperature traces were digit ized at each of the sighting times and 125 coded with that data. The analysis of this data is detailed in Appendices I and J. B.2.7 Acoustic Sounder The atmosphere can be probed remotely and continuously in a semi-quantitative manner by a monostatic acoustic sounder (McAllister, 1968; Beran and Hal l , 1974). The instrument in i ts most basic form trans-mits a pulse of sound vert ical ly into the atmosphere and then detects any echoes scattered by thermal structures. A commercially available model (Aerovironment, Model 300) was used in this study to produce a con-tinuous record of the height of thermal turbulence structure above the s i te . The transmit/receive unit, was located inside the transformer station approximately 10 m from the tower. The transceiver and display unit were set up inside the instrument t ra i le r . The instrument used in this study produced a 25 W pulse of sound at 16 Hz every 18 s. The recording system was adjusted to display the lowest 1000 m of the atmos-phere on a time base of 30.5 mm per hour. This form of sounding has been used in an urban environment (Bennett, 1975; Melling, 1979; Jensen and Petersen, 1979) where the major problem is interference by ambient (part icularly t ra f f i c ) noise. With the above settings this interference was at an acceptable minimum, pro-ducing a l ight but continuous darkening at upper levels. Apart from chart paper changes every 28 days, the instrument operated without attention. The traces from the sounder were digit ized and analysed as detailed in Appendices I and J. 126 C. Synoptic Background to the Observational Period The climate of Vancouver ( 4 9 ° 13' 37" N, 123° 4 ' 37" W) is characterised by i ts mid-latitude location on the west coast of a large continent with a very mountainous hinterland (Hay and Oke, 1976). This study was undertaken from mid-July to mid-August of 1978 in what was an extreme example of the typical anti-cyclonic regime which dominates the summer weather in this region. The following is a synoptic sketch of the weather during the observation period extracted from surface, and 500 mb charts, hourly observation sheets from Vancouver International Airport observing station (for location see Figure A.l) and visible band sate l l i te imagery (data provided by the Atmospheric Environment Service, Dept. of the Environment, Canada). The weather over the f i r s t half of the study period was dom-inated by a broad anticyclone centred at approximately 150°W, 50°N and covering a l l of the Eastern Pacific Ocean, with a dry thermal trough over the western United States of America. Associated with these surface features was an upper level ridge parallel to the west coast. This regime brought clear skies to the region with only occasional bursts of marine stratus advected up the coastal inlets (Spagnol, 1978). During this period a weak short wave moved through the long wave ridge on the 23rd of July bringing some scattered cloud but no precipitation. This regime persisted unt i l the 26th/27th when a deepening closed vortex over the Pacific Ocean began to dominate the flow at a l l levels. In the transit ion between these two regimes, large-scale motions without frontal origin realised potential ins tab i l i t y , producing wide-spread convective act iv i ty over the entire region. Recording stations in Vancouver reported 4 to 6 mm of rain the 26th (Haering, 1978). After the 27th, the cold Low persisted and remained stationary, with an associated front taking on a 127 N-S orientation some distance to the west of the coastline. In response to this cold Low, a surface ridge of moderate amplitude developed bring-ing further clear skies and continued subsidence of warm air to the South-western Br i t ish Columbia region. This regime was remarkably persistent, and lasted from the 28th July to the 9th of August when a strengthening westerly upper flow over the Pacific f ina l l y drove the Low over the coast bringing cloud and precipitation to the region and heralding the end of both the summer and of the observational phase of this study. 128 D. The Data-Set The data gathered in this study (by the instrumentation detailed in Appendix B) w i l l be compiled and prepared for general teaching and research use by interested parties after the completion of the thesis. This appendix serves as an outline of the scope and extent of the data which consists of: Digitized (at varying sampling rates).measurements of daytime (0500-1900 Solar Time) mixed layer depths for July 20th, 22nd, 23rd, 28th, 29th to August 8th. Complete hourly averaged surface radiation and energy budgets from July 16th to August 8th with occasional missing data points in some of the earl ier days. Mean hourly averaged wind speed and direction at level 4 (Figure B.l) from July 16th to August 8th. Potential temperature profi les of the planetary boundary layer taken intermittently throughout each of the days with at least three f l ights on each day. Hourly blocks of three-dimensional turbulence stat is t ics taken intermittently throughout each of the days (a total of 62 blocks of useable data were taken). These data are a l l stored in data f i les in The University of Br i t ish Columbia Computing Centre where they were subjected to the analyses described in this thesis. 