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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  this  an a d v a n c e d  degree  the  shall  Library  I further for  scholarly  by h i s of  agree  this  written  thesis at  the U n i v e r s i t y  make  that  it  purposes  for  freely  permission  It  financial  of  British  of  Columbia,  British for  gain  Columbia  the r e q u i r e m e n t s  reference copying  of  I agree and this  shall  that  copying  n o t be a l l o w e d  or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  Depa r t m e n t  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  of  for extensive  permission.  The U n i v e r s i t y  fulfilment  available  may be g r a n t e d  representatives. thesis  in p a r t i a l  or  publication  w i t h o u t my  ABSTRACT The rate of dispersion of atmospheric pollutants and the volume of atmosphere available for the dilution of pollutants are examined in an unstable suburban atmosphere at a coastal location. Within the framework of the s t a t i s t i c a l theory of diffusion, i t can be shown that the non-dimensional dispersion functions y / ° t a  v  a n c  '  a / a t can be determined by integration of the Eulerian spectral functions z  w  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 time scale.  = z/o^ which is simply related to the Lagrangian integral  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.  S (t*)  = (1.0 + 1.21/t*) - l  The spectra and integral turbulence s t a t i s t i c s 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 dilution of pollutants is controlled primarily by the mean wind speed and mixed-layer depth.  This  l a t t e r 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  iii  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 l a t t e r phenomenon being linked to thermally driven circulatory systems.  The  inclusion of these processes into the mixed-layer depth model allows i t s application to areas in which meso-scale phenomena may have a considerable effect on the diurnal behaviour of the mixed-layer depth. The model thus derived consists of a system of non-linear different 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 t y .  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 r e a l i t y in the presence of synoptic-scale non-stationarities.  The input requirements of the model  are hourly values of surface sensible heat f l u x , mean wind speed and upwind distance to the surface giving rise to the advected heat flux (usually a coastline or urban-rural boundary), and estimates of the intensity of the capping inversion and horizontal divergence.  The model is sensitive to all  input variables, the degree of sensitivity being indicated by the dependence of the maximum mixed-layer depth on the measured boundary conditions.  iv TABLE OF CONTENTS Page Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii  Acknowledgements  xii  1.  Preface  1  1.1  Rationale  1  1.2  Objectives  1  Turbulent Diffusion  3  Introduction  3  2.1  Specification of Diffusion Parameters  4  2.2  Gaussian Plume Model Parameters and the  Part One: . 2.  Statistical Theory of Diffusion 3.  6  An Application of the Statistical Theory of Diffusion to Measured Flows 3.1  3.2  3.3  10  The Dispersion Functions from Measured Eulerian Spectra  10  3.1.1  Eulerian-Lagrangian Transformations  10  3.1.2  Transformation of the Dispersion Integral  12  Eulerian Turbulence Functions over a Suburban Surface in Unstable Conditions  13  3.2.1  Integral Statistics  13  3.2.2  Spectra  22  Computation of the Dispersion Functions  25  3.3.1  25  Computational Details  V  4. Part Two: 5.  6.  7.  3.3.2  Crosswind Spread  26  3.3.3  Vertical Spread  33  Conclusion  36  The Depth of the Daytime Mixed Layer Introduction  39  5.1  Specification of the Mixed Layer Depth  39  5.2  Mathematical Modelling of the Mixed Layer Depth  41  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  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  L.  Profiles  155  Mixed Layer Depth Program and Sample Data  158  vii LIST OF TABLES Page, Table 3.1a  vs o^/u^; i = u,v,w.  Table 3.1b  t, vs a^/u^: i = u,v,w.  Table 3.2  Relation Between z/L classes and Pasquill-Gifford Classes.  Table 3.3  16  17  Least Squares Fitted Parameters to Equation (3.7) For Three Components.  Table 3.4  16  17  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 Sensitivity.  81  Table A.l  Sectorial Analysis of Roughness Length.  115  viii  LIST OF FIGURES Page Figure 3. 1  Non-dimensional Integral Alongstream Turbulence Statistics as Functions of Surface Layer Similarity Variables  Figure 3. 2  Non-dimensional Integral Crosswind Turbulence Statistics as Functions of Surface Layer Similarity Variables  Figure 3. 3  19  20  Non-dimensional Integral Vertical Turbulence Statistics 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 vs Maximum Surface Sensible Heat Flux  78  Figure 7. 15 Maximum Inversion Height vs Mean Wind Speed in Mixed Layer  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  no  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 S t a b i l i t y Class  142  Figure F.,2 a,b,<  Construction of Composite Spectra  143  X  Figure 1.1  Mean Wind from Tower and Balloon Sonde  Figure J.l  Inversion Height from Acoustic Sounder and Potential Temperature Profile  Figure K.l  152  154  Subsidence in Potential Temperature Profiles on August 8th  156  xi ACKNOWLEDGEMENTS I am grateful to the academic community of The University of British Columbia and in particular my colleagues in the Geography Department of that i n s t i t u t e for providing an eclectic and stimulating intellectual environment in which to conduct my studies.  