129 E. Estimation of the Surface Energy Budget E.I Budget Closure by Distribution of Residuals As described in Sections B.2.2, 3 and 5, turbulent sensible heat f lux, Bowen rat io and net all-wave radiant heat f lux were independently measured within the surface layer. From these three quantities an estimate of the four terms in the following idealized energy budget had to be made: Q* = Q H + Q E + A Q S (E .I) Q* is the net all-wave radiant f lux density, the turbulent sensible heat flux density, Q E the turbulent latent heat f lux density and A Q s the canopy layer heat storage. At f i r s t sight the above would appear to be no more than a t r i v i a l algebraic problem, but i t must be remembered that the measure-ments of the turbulent fluxes are subject to relat ively large errors which would appear in the residual (canopy layer storage term) and mask most of i t s real variations. A more detailed look must be taken at the various possible ways of estimating the energy budget terms. Because of the extreme inhomogeneity (in both material and conformational senses) of the suburban surface, the direct measurement of canopy layer storage is an a l l but impossible task, and was not attempted in this study. Fortunately the canopy layer storage is expected to be a very conservative variable, and can be parameterized from the net all-wave radiation (Kalanda, 1979; and Oke, et a l . , 1979). This parameteri-zation has the form: A Q S = 0.24(Q* - 17.0) (E.2) when Q* * 5.5 Wm 2 130 and A Q S = 0.70 Q* (E.3) otherwise. Given the three measured quantities and this parameterization, there are f ive dif ferent energy budgets that can be constructed (three closed and two open). The following (unfortunately somewhat complicated) notation is introduced in order to i l lus t ra te the f ive budgets. Q* - net radiation as measured. A Q s - canopy layer heat storage, parameter!'sed as eqns (E.2) and (E.3). - turbulent sensible heat flux as measured by the YST system. g - Bowen rat io as measured by the di f ferent ia l psychrometer system. - turbulent sensible heat f lux, calculated as jj^y ( Q * - A Q S ) I 1 ^ Q E - turbulent latent heat f lux , calculated as (i+g) (Q " A Q S ^ Q^' - turbulent latent heat f lux, calculated as Q^/g. Q E'' - turbulent latent heat f lux, calculated as Q* - - AQ S. 1 * i i A Q S - canopy layer heat storage calculated as Q - - . The three closed budgets are: Q* = Q|!) + Q E + A Q S (E.4) This budget is independent of the YST system. Q* = Q H + Q E' + A Q J < E- 5) 131 This budget uses data from both the di f ferent ial psychrometer and YST systems and does not use the parameter!'zations for canopy layer heat storage. Q* = Q H + QE" + A Q S (E.6) This budget is independent of the di f ferent ia l psychrometer system. The two open budgets are: Q* = Q H + Q E + Q S + e-, (E-7) and Q* = Q H + QE' + Q S + ^ ( E - 8 ) Both open budgets u t i l i ze data from both the di f ferent ia l psychrometer and yaw sphere-thermometer systems and are closed by the residuals e-j and e^. The core observational period consisted of some 480 hourly intervals. I f , during a given interval , the yaw sphere-thermometer system was not operative, the budget shown in (E.4) had to be used. Similarly (E.6) was used when the di f ferent ia l psychrometer system was not operative. These cases covered 220 hourly Intervals. Some decision network had to be set up in order to decide on the best estimates of the various fluxes during the remaining intervals. One rational approach to this problem was to divide these 260 intervals into four classes as follows (based on the open budgets), i ) Complete agreement. The two budgets given by (E.5) and (E.6) were judged to be in complete agreement when there was termwise agreement of the three r ight hand terms to within a (small) error, here taken to be 0.125Q + 10.0 W n r 2 . 