In particular, 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 gathering phase. appreciated.  Their efforts with the tracking theodolites are especially Sheila Loudon served ably as a computing assistant.  Brian  Kalanda provided valuable advice on the operation of the differential psychrometer system. Research grants by the Natural Sciences and Engineering Research Council of Canada and a Scientific Subvention from the Atmospheric Environment Service of Environment Canada to Dr. T.R. Oke covered the considerable funding requirements of the f i e l d work.  I was personally supported  by teaching assistantships, a summer research fellowship and l a t t e r l y by a Killam Predoctoral Fellowship, a l l from The University of British Columbia. The British Columbia Hydro and Power Authority gave permission to use their Mainwaring Substation as a research s i t e .  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 t s 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 turbulence data and the Institute (now Department) of Oceanography at The University of British Columbia allowed free access to their analogue to d i g i t a l convertor and mini-computer for the digitization of those data. Richard Leslie w i l l i n g l y b u i l t the bridge-amplifier and active f i l t e r s and provided much guidance on matters electronic.  1 1.  Preface 1.1  Rationale The concentration of pollutants in the atmosphere and at the  surface of the Earth is determined primarily (apart from source strength v a r i a b i l i t y ) by the rate of dispersion into the atmosphere and by the volume of atmosphere available for d i l u t i o n .  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 w i 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 city from which the study w i l l draw i t s data is Vancouver,  British Columbia, Canada, which has a mid-latitude coastal location.  The  results w i l l thus be characteristic of this situation, but are not expected to be specific to any particular feature of the chosen c i t y .  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 r e a l i t y 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 t a t i s t i c a 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 w i l l be to develop a mathematical 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 t e .  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 t s 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 d i s t i n c t lines, each developed from a different theoretical base.  The Gradient. Transfer approach is really a first-order closure  scheme which relates mass fluxes to mean velocity gradients by an eddy diffusivity.  The approach has a comforting feel because of the similarity  i t bears to the classic Fickian (molecular) diffusion framework.  The  crippling flaw of this approach is that the eddy d i f f u s i v i t y 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 diffusion, even i f i t s influence lingers strongly enough to prompt Scorer s (1976) warnings 1  against i t s 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 specification of the three component K's remains a problem.  A variety of more  or less r e a l i s t i c 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 similarity arguments to derive forms of K based on nondimensional 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 Willis (1975).  An alternative  view of this approach is to treat the concentration distribution as a  5 function of the chosen non-dimensional groups (Gifford, 1975).  However,  this a b i l i t y to short-circuit process is a property of the similarity theory, rather than the phenomenon. Diffusion from a continuous source may be treated from a purely s t a t i s t i c a l viewpoint in what is known as the Taylor (1921) Statistical Theory.  This approach resolves many of the d i f f i c u l t i e s of the  other two p o s s i b i l i t i e s , 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 distribution.  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 t a b i l i t y type.  The six Pasquill turbulence types (Pasquill,  1961) form the most convenient operational scheme and can be related (Golder, 1972) to more basic s t a b 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 different surfaces to be represented by different 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 curves in analytic form for diffusion over urban surfaces.  z  Within this  formulation, the need to derive diffusion parameters directly from observations of atmospheric diffusion makes their determination tedious, time consuming and subject to large s t a t i s t i c a l v a r i a b i l i t y (all these factors being inherent drawbacks of that kind of observation).  The Gaussian  model i t s e l f remains ( i f properly used) a peerless mathematical tool for estimating diffusion because i t is simple, flexible 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 Statistical Theory of Diffusion Taylor (1921) in his s t a t i s t i c a l theory showed that an ensemble  average of particle displacement under the influence of a stationary, homogeneous turbulent flow w i l l have a variance given by: V  aj=2af,/  /  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 i m 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  V[  l a'  =^  2  \  ' * „ i (n){stnUnt)/(Trnt)} dn \,L<" 2  (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 diffusion, 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/tL)  (2-3)  where t^, the Lagrangian integral time, scale is given by: t= t  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 s t a t i s t i c a l theory, and has met with general approval among the research community active in this f i e l d (Hanna et a l . , 1977 and Randerson, 1979). The detailed specification of S thus remains the major objective.  Two limiting values of S are: S+1.0 as t*0 S^(2t /t) / 1  L  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 distinct ways (Pasquill, 1975b).  8  Closed mathematical forms for $ (n) or R(x) may be substituted L  into the integral in equation (2.2), which w i l l then yield s(t/t ). L  This method has been illustrated by Pasquill (1975b) who uses a variety of forms suggested for R(x) and tabulates s(t/t ). L  Direct observation of y / ° t over a range of values of t so a  v  that the large t l i m i t can be used to find 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 distinct  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 vertical) 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 ratio of Lagrangian to Eulerian integral time scales.  This  method has been applied (Hay and Pasquill, 1959, and Haugen, 1966) in order to test the v a l i d i t y of a particular form of the Eulerian-Lagrangian transform, u t i l i z i n g 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 ( t / t ) over f l a t grassland and shows that y  L  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 Statistical 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 t u a l l y a l l atmospheric measurements are Eulerian in nature.  This  conflict of viewpoint would be easily resolved i f some theoretical transformation 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 i n t u i t i v e hypotheses, each having i t s own set of (often unclear) limitations.  For the purposes of this study, the most convenient  formulation of an Eulerian-Lagrangrian transform is one which addresses the ratio of the integral time scales from the two frames of reference. The simplest approach to the integral time scale ratio is provided by the "frozen eddy" hypothesis which suggests (Pasquill, 1974) that: t /t L  E  = 1/i = U/a  u  where t ^ / t ^ is the ratio of the Lagrangian to Eulerian integral time scales, i is the turbulent intensity and is equal to the r a t i o 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 s u f f i c i e n t l y 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 /t L  where  E  = gu/a  (3.1)  u  3 = 0.80  (Saffman)  g = 0.35  (Philip)  (note that Hay and Pasquill (1959) use a different g, v i z ; g = t / t ) . L  £  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 similarity  theory and s t a t i s t i c a l theory forms of eddy d i f f u s i v i t y 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 t a t i s t i c a l approach introduced by Ariel and Buttener (1966) to determine the  Lagrangian velocity autocorrelation  function from Eulerian wind s t a t i s t i c s 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 t a b 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  ( n ) { s i n U n t ) / U n t ) } dn  (3.2)  2  v > L  •'0  The Lagrangian-Eulerian transform takes the form (Pasqui11,.1974) $l^(n) = r $ ( r n ) £  where  r = t ^ / t ^ = u/2a  u  Applying this transform to (3.2) gives: S j ( t ) =j where  $  F  $ (n){sin(2Trta n/u)/(27rta n/u} dn  (3.3)  2  VjE  u  u  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 s t a t i s t i c s 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) = / •'o  $  V5?£  (f){sin  (2 rta f/z)/(2TTta f/z)Fdf 1  u  u  this may be rewritten as: r  Sj(t*). =•/ * o;  oo  $  ( f ) { s i n (2Trft*)/(2^ft*)} df 2  Vj[£  (3.4)  13  where t* = t / t t  = t a / z is the non-dimensional travel time scaled by u  = / o - This form of scaling arises naturally in the integral, and is z  u  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 f e r  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 s t a t i s t i c s 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 Statistics  In accordance with the Monin-Obukhov similarity 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  where u* is the surface f r i c t i o n velocity, the dimensional functions and x,  =  (3.5) are a set of non-  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  / u  *  =  Vv i ( z  (3.6b)  / L )  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 i m i t o f t h e ratios o^/v* and Counihan (1975) summarises the values as 2.5, 1.9 and 1.3 for u, v and w respectively. not appear to depend on height. moment closure,  These ratios do  Binkowski (1979) develops a simple second-  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 e x p l i c i t reference to the character of the underlying surface.  His func-  tions compare well with the data from two independent sets of f i e l d measurements which show wide scatter for the horizontal components in unstable cases (as pointed out above). A fundamental bar to compilations and comparisons of integral s t a t i s t i c s 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 f u l l range of micro-meteorologic fluctuations ( i . e . , from the centre of the spectral gap at f ^ 6xl0~ (Smedman-HOgstrom and 5  Hogstrom, 1975) to the high frequency end of the i n e r t i a 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. Turbulence measurements in this study were made over a carefully selected suburban s i t e , using a Gill UVW anemometer mounted on a freestanding steel tower at an effective height of ^ 20 m into the surface  15 layer (see Appendices A and B).  The integral s t a t i s t i c s were calculated  directly from 62 blocks of data each containing 8192 data points sampled at 2.5 Hz (see Appendix F for details of the analysis). yield a non-dimensional frequency range of 3.0xl0" <f<50. 