132 i i ) Obvious error in one term. A budget was judged to be obviously in error i f i t met any of the following conditions a) positive canopy layer heat storage with negative net radiation (or the converse). b) turbulent sensible heat f lux greater than net radiation. c) turbulent latent heat f lux greater than 1.25 times net radiation. i i i ) Incomplete Agreement. The two budgets were judged to be in incomplete agreement i f they passed the "obvious error" f i l t e r , had both turbulent terms less than 0.70Q*, and did not f i t into class i ) . iv) All cases not f i t t i n g into the above classes. In the class i ) budgets, the best estimates of the fluxes were taken to be the means of the fluxes in the budgets (E.5) and (E.6). In the class i i ) budgets, the budget obviously in error was rejected and the alternative one used as the best estimate. The class iv) budgets were generally for intervals when Q* was very small, the ambiguities being the result of measurement errors masking the actual fluxes. An examination of the terms usually showed one or the other budget to be in error. The class i i i ) budgets were subjected to scheme whereby the residuals e-j and £^ w e r e distributed into the turbulent terms in the rat io of the absolute magnitude of estimated errors in those terms. The decision tree whereby the budgets were determined is schematically shown in Figure E.I , the bracketed numbers being the number of cases (hourly averages) that f e l l into each category. 133 The open budgets were considered to be open because of measure-ment errors rather than advective e f fec ts . This is consistent with our assumption of e f fec t i ve surface homogeneity (see Appendix A) and is supported by Figure E.2 which shows the residual E-| to be independent of wind d i rec t ion {z^ has a s i m i l a r l y random d i s t r i b u t i o n ) . For these reasons the residuals are considered to be energy which must be d i s t r i b u -ted amongst the two turbulent terms, these being subject to the most uncertainty in measurement. Figure E.3 shows the frequency d i s t r i b u t i o n of the residuals e-| and They both have near-zero means and are strongly l ep toku r t i c , e-j being the more extreme. For th is reason ( E . 7 ) was chosen as the open budget, and e-j was d is t r ibu ted between and Q E according to the fo l lowing scheme. Consider the fo l lowing open energy budget: Q* = Q H + Q E + A Q s + e (the complex notat ion has been dropped fo r c l a r i t y ) . The residual e is assumed to be composed of two port ions ar is ing from measurement errors in the two turbulent terms only v i z : Q* = ( Q H + e H ) + ( Q £ + e £ ) + A Q s where e u + e r = e , and our estimated turbulent f luxes are: n h % = % + £H Q E = Q E + E e . 134 No Q , Budgets 1, 2 and 3 agree to within 6Q 1 I f Q * > 0 and H Q u > Q * I f 0* > 0 and Q E > 1.25 Q * I f Q * < 0 and AQ'S > 0 I f Q * > 0 and AQ'S < 0 Use Q* = Q H + Q E + A Q S Qu + Qu Qp + Q'r A Q S + A Q ' Use Q * = H 0 H + E 0 E + — ^ >^ •Use Q* ='Q|!| + QE + A Q S Use Q* = Q U + Q E' 1 + A Q S Use Q * = Q u + Q E + A Q $ •Use Q * = Q H + Q E + A Q S Use distributed residual budget. Class IV budget (subjective decision necessary) Figure E.1: Decision tree for closure of energy budget. The number of cases in each category is given on the r ight. Notes: 1. SQ = 0.125 Q * + 10.0 W m 2 2. 6Q' = 0.7 Q * (68) (8) (41) (19) (1) (14) (62) (35) 360 OO O O O O o 240 c o •j: u z I 60 -250 OO O O O OOi O O OOO O O OODOOOO O (3D QD OOO °S O CD 0 ODOO O OOO O OO OOO O © O CD (BD CD) 0 O CD O O OO O O O O O O O GOO o o oo oc&xr) o OO CD O OO OOO GEO O © O O ® OO OOO O oo o o o o oo o o oo o C5DOO O o o o o o o o O QD CD O O O O -150 -50 50 Residual (Wm"a) 150 cn 250 Figure E.2: Residual vs Wind Direction 136 -600 -400 -200 200 400 600 <?, [ W m ' ) -600 -400 -200 200 400 600 ej [ W m J ] Figure E.3: Frequency Distribution of e1 and e 2 . 137 A s tat is t ica l treatment of experimental data yields probable errors <5QH and <$QE in the two fluxes (Fuchs and Tanner, 1970; Bailey, 1977; Kalanda, 1979). We require that: e u / e E = ± 6QH/6QE = ± r Solving for and e E leads to e u = e / ( l ± r) e£ = e / ( l ± 1/r) where the positive sign is for <5Q^  and 6QE having the same sign and the negative for their having opposite signs. Since the sign of the error cannot be determined from the error analysis, the residuals must be calculated for both cases and the resulting energy budgets examined. The most reasonable estimate of fluxes wi l l always be obvious. An example is drawn from the data on the 27th of July for the hour ending 1700 LST (Local Solar Time). Q* QR QE Qs e 6QU 6QE 149.1 = 95.0 + 67.8 + 34.3 - 48.0; 17.3, 19.2 (al1 fluxes in W rn"2). For the positive sign, 149.1 = 69.7 + 45.8 + 34.3 For the negative sign, 149.1 = 389.8 - 504.5 + 34.3. Quite obviously the f i r s t form is the more real is t ic one, and both fluxes had been overestimated by the instrumentation. A second example is from the hour ending 0900 LST on the 5th of August. 138 Q* QH QE Qs e 6 Q H 6 Q E 382.5 = 313.0 + 46.1 + 87.7 - 64.3; 17.1; 36.3 Positive r: 382.5 = 269.3 + 25.5 + 87.7 Negative r: 382.5 = 191.7 + 103.1 + 87.7. The second case is taken to be the best estimate as the Bowen's rat io, in the f i r s t case is unreal ist ical ly high. The instrumentation overestim-ated and underestimated in this case. E.2 Examples of an Urban Surface Energy Budget A previous energy budget study at this site (Kalanda, 1979; Kalanda et a l . , 1980) ut i l ized only the di f ferent ial psychrometer system for estimating the turbulent fluxes (the budget is as eqn. (E.4)). Energy budgets measured in that study are not signif icantly different from those of the present study and exhibit the usual temporal variation of fluxes and the relative magnitudes of Q„, Qr and AC) now known to n t S typi fy urban surfaces (Oke, 1978). Figure E.4 is an example of a 24 h surface energy budget for the urban surfaces surrounding the Mainwaring Substation. The entire energy budget is not expl ic i t ly ut i l ized in this study, but the surface turbulent sensible heat f lux is a crucial parameter in both Parts One and Two. The extra ef for t involved in determining the entire budget was deemed jus t i f ied by the increased confidence in Qu when the other three terms were determined and shown to be reasonable. CO Local solar time (h) Figure E.4: Suburban Surface Energy Budget. 140 F. Spectral Analysis The magnetic tapes containing the Gil l UVW signals, collected, f i l te red and recorded as in Section B.2.1 were transferred to The Univer-si ty of Br i t ish Columbia Computing Centre for analysis. The f i r s t step in this analysis was to sweep the data off the PDP tapes and demultiplex the three velocity signals. At this step, single- and double-point data spikes were removed by replacing them with adjacent values. Each t r i p l e t (u, v and w) was then transformed according to Horst (1973) to remove the by now well-known response errors inherent in this instrument (Drinkrow, 1972; Hicks, 1972; G i l l , 1973; and Fichtl and Kumar, 1974). The data thus transformed was calibrated using coefficients determined before and checked after the study period, and written to magnetic tape in blocks of 2048 t r i p le ts . Each raw data block was s l ight ly over an hour in extent. At a sampling rate of 2.5 Hz, four of these smaller blocks gave some 54 min of data, allowing leading and t ra i l ing discards of approximately 5 min each. This selected subset of data was then examined for trends and discontinuities and those hours with strong trends or marked discon-t inu i t ies discarded, leaving 62 "hours" of usable data for further analysis. This selected data was then transformed into flow co-ordinates using the means produced by the previous program and standard co-ordinate rotation forms. After transformation,the time series was sp l i t into mean and fluctuating parts by subtraction of a linear trend, the fluctuating part being saved for further analysis. The signals produced by this process were used to generate the integral s tat is t ics presented in Section 3.2.1. The next stage of analysis involved the use of a standard Fast Fourier Transform (FFT) routine to produce energy density spectra of the three velocity components. The data blocks were grouped into 141 s tab i l i t y (z/L) classes as described in Section 3.2.1, and the spectra combined to produce ones representative of each s tab i l i t y class. This was achieved by averaging the spectral amplitudes from al l eight data blocks in each z/L class into bands, each of width 0.1 units of non-dimen-sional frequency ( f = nz/uf) in log space. Because of this form of band-averaging, the high frequency points are averages of large numbers of determinations 4000), the low frequency points being derived from a much smaller number 8) , thus resulting in some scatter at the low frequency end. The spectra were then plotted in a variance-preserving form and a smooth curve drawn by eye. Figure F.l shows a typical spec-trum, i l lus t ra t ing the low frequency scatter. The sl ight scatter in the high frequency points is due to a small amount of aliasing in the spectral analysis of some of the runs. These smoothed spectra were then replotted (on one set of axes for each component). Figure F.2 (a, b, and c) shows the results of this replot t ing, and indicates that spectral forms are largely independent of s tab i l i t y . No systematic ordering of curves with s tab i l i t y could be discerned, presumably due to purely stat is-t ical f luctuation masking the weak trends with s tab i l i t y mentioned in Section 3.2.2. A single smoothed spectrum was drawn by eye from the mean position of the cluster of lines in Figure F.2, and digit ized for use in determining the dispersion functions. These are the spectra shown in Figure 3.4. For the reasons given in Section 3.3.2, no attempt was made to correct the spectra for the less than perfect high-frequency response of the sensors. •3-0 -2 0 -1-0 0 0 1 0 20 log (f) Figure F .2b: Construction of Composite Spectrum. Crosswind Component. 3-0 -2 0 -1-0 0 0 1 0 2-0 l o g (f) Figure F.2c: Construction of Composite Spectrum. Vertical component. 