3  These data 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 t a b 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 verified 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 increasing i n s t a b i l i t y with a great amount of scatter.  This scatter is inherent  in a l 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 t s mean ratio of o^/u* and of  the analysis being repeated to give seven classes  This process of combining s t a t i s t i c s 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. this study the range of U was low (a mean of 2.5 m s standard deviation of 1.1 m s "  1  - 1  However, and a  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 corresponding means and standard deviations of the ratios a - / u * .  16  Table 3.1a: e-vs o . / u * ; i = u,v,w Class  C  a /u^  a /u^  a /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:  i  u  v  w  ?vs a . / u * ; i = u,v,w.  Class  a  u * /u  V * u  a  ,w/ * * u  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 t a b i l i t y 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)  (3.7)  C  can be f i t t e d 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 e d 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*  ?  5.57 3.18  17.94 0.02  0.30 0.75  a /u* v  c  0.00  36.81  0.27  ?.  5.23  0.10  0.51  ?  0.00  19.05  0.25  ?,  2.49  0.41  0.35  u  w  a /u*  w *  18 The adiabatic limits for the ratios a^/u^ are not e x p l i c i t in the data since no t r u l y 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~ , 15.0) (Clarke, pers. comm.) 3  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-dimensional Wind Velocity Standard Deviations. This Study  <V * U  ° / * u  Clarke et a l .  Coppin  Counihan  (1975)  (1979)  (1975)  2.H0.5  2.39  2.5  2.5  1.7+0.5  1.79  -  1.88  1.3±0.2.  1.26  1.1  1.25  v  °w * / u  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  2-0  3*0  log(" / ] z  L  0-0  1«0 log(' / ) Z|  L  Figure 3 . 1 :  Non-dimensional Integral Alongstream Turbulence Statistics 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  log("\)  Figure 3.2:  Non-dimensional Integral Crosswind Turbulence Statistics as Functions of Surface Layer Similarity Variables. (solid lines are equation 3.7 with parameters given in Table 3.3)  21  0-0  log{  log(  Figure 3.3:  _  z  \)  1-0  20  \ )  Non-dimensional Integral Vertical Turbulence Statistics 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 fields 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 representation 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 l d .  Surface layer similarity theory predicts that the normalized  spectra w i l l be functions of a dimensionless frequency f = nz'/u (where n is frequency and z a length scale). 1  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. ized by division by u ^ ^ 2  and  3  His spectra are normal-  where u* is the surface layer f r i c t i o n velocity  is the non-dimensional dissipation rate, the data being collected  over " f l a t featureless t e r r a i n " .  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), DCicheneMarullaz (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 fluctuations have similar details of structure as do those over f l a t (low roughness 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 frequencies 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 d e t a i l ) .  24  0-0  1 U  region  3  -  i region 2 X=25z  x=42z ^  ^  ^  1  1 region 1  ^ „X = 4z  -  -2-0  -3-0  I  'V'  i  I I  1  region  - / ^  3  I I region 1  i region 2 \ = U7z  •5z  X=72z  -  I I  -3-0 'W  '  x  = 1 0  1 I  z  -1-0  -  -2-0  1 -3-0  -2-0  1 -1-0  0-0  1-0  2-0  log(f)  Figure 3.4:  Energy Density Spectra f o r 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  "u" Kaimal (1978)  2z  0.67z  0.5z  47z  "v" Kaimal (1978)  Iz  0.25z  "w" This Study  -  -  lOz  "w" Kaimal (1978)  -  -  6z  "v" This Study  42z i  1.6z. 72z  i  1.6z.  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 digitised 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 library routine in The Univers i t y of British 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 forwardand backward-difference schemes (Madderom, 1978).  The l i s t i n g 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 corresponding to t = 54 min (the length of the turbulence data blocks) with a representative upper l i m i t for a of 0.8 m s " u  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. 3.3.2  This w i l l be referred to later.  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 linearly extended (in log-log space) to an amplitude of -1.75 at a f r e 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 r e quency end.  As i n t u i t i v e l y 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  I  L  100-0  125-0  150-0  t*  Figure 3.5: O a  Crosswind Dispersion Function.  Draxler's (1976) function Measured urban spectrum  •  b  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 l a t t e r in a simple manner.  Hunt and  Weber (1979) present the relation: t, =0.33 z/cr L  W  Busch (1973) reports a Ja„ to be constant at 0.52 for a l l unstable condiw u tions.  The data of this study show this ratio to scatter widely between  0.4 and 0.8.  Taking the value of 0.52 leads to t  where t  s  = 1.58t  (3.7)  L  = z / a , our scaling time.  From this i t can be seen that t  u  L  (and t- ) is i m p l i c i t l y a function of s t a b i l i t y as suggested in Section 3.2. s  Draxler (1976) scales his travel time with t^, the time at which the dispersion function equals 0.5.  