146 G. Program to Compute Dispersion Function from Digitized Spectra CCCCCCC I N T E G R A T E D I S P E R S I O N F U N C T I O N FPOM D I G I T I Z E D SPECTRUM D I M E N S I O N F ( 1 0 0 ) , Y ( 1 0 0 ) , F N < 1 0 0 ) , F 1 ( 1 0 0 ) , T I ( 1 0 0 ) , A ( 3 ) CCCCCCC READ H E A D I N G CFF D I G I T I Z E D S P E C T R U M F I L E R E A D < 7 , 1 0 3 ) A ( 1 ) , A < 2 ) , A ( 3 ) 1 0 3 F O R M A T ( 3 A 4 ) N = l CCCCCCC READ D I G I T I Z E D S P E C T R A L C O - O R D I N A T E S IN LOG SPACE 6 6 R E A D C 7 , 1 0 0 , E N D = 1 1 ) D X , D Y C C C C C C C CONVERT C O - O R C I N A T E S TO L I N E A R S P A C E F ( N) = 1 0 . 0 * * D X YCN )= U 0 . 0 * * D Y ) / F ( N ) N=N + 1 GO TO 6 6 11 N=N-1 C C C C C C C LOOP THROUGH TRAVEL T I M E 0 TO 1 5 0 I N 3S DO 22 J T = 1 , 5 1 T I ( J T ) = F L O A T ( J T - 1 ) * 3 . 0 CCCCCCC LOOP THROUGH S P E C T R A L C O - O R D I N A T E S DO 3 3 J K = 1 , N F 1 = T I ( J T ) * F ( J K ) * 6 . 2 8 3 1 8 5 F 2 = S I N ( F 1 ) I F ( F 1 . E Q . O . O ) G O TO 5 5 F 3 = F 2 / F l GO TO 3 3 5 5 F 3 = 1 . 0 CCCCCCC COMPUTE INTEGRAND 3 3 F N ( J K ) = Y ( J K ) * F 3 * F 3 C C C C C C C I N T E G R A T E FUNCTION F I ( J T ) = S Q R T ( Q I N T 4 P ( F , F N , N , 1 , N H 22 CONTINUE C C C C C C C CHECK OUTPUT S P E C T R A L I N T E G R A L A I N T = F I ( I t W R I T E ( 6 , 1 0 1 ) A I N T 101 F 0 R M A T ( 1 X , » T * F ( T * ) • , 3 X , F 7 . 5 ) C C C C C C C OUTPUT T A B L E AND P L O T F I T * ) DO 4 4 J K = 1 , 5 1 F I ( J K ) = F I ( J K l / A I N T WRITE ( 6 , 1 0 2 ) T I ( J K ) , F I < J K ) T I C J K ) = T I ( J K l / 2 5 . 0 4 4 F I ( J K > = F I ( J K ) * 5 . C 1 0 2 F 0 R M A T ( 2 X , F 6 . 0 , 3 X , F 6 . 4 ) C A L L A X I S ( 0 . 0 , 0 . 0 , « T * » , - 2 , 6 . 0 , 0 . 0 , 0 . 0 , 2 5 . 0 ) C A L L A X I S ( 0 . 0 , 0 . 0 , » F ( T * » • , * 5 , 5 . 0 , 9 0 . 0 , 0 . 0 , 0 . 2 0) C A L L L I N E ( T I , F I , 5 1 , U ) C A L L S Y M 8 0 L ( 0 . 5 , 4 . 6 , 0 . 1 4 , A , 0 . 0 , 1 2 ) C A L L PLOTND STOP 100 F 0 R M A T ( 2 F 9 . 3 ) END 147 H. Application of the Dispersion Functions The dispersion functions given in Equations (4.1) and (4.2) may be used to estimate plume spread from basic meteorologic variables, and u t i l i z ing accepted relations between these variables. Three examples follow, relying to varying degrees on direct measurement. Example 1. I f the mean wind speed and variances of the horizontal wind components are known from measurement, the plume width at a given height may be estimated as follows: TJ = 5.0 m s _ 1 a u = 0.50 m s _ 1 a v = 0.38 m s _ 1 z = 20.0 m At a travel time of 60s for example (amounting to a downwind distance of 300 m), the non-dimensional travel time i s : t* = t a / z =1.50 The dispersion function (from Equation 4.1) i s : S = 0.57 y so ay = S y t/a v = 90 m Example 2. The above example requires considerable measurement. In the absence of that degree of knowledge of atmospheric variables, a neutrally s t ra t i f ied atmosphere or one with high wind speed (> 10 m s _ 1 ; Pasquill (1974)) 148 may be quite easily treated, requiring knowledge of the mean wind speed and Counihan's (1975) equations. An estimate of the aerodynamic roughness length must be made on the basis of the surface type. This estimate may be based on Counihan's (1975) Figure 8, a mean wind speed of 10 m s - 1 over surface type 3 gives the following results. 0.2 < zQ < 1.0 m, take Z q = 0.6 m. From Counihan's (1975) Equation 4 a u = 2.28 m s _ 1 for z = 20 m and a v = 1.71 m s - 1 from Counihan's (1975) Equation 3. The values of TJ, a , a v and z can now be applied as in Example 1 to provide an estimate of a (the result f o r # t = 60 s is a y = 9.7 m). Example 3. In an unstable atmosphere, in the absence of measured wind variances, some estimate of the turbulent sensible heat f lux must be available, either through direct measurement or parameterization. The former method is demanding in terms of instrumentation and operational requirements and the lat ter is at best rough with the currently avai l-able schemes. Given a measured value of Q ,^ a value for u*/u may be es t i -mated from Pasquill's (1974) Figures 6.3 and 6.4, using a value of zQ obtained as in example 1. Pasquill's (1974) Figure 6.5 may then be used to estimate the Monin-Obukhov length L. Panofsky et a l . ' s (1977) forms for a./u* as functions of z/L and z^/L w i l l provide estimates of and G v which w i l l allow estimation of ay as in example 1. for a given wind speed. 149 I. Theodolite-Tracked, Balloon-Borne Temperature Soundings Data from the theodolite sightings (alt i tude and azimuth) together with temperatures from the mini-sonde sensors (see Appendix B.2.6) were fed into a computer program based on the method of Thyer (1962). This program generated profi les of wind-speed, wind-direction and temperature and provided estimates of the height of the balloon at each sighting. The positions of the balloon as determined by this method are subject to errors related to the geometry of the tracking system and can be prohibit ively large, as indicated by Schaefer and Doswell (1978) and Nettervi l le and Djurfors (1979). These errors are inherent in the technique and can be minimized by ensuring that the apex angle of the theodolite-balloon-theodolite triangle does not become too small. This was not a problem in this study as the most useful information came from the sonde at altidues of less than 700 m which (with a 300 m base-line) was well within the region of acceptable errors. With each height determination, the program computes two forms of error estimate. One is based on the analysis of Schaefer and Doswell (1978) which gives an estimate of the maximum probable error in height from the tracking system geometry. The second error estimate is the length of the "short l ine" which is the shortest distance between the sighting lines from the two theodolites. This quantity is a measure of the overall consistency of the sighting. These two error estimates were used as guides in inter-preting profi les from these soundings. In order to produce detailed potential temperature profi les the temperature-time output of the radiosonde receiver was digit ized at approximately 0.2 min intervals and those values fed into a computer program together with the theodolite-determined heights for the f i r s t 150 six minutes of that f l i gh t . This corresponds to approximately 1100 m of rise at the mean rise rate of 3.1 m s " 1 . The temperature-time pairs were then converted to temperature-height pairs using l inearly interpo-lated heights between the theodolite determined ones. The temperatures were then converted to potential temperature (adjusted for mean pressure changes as measured at Vancouver International Airport) and plotted as potential temperature prof i les, examples of which are shown in Figures 6.2 and K.l. The intensity of the inversion immediately above the mixed layer was extracted graphically from these profi les and used as input to the mixed layer depth model (see Sections 7.1 and 7.2). The temperature of the mixed layer was estimated from the approximately adiabatic portion of the prof i le and used for comparison with the model's prediction. The i n i t i a l temperature of the mixed layer was determined by extending the early morning capping inversion prof i le down to the measured (by acoustic sounder) i n i t i a l inversion height. This temperature was usually found to be in good agreement with the minimum morning temperature measured at the top of the tower. The use of these profi les in determining the subsi-dence rate is i l lustrated in Appendix K. One of the required variables in the implementation of the mixed layer depth model is the mean wind speed in the mixed layer. Point estimates of this quantity are available from the position of the balloon at each pair of sightings. The theodolite data is analysed by a program which provides profi les of wind speed, from which reasonable estimates for the height-averaged mean may be drawn. The hourly mean wind speeds at the top of the tower are plotted against these mixed layer values on Figure 1.1 which shows a strong relationship between the two. The linear regression equation being 151 TJ. = -0.32 + 1.02 u L tower balloon On the strength of this result , the model input is simply the hourly averaged mean wind speeds from the top of the tower. 152 Figure 1.1 : Hourly mean wind speeds from the top of the tower and from the balloon sonde. 153 J. Comparison of Acoustic and Balloon Soundings As well as providing valuable information about subsidence rates and inversion intensities the potential temperature profi les were used to indicate the position of the inversion base in relation to the wide and often diffuse band on the acoustic sounder trace. The potential temperature profi les generally showed clear discontinuities which were taken to be the inversion base (Coulter, 1979). This level was generally found to be within the elevated scattering band from the acoustic sounder. Figure J. l shows a comparison of the inversion height as measured by the acoustic sounder and as determined from the potential temperature prof i les. The diagonal line represents agreement. The data show no obvious trend and indicate that s ta t is t i ca l l y the best estimate of inver-sion height w i l l be given by the centre of the scattering band on the acoustic sounder record. 154 0-0 0-1 0-2 0-3 0-4 0-5 0-6 0-7 z i s o u n d e r , k m ' Figure J . l : Inversion height from acoustic sounder and potential temperature prof i le . 