From Figure 3.5, S(47.17) = 0.5000, which  means that t  = 47.17 t  s  (3.8)  t . = 74.53 t  L  (3.9)  i  or, using (3.7),  Equation (3.8) was used to transform Draxler's (1976) recommended function. S U / ^ . ) = (1.0 + 0 . 9 ( t / t ) / ) " 1  2  1  i  at selected points onto Figure 3.5.  The agreement between these points and  curve b is excellent, with only very slight 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 i m i t of his function to relate t^ to t^ and shows that t . = 1.64t in conflict with (3.9).  (3.10)  L  A possible reason for this disagreement is the  uncertainty inherent in extrapolating an empirically f i t t e d function to i t s limiting value at i n f i n i t y . implying  Draxler (1976) suggests t^ = 1000s, thus  = 610s by equation (3.10), or t  L  = 13s by equation (3.9).  While their constancy is an unrealistic simplification, the l a t t e r value is in good agreement with the prediction of equation (3.7), using a typical value of CT = 0.75 m s " u  1  (this corresponds to Ti" = 2.5 m s  lent intensity of 0.30), giving t  = 12.7 s.  L  _ 1  and a turbu-  Fichtl and McVehil  (1969)  suggest that' t^ may be approximated by: t  ,  L  _ 'Hnax 2nu  where x , „ is the wavelength maximum of the Lagrangian "u" specturm. max From the spectra of this study (Table 3.4), this leads to m  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 : L  L  =  42XL58  i 2  IT  This r a t i o 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 time scale.  as a  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 <f<0.2, the.same width (in log space) and extending to 2.5 _lf  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 t a 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. Since  We want x = TTt. t* = t a / z ,  we have  u  x = yt*  which for this study is equivalent to x = 66.7 t* i f z = 20.1 m and i = 0.30.  (3.11)  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 i o n , 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 s t a t i s t i c a 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 universality, 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 i m i l a r i t y 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 r o l l off as approximately of the i n e r t i a l subrange.  ra  t h e r than the expected -2/3 behaviour  This is presumably due to the i n a b i l i t y of the  sensors to respond to high frequency fluctuations.  In order to quantify  this shortcoming, the "v" spectrum was extended linearly (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 significant 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 ( t * ) = (1.0 + 0.16 T t * ) "  1  y  This form may then be used to determine crosswind plume width (see Appendix 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 t s use for determining a . z y  The  3  major problem lies in the vertical inhomogeneity of the atmosphere which reduces the results to approximations at best.  These approximations are  l i k e l y to be good for elevated releases in unstable atmospheres, and poor for ground-level releases in unstable atmospheres and elevated releases in stable atmospheres. time.  A secondary problem lies in the choice of scaling  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 Willis (1975) and Hanna et a l .  (1977)) where z^ is the depth of the mixed layer, and w* = (|: w e 1  1  z^.) / 1  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 flux.  body of data to relate c /w* to z/z^ in the form w  Irwin (1979) collects a  3  34  <V * W  =  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 find that 1.92a (z/z.) 1-b  t /t u s  Our data are generally in the range 0.03<z/z^<0.3  (see Part Two),  where a = 0.72, b = 0.21, giving y t  = 1.38(z/ ) 0.79  s  Zi  yielding 0 . l < t / t < 0 . 5 u  s  A s t a t i s t i c a l analysis of data from the 62 data blocks shows this ratio 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 ratio 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. The curve on Figure 3.7 is well represented by  which may be used to determine vertical plume dimensions for input into Gaussian plume model dispersion calculations.  O-o  1  1  0-0  25-0  1  '  50-0  75-0 t*  Figure 3.7:  Vertical Dispersion Function. O - Irwin's (1979) values.  1  100-0  '  125-0  •  150  36  4.  Conclusion The results of this part of the study have shown that the Gill  UVW anemometer can be successfully used to measure the turbulent structure of the unstable atmosphere over a very rough surface (z ^0.5m). Q  Its  response shortcomings, particularly in the vertical sensor, are partly ameliorated by the relatively 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 t a b i l i t y 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 t a t i s t i c a l theory of diffusion, i t was shown that the non-dimensional dispersion functions y/° ^ a  v  and  a / a t can be determined by integration of the Eulerian spectral functions z  w  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 fields whose mean properties are uniform in both space and time, and does not take account of wind shear as a means of crosswind diffusion.  The dispersion functions so produced agree very well with  previous forms and are:  37  and  sy(t*)  = (.1.0 4- O.T6/t^)"  C4.1)  s (t*)  (1.0+ 1.21/t*) - l  (4.2)  z  The numerical coefficients being least squares f i t t e d 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 unrealistically 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 and y  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 s e 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 altitudes.  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 i c t i o n a l and thermal effects of the underlying surface extend.  This layer is commonly in a state of highly turbulent  motion which f a c i l i t a t e s the uniform mixing of entropy and gaseous atmospheric 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 s e 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 i m i t but can  be defined as the height at which the turbulent fluxes (resulting from surface effects) have fallen 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 i m i 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 t s own definition of the exact height (Coulter, 1979).  The principal methods of measurement being to sound directly some  mixed layer parameter (usually temperature), and thereby detect the discontinuity at the inversion base, or to remotely detect some effect of the entrainment process.  Rather than full-scale f i e l d measurement programmes,  the atmospheric boundary layer can also be modelled on a laboratory scale so as to elucidate i t s 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; M i l l e r , 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, inversioncapped, 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 Tennekes (1976) approval arose from 1  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 flux at the inversion 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 e n t r a p 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 i o n velocity (see Davidson et a l . , 1980). The model equations are: (w'e')  s  (w'e')  -  i  (5.1) (5.2)  dz. dt  '.dA  dt  -  c(w'e')  s  (5.3) (5.4)  42 where e is the mean potential temperature of the mixed layer, w'e' is a kinematic sensible heat flux where e and w' are the fluctuating compon1  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 (vertical) 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 geometry of an idealized mixed layer potential temperature p r o f i l e .  Equation  (5.4) is a parameterization of the heat flux at the inversion base from the surface layer heat f l u x , 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 t u 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 t u l l , 1976a). Carson (1973) shows how equations (5.1) to (5.3) can be solved analytically using a simple sinusoidal surface heat flux.  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 distinct phases in the evolution 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. for c are in the range (0.1 - 0.2).  His values  Most recently Caughey and Palmer  (1979) present a direct measurement of the vertical profile of turbulent sensible heat flux 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 first-order model. The basic ideas of the model have been applied to idealized metropolitan areas by including an advected heat flux term in equation (5.0)(Barnum and Rao, 1975) in order to simulate thermal internal boundarylayer 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 t u 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 until 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  until approximately eight hours after sunrise, by which time the inversion has risen to i t s 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 i n a b i l i t y of the sensor to respond to signals scattered from upper parts of these "plumes" which are presumably decreasing in a c t i v i t y 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  Figure 6.1:  TIME  (HOURS)  Acoustic sounder trace for August 1st.  46  Z: (X ) 3  .C  1  Zilx.J  eix,) Potential  etxj  e(x,j  Temperature  Figure 6.3: Potential Temperature Profiles at Various Distances Distances from the Upwind edge of a Thermal Internal Boundary-Layer. 1000-0  800-0  h  600-0  h  40 0-0  h  20  h  0-0  0-0 291-0  299-0  Figure 6.2: Potential Temperature Profiles for August 1st.  47 S t u 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 f a 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 i m i 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 digitized at approximately ten minute intervals for comparison with the model results. Potential temperature p r o f i l e s , such as those shown in Figure 6.2, were used as verification 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 latitude.  The mean inversion intensity (immed-  iately above the mixed layer) from a l l balloon soundings was 0.019 ± 0.009 K m" , in good agreement with the San Jose figure of 0.012 K m . 1  -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. characteristic;  Equation (5.1) expresses the thermal component of this 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 possibility of non-zero horizontal divergence due to advected heat fluxes.  Following Barnum and Rao (1975), we may  rewrite equation (5.1) as:  z  where Q = (w'e )  iljt:  and  1  =  ( 1  +  _=  Q  ( 6  + TJ^- ,  L)t  S  c )  ot  -  n  x being the upwind distance to  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. ~DT  A  where  vi{z.)  =  c Q  +  A w ( z  i)  ^6.2)  is a vertical velocity of as yet unspecified origin. 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 ,  (6.3)  0  where ?  0  is the early morning value of the potential temperature at what  w i l l become the lower l i m i t of the mixed layer.  