155 K. Subsidence Estimation from Potential Temperature Profiles Estimates of the horizontal divergence (see Section 6.3.1) can be obtained by observing the subsidence of features in the potential temperature profi les above the inversion base. A pair of;such profi les is shown in Figure K.l, where a "kink" in the potential prof i le is clearly defined in two soundings separated by 2 h, in this time the "kink" had subsided by 154 m. This thermal feature was clearly evident through-out that day (August 8th), exhibiting a slow downward movement. Equation (6.12) is an exact form for the subsidence, followed by the successive approximations of (6.13) and (6.14). Representing these forms by the general function w(e ,Y>a»3»z), we want to solve Clearly the f i r s t four arguments of w are unknown functions of time and the equation is insoluble. I f we presume them to be approximately constant and replace them by their mean values from Figure K.l , then substitution of (.6.13) for w renders(.K.l)soluble. The use of the approximate form (6.13) is jus t i f ied here since the conditions of the approximation are met (see Section 6.3.1). ^JT = w(e 0,Y,a,3 ,z) . (K.l) eQb + 2Y(1 -b ( e 0 + Y ( l - a)z) Integrating this leads to: b ( e o l n ( z 2 / z 1 ) + Y ( l - ct)(z 2 - 2 ] ) ) (K.2) (e Q b + 2Y(1 - a ) ) ( t 1 - t 2 ) e (K) Figure K.1: Subsidence in potential temperature profi les on August 8th. The subsiding "kink" is indicated by an arrow, in each case, with i ts height in metres. Relevant parameters are as follows -Time (LST) 1000 1200 zkink (m) 850 696 Y (K m-1) 0.0176 0.0171 ? (K) 300.1 303.1 z i (m) 315 370 157 Substituting the mean values of e , y and a and the values for z^, t-| and from Figure K.l leads to: 3 = 1.74 x IO" 5 s " 1 , a typical value in this study. I f the second (and also jus t i f iab le) approximation (Equation 6.14) is taken, a similar analysis results in 3 = 2.78 x IO ' 5 s" 1 . Similar analyses were performed for as many pairs of f l ights as possible on each day on which the inversion height model was tested, a single constant value of 3 being input for the synoptic scale subsidence for each day. • 158 L. Mixed Layer Depth Program and Sample Data FORTRAN IV program to simulate inversion r ise. D I M E N S I O N Q H Q 4 ) , A < 7 ) , Z I ( 1 4 0 ) , T H ( 1 4 0 ) , D T ( 1 4 0 ) , V P ( 3 ) , T V ( 6 ) D I M E N S I O N T H E ( 1 4 ) D I M E N S I O N T ( 1 4 0 ) , S ( 2 ) , G A M ( 1 4 ) , U B ( 1 4 ) , V ( 3 ) , Q H I ( 1 0 ) , T H V ( 6 ) C O M M O N A , P 2 , T I S , B E X T E R N A L F C C C C C C C B O Y L A Y SIM P I ^ S Y , B A C K P L O T , W I T H S U B S I D E N C E H T N 3 . C C C C C C C R E A D B O U N D A R Y C O N D I T I O N S A S H O U R L Y A V E R A G E D V A L U E S O F S E N S I B C C C C C C C L E H E A T F L U X , L A P S E R A T E , M E A N W I N D S P E E D , M E A N W I N D D I R E C T I C C C C C C C N , A N D I N I T I A L I N V E R S I O N H E I G H T , M I X E D L A Y E R T E M P E R A T U R E , D A C C C C C C C T E , TWO P A R A M E T E R S , S T A R T T I M E A N D E X P O N E N T F O R M E S 3 S C A L E C C C C C C C ' SJoS I D E N C - E , A N D M E A S U R E D T I M E A N D M E A N T E M P E R A T U R E O F M I X E D • C C C C C C C L A Y E R . R ' € A D < 5 , 1 3 0 ) Q I , ( Q H ( I ) , I = 1 , 1 4 ) , 3 F , ( G A M ( I ) , I = 1 , 1 4 1 , ( U B ( I ) , I = 1 , 1 4 ) 1 , I T H E ( I ) , 1 = 1 , 1 4 ) , 1 1 0 , T H Q , B , S ( l ) , S ( 2 ) , P l , P 2 , I T S f F B , G 0 2 , ( T V ( I ) , T H V ( I ) , 1 = 1 , 6 ) C C C C C C W R I T E ( 6 , 1 0 3 ) S ( 1 ) , S { 2 ) C = 0 . 2 0 N = 3 R E L = 1 . 0 E - 1 0 A b S = 1 . 0 E - 1 0 D T O = Q . 1 C C C C C C C L O O P T H R O U G H F O U R T E E N H O U R S DO 2 2 J = l , 1 4 C C C C C C C I N T E R P O L A T E Q H A T T E N P O I N T S W I T H I N T H I S H O U R . DO 7 7 I K = 1 , 1 0 I F ( J . £ Q . 1 ) G 0 T O 9 9 Q i = Q H < J - 1 ) GO TO 5 5 9 9 Q 1 = Q I 5 5 Q 2 = Q H ( J ) I F ( J . E Q . 1 4 J G 0 T O 4 4 Q 3 = Q H ( J + l ) GO TO 2 2 2 4 4 Q 3 = Q F 2 2 2 I F ( I K . G T . 5 J G 0 TO 8 8 D Q = ( Q 2 - Q 1 ) / 1 0 . 0 Q H I ( I . < J = Q 1 + D Q * F L 0 A T ( I K + 5 ) GO T O 7 7 8 8 U Q = 1 Q 3 - Q 2 ) / 1 0 . 0 Q r l I ( I O = Q 2 + D Q * F L 0 A T ( I K - 5 ) 7 7 C O N T I N U E C C C C L . C L O O P T H R O U G H T E N S I X M I N U T E I N T E R V A L S DO 1 1 K = l , 1 0 J K = i O * ( J - 1 ) + K J K 1 = J K - 1 I F ( Q H H K ) . L T . 0 . 0 . A N D . J . E Q . 1 J G 0 T O 6 6 I F ( Q r i K K ) , L T . 0 . 0 ) G 0 TO 8 7 8 i F L A G = 1 C C C C L C C C O M P U T E F E T C H O F U R B A N S U R F A C E F O R E L L I P T I C A L C I T Y R S = 7 . U 0 T H E S = 0 . 3 9 3 7 I F ( T H E ( J ) . L T . 9 0 . 0 ) GO TO 9 0 1 159 P H I = ( T H E ( J ) - 9 0 . 0 ) * 1 . 7 4 5 3 E - 0 2 G O T O 9 0 2 9 0 1 P H I = ( 2 7 0 . 0 + T H E ( J ) ) * 1 . 7 4 5 3 E - 0 3 * 0 2 C S = C G S ( P H I ) * * 2 S I = S I N ( P H I ) * * 2 R P = S i R T ( 7 0 5 6 . 0 / ( 1 9 6 . 0 * S I + 3 6 . 0 * C S ) ) D X = 1 0 0 0 . 0 * ( ( R P * C O S ( P H I ) + R S * C O S { T H E S ) ) * * 2 + + ( R P * S i N ( P H I ) + R S * S I N ( T H E S ) ) * * 2 ) * * 0 . 5 : C C C C C C S E T C D E F F I C I E N T S A l T O A7 A ( 1 ) = 3 H I ( K ) * ( 1 . 0 + 0 / 1 2 1 2 . 0 A ( 2 ) = 0 . 7 i 7 1 * S Q R T(UtJ{ J ) * Q H I ( K ) * P l * G A M ( J ) / ( 1 2 1 2 . 0 * D X ) ) A ( 3 ) = C * Q H I ( K ) / 1 2 1 2 . 0 I F ( J K . G E . I T S J G O T O 6 A ( 4 ) = B G O T O 7 b A ( 4 ) = B * E X P ( F 8 * 0 . 1 * F L 0 A T ( J K - I T S ) ) 7 At 5 ) = S c 3 R T(UB( J ) * Q H I ( K ) * ( 1 . 0 + 2 . 0 * 0 / ( 2 4 2 4 . 0 * G A M { J ) * P 1 * D X ) ) A ( 6 ) = G AM { J ) * P 1 A ( 7 ) = B * G 0 T I S = 3 6 Q . 0 * F L O A T ( J K ) T I = 0 . 0 T F = 3 6 0 . 0 ; c c : c c c S E T A R R A Y V A L U E S O F I N V E R S I O N H E I G H T , M E A N T E M P E R A T U R E A N D C C C C C C C T E M P E R A T U R E S T E P T O I N I T I A L O R C O M P U T E D V A L U E S . I F ( J K 1 . G T . O J G O T O 6 7 V ( 1 ) = T H 0 V ( 2 ) = Z I 0 V ( 3 ) = D T 0 G O T O 1 67 V ( 1 ) = T H ( J K 1 ) V< 2 ) = Z I ( J K 1 ) V ( 3 ) = D T ( J K 1 ) C C C C C C C C O M P U T E D E R I V A T I V E S FOR O U T P U T V P ( 2 ) = A ( 3 ) / V ( 3 ) - A ( 4 ) * V ( 2 ) - A ( 5 ) V P ( 1 ) = A ( 1 ) / V ( 2 ) - A ( 2 ) V P ( 3 ) = A ( 6 ) * V P ( 2 ) - V P ( 1 ) + A ( 7 ) * V ( 2 ) * E X P ( B * T I S ) * R I T E ( 6 , 1 0 4 ) J , K , ( V ( I J , 1 = 1 , 3 ) , ( V P ( I ) , 1 = 1 , 3 ) , ( A ( I ) , 1 = 1 , 7 } C C C C C C C U S E N U M E R I C A L S O L U T I O N O F D I F F E R E N T I A L E Q U A T I O N S TO C O M P U T E C C C C C C C NErt V A L U E S OF T H E T H R E E V A R I A B L E S . I C A L L D E ( F , N , V , T I , T F , R E L , A B S , I F L A G ) G O T Q < 3 , 4 , 1 , 1 , 1 , 3 ) , I F L A G 3 W R I T E ( 6 , 1 0 1 ) J , K , I F L A G , T I , T F , R E L , A B S , N G O TO 3 3 4 T H ( J K ) = V ( 1 ) Z I ( J K ) = V ( 2 ) , D T ( J K ) = V ( 3 ) G O TO 11 6 6 D T ( J K ) = D T O Z I ( J K ) = Z I O T H ( J K ) = T H O I I C O N T I N U E 22 C O N T I N U E C C C C C C C L O O P T H R O U G H A R R A Y S FOR P L D T T I M G . 8 7 8 D O 111 M= 1 , J K 1 T H S = T H 0 - 2 . 0 T H F = T H 0 + 1 8 . 0 160 T ( M ) = 1 . 0 + F L O A T ( M ) / 2 0 . 0 I F ( T H ( M ) o L T . T H S ) T H ( M ) = THS I F ( T H ( M ) . G T . T H F ) T H ( M ) = T H F T H ( M ) = 6 . 0 + 1 T H < M ) - T H S ) / 1 0 . 0 111 Z I ( M ) = 1 . 0 + Z I ( M ) / 2 0 0 . 0 CCCCCCC ALL 0ASHL;M( . 0 7 , . 0 7 , . 0 7 , . 0 7 ) CALL A X I S t 1 . 0 , 1 . 0 , • I N V E R S I O N HEIGHT < M ) • , 2 0 , 5 . , 9 0 . , 0 . , 2 0 0 . ) CALL AXIS< 1 . 0 , 1 . 0 S O L A R TIME ( H ) * , - 1 4 , 7 . 0 , 0 . 0 , 5 . 0 , 2 . 0 ) CALL A X I S ( 1 . 0 , 6 . 0 , ' T H E T A ( K ) • , 9 , 2 . , 9 0 . , T H S , 1 0 . ) CALL P L O T ( 1 . 0 , 8 . 0 , + 3 ) CALL P L O T ( 3 . 0 , 8 . 0 , + 2 ) C A L L P L 0 T ( 8 . 0 , 1 . 0 , + 2 ) CALL P L O T ( 8 . 0 , 6 . 0 , + 3 ) CALL P L O T ( 1 . 0 , 6 . 0 , + 2 ) CCCCCCC READ D I G I T I S E D VALUES OF INVERSION HEIGHT FOR PLOTTING. 444 READ( 5 » 1 0 5 , E N D = 3 3 3 ) T D , Z D , N P E N T D = ( T D - 5 . 0 ) * 0 . 5 + 1 . 0 Z D = Z D / 2 0 0 . 0 + 1 . 0 CALL PLOT(TD,ZD,NPEN) GO TO 444 333 CALL L I N E ( T , Z I , J K 1 , + 1 ) 1 = 0 777 1=1+1 I F ( THV( I ) . EQ.0.0.).GQ_TQ 888 T V T = ( T V ( I ) - 5 . 0 ) * 0 . 5 + 1 . 0 T H T = 6 . 0 + ( T H V ( I ) - T H S ) / 1 0 . 0 CALL S Y M B O L ( T V T , T H T , 0 . 0 7 , 1 , 0 . 0 , - 1 ) GO TO 777 888 CALL L I N E ( T , T H , J K 1 , + 1 ) CALL S Y M B O L ( 1 . 2 , 7 . 6 , 0 . 1 4 , S , 0 . 0 , 8 ) CALL PLOTND 33 STOP 100 F 0 K M A T ( 1 0 F 6 . 1 , / , 6 F 6 . 1 , / , 3 ( 1 0 F S . 1 , / , 4 F 6 . 1 , / ) , 2 ( 1 X , F 5 . 1 ) , 1 X , 1 E 7 . 1 , 2 A 4 , 2 ( 1 X , F 4 . 2 ) , 1 X , I 3 , 1 X , F 4 . 2 , 2 X , F 6 . 1 , / , 6 ( F 6 . 2 , F 6 . 1 ) ) 131 F O R M A T ( 1 0 X , 2 ( 1 2 , 2 X ) , « E R R O R IN DE - CODE • , 1 2 , 2 ( 2 X , F 6 . 1 ) 1 , 2 ( 2 X , E 1 4 . 4 ) , 2 X , I 3 ) 103 FORMAT(2X, • INVERSION RISE FOR » , 2 A 4 ) 13 4 F O R M A T ! 2 ( 1 X , I 2 ) , 6 X , 2 ( 1 X , F 6 . 1 ) , 1 X , F 6 . 3 , 3 ( 1 X , E 1 4 . 7 ) 1 , / , 7 ( 1 X , E 1 4 . 7 ) ) 105 F 0 R M A T ( 2 F 9 . 3 , I 2 ) END SUBROUTINE F ( T I , V , V P ) CCCCCCC ROUTINE FOR COMPUTING DERIVATIVES (CALLED BY D E ) . COMMON A , P 2 , T I S , B DIMENSION V ( 3 ) , V P ( 3 ) , A ( 7 ) V P ( 1 ) = A ( 1 ) / V ( 2 ) - A ( 2 ) V P ( 2 ) = A ( 3 ) / V ( 3 ) - A ( 4 ) * V < 2 ) - A < 5 ) VP( 3 ) = A ( 6 ) * V P ( 2 ) - V P ( 1 ) + A ( 7 ) * V ( 2 ) * E X P ( B * T I S ) RETURN END 161 -26.7 24.3 67.4 104.6 170.9 170.4 210.3 213.7 147.7 239.2 369.3 261.8 161.0 113.7 10.8 -14.3 .0200 .0320 .0440 .0368 .0248 .0128 .0134 .0144 .0154 .0165 .0175 .0185 .0195 .0196 1.4 2.4 2.6 2.8 2.4 2.6 2.6 2.7 3.3 4.9 4.3 4.2 4.6 3.9 134.0 145.0 162.0 158.0 166.0 182.0 179.0 2C4.0 224.0 152.0 150.0 149.0 146.0 138.0 35.0 290.0 5.2E-06 AUG 01 0.70 1.00 65 0.45 0.0124 7.70 291.4 10.53 294.4 15.62 296.4 5.081 45.841 3 5.127 64.558 4 Sample input data for inversion rise simulation. 


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