I t (~e ) is assumed inde0  pendent of space and time, so that: DA Dt  (6.4)  where W(z -,t) is the heating effect of synoptic scale processes experi1  enced at the top of the mixed layer. In order to find the spatial behaviour of z^ and e , an approximate f i r s t integral of equations (6.1), (6.2) and (6.4) must be found. This should produce more r e a l i s t i c 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: x  then results in  _9_ 3t  _3_ 9T  and  _9_ 8X  = t - x/u, 1 9  _D_ Dt  with u constant. The model equations therefore become:  _9_  ay  50  9A  _ dQ_  =  ay  (6.7)  9y " a y  Y  (6.5) and (6.7) give:  z^y-^-  9 z  Subtracting (6.5) gives:  1^^—  - fy) =  i  d  - —  ( 1  * °  ) Q  (AZ^  u  Which upon integration yields: : i z  - AZ. = ^ + f(x) u  2  Y  (6.8)  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 i s :  z  ° = /2Qy  "V  1  yYUU  replacing this in (6.6) yields  A  o  =  / 2 c%  V  Y  2A°  The ratio 375= 2c is less than unity, j u s t i f y i n g our approximation. YZV '1  Replacing A Z ^ in (6.8) by A z^ yields:  .2cjQy_ 1  YU  This form is in accord with Carson's (1973) integration of the non-advective equations.  The quadratic spatial behaviour of z^ is supported by  51 the observations of Wiseman and Hirt (1975), Raynor et al_. (1979) and Portelli (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:  whi le  3X  T 2ux  ! f i  -  9X  S  H + 2c '  f i n  -  Vl +  2c J  2c)Q  2y 2yux 8 z  i  Using these forms for  9?  and — O A  , equations (6.5) and (6.6) can be  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 i e l d .  The equation of continuity has:  v.(pu) = 0  52 or, s p l i t t i n g the horizontal and vertical components  pV  where  H "  U  "  =  3z  ^ ^ pW  is the horizontal divergence operator.  Since the synoptic condi-  tions were largely stationary we may, without much fear of oversimplificat i o n , assume the horizontal divergence to be constant over any given day, so  £ (PW) = -  P  (6.9)  3  where 3 = ^(y) is a constant (often erroneously called the subsidence v  parameter).  A convenient formulation for the density of the atmosphere  is (Schmidt, 1946): p(z) = e(z) ^ e " °o  b  (6.10)  z  where p and e are the density and potential temperature at some reference Q  Q  level and b = 10 m _1+  _1  is approximately constant.  Using a two-layer thermal  atmosphere, e(z) becomes e(z) = e  0 < z < z.  Q  e(z) = e + ( z - z . ) 0  Y  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 subsidence velocity is given by:  w  =  \^ -V l  b(e " ( i - a)z) Q  +  bZ  r  +  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: - eze  w  (6.13)  0  (e + ( l - a)z) o  Y  I f , in addition. Y(1 - a)z << e and  be  <<  YO  - a)  0  the subsidence velocity is given by (6.14)  w = - ez  This approximation w i 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 profiles (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 substituted into equation (6.7) and used in the model.  In addition to imposing a vertical velocity at the entrainment 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 w i 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 ) . t  Q  This "parcel" of air started i t s subsidence at a time  when i t was at a height z . Q  (Note that this i n i t i a l height z , bears Q  no relation to the surface roughness length usually given this symbol). The inversion must steepening since the subsidence velocity increases with height.  In i t s simplest form,  ws integration leads to  (6.15)  55  Figure 6.4:  Subsidence Warming; for explanation see the text.  56  Now,  e(t)  = y (z 0  0  +e  - z\)  Q  substituting (6.15) and differentiating leads to: de dt  =  6 y 6  ' 3(t - t ) V i z  e  e  0  The value of Y w i l l be chosen as the mean inversion intensity of the q  lowest 650 m of the atmosphere in the early morning temperature sounding. The figure of 650 m was chosen since this is the height z  Q  that would be a  typical maximum for the conditions encountered in this study. t = 7h, 3 = 10" s " ) . 5  1  (z.j = 500 m,  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 Portelli (1979) at a lakeshore s i t e .  I t is  probably associated with the dynamics of a meso-scale sea-breeze circulation. 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 literature on sea and land-breeze circulations Emslie (1968), Hoos and Packman (1974), Hay and Oke (1976) and Kalanda (1979) deal directly or indirectly with those phenomena in the Vancouver/Fraser Valley region.  Guy (1979) uses wind-speed and direction  profiles from 23 mini-sonde f l i g h t s from this experiment (see Appendices B and I) to characterise the structure of the meso-scale circulations over the c i t y .  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 circulation.  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 direction, 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 i g h t 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 illustrated by the tetroon f l i g h t patterns of Lyons and Olsson (1973), sea breeze circulations have regions of u p 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 fields  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 i e l d for 1700 h shows that the region of maximum horizontal v o r t i c i t y 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 s , approximately an order of magnitude larger than that due to - t t  _ 1  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 u n t 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  t < 1130  S  B(t) = 6 e ° s  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  and from (6.3)) and substituting the derived forms for  i  3?  ' — O  O A  X  '  w(z^.) and W(z-,t) produces the following system of f i r s t order nonlinear d i f f e r e n t i a l equations:  ___ 3t  3 X a  a,z. - a " 4 i " "5  (7.2)  c  fc  ^7  where  3t  "  a-j  =  a  6 ^ F " 3t " 7 i a  3  ( 7  (1 +c)Q  yi + 2c  '2  a  z  = cQ  J  -  3 )  60 JO  K  a5  a,6  a  / 7  + 2c) 2x Y  Y  = gy e ett - t ) o  0  The coefficients a-, to a are a l l in general time-dependent 7  and their values w i 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 flux values being linearly interpolated for each six minute interval, and the horizontal divergence being set according to equation (6.17).  The solution to the system of differential equations  was provided by a library program in The University of British 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 i s t i n g 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 flux values (Wm" )(see Appendix E). 2  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 ) interpolated linearly from the tempera1  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 profiles 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. directions are used to calculate the distance x in coefficients equation (7.1) and (.7.3).  These anc  * 5» a  The calculation was made on the basis of an  assumed e l l i p t i c a l plan of the urbanized part of Vancouver (see Figure A.l).  The next line contains the i n i t i a l inversion height, mixed layer  temperature (K), synoptic horizontal divergence ( s ) , a data level, _ 1  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 ) for calculating -1  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 digitized mixed layer  depth and times for model validation. step was set to 0.1 K. on a l l days.  At i n i t i a t i o n the temperature  62 7.2  Results of Mixed Layer Modelling Complete data sets for mixed layer modelling were available  on thirteen days during the study period. results in graphic form.  Figures 7.1 to 7.13 show the  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 digitized so as to include fluctuations with characteristic times slower than roughly 10 min, which is much faster than the characteristic times of the modelled inversion 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 e x p l i c i t l y 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 o r i g i n , 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 profile and would require much more frequent soundings than available in this study.  Its general behaviour i s , however, quite conservative  and displays a gentle rise from i t s 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.  It  remains steady at this value usually for about four hours and then begins a slow decline .  63  20  g  800  r  600 -  E  50  70  11-0  9-0  Local solar  Figure 7.1:  1*0  time (h)  15-0  170  19-0  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 i g h t line 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 5  0  1  1  1  1  i  ,  I  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 (? = 292.5K) 0  The overall behaviour of the model is good with an underestimation of inversion height and temperature in the l a t t e r part of the day.  65  0  o ©  9-0  Figure 7.3:  11-0 13^ Local solar time (h)  19-0  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 (? = 286.7K) 0  This day was marked by unsettled synoptic conditions, as a surface ridge developed. This non-stationarity is reflected in the relatively 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  20  0  1  1  1  1  50  70  9-0  HO  1  •  13-0  15-0  I 170  19-0  Local solar time (h)  Figure 7.5:  Inversion Rise Modelling for July 29th. Symbols as for Figure 7.1 (? = 288.6K) 0  The model behaves f a 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 l a t t e r half of this day.  68  20  g  0  I 50  1— 7-0  1  1  i  i  ,  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 (J = 290.3K) Q  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 origin.  69  aoo f-  Figure 7.7:  Inversion Rise Modelling for July 31st. Symbols as for Figure 7.1 (? = 284.OK) 0  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 disagreement is not fundamental.  70  600 r  0  "  1  1  1  1  1  I  I  50  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" = 288.OK) 0  The remarks for July 31st apply to this simulation as well.  *  r  71  20  Local solar time (h)  Figure 7.9:  Inversion Rise Modelling f o r August 2nd. Symbols as f o r Figure 7.1 ( ?  = 286.7K)  In s p i t e of the rather complex behaviour of the inversion height, the model tracks very w e l l .  72  20  0 I 50  1  70  Figure 7.10:  1  9-0  1 i 110 130 Local solar time th)  i 15-0  i 170  I 19-0  Inversion Rise Modelling for August 3rd. Symbols as for Figure 7.1 (? = 290.OK) 0  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  600 \-  i.00  200  70  Figure 7.11:  9-0  110 130 Local solar time (h)  150  17-0  19-0  Inversion Rise Modelling for August 4th. Symbols as for Figure 7.1 (? - 284.3K) 0  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 5  0  Figure 7.12:  1  ™  1  9-0  1 _i HO 130 Local solar time (h)  . 15-0  17-0  | 19-0  Inversion Rise Modelling for August 5th. Symbols as for Figure 7.1 (? = 283.2K) 0  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 particularly 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 profile immediately 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 multiplicative 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. for a l l days modelled.  This adjustment is used  The overall sensitivity 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 t s 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 sensitivity 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 flux given by:  «H - « H  with Q = 340 W r max nH  2  m a x  sinfliV1'  77 The inversion intensity (y) was set constant at 0.015 K rrr , the mean 1  wind speed (u") constant at 2.5 m s , the wind direction constant at 180°, - 1  an i n i t i a l inversion height ( z i ) of 10 m and mixed layer temperature ( e ) Q  0  of 290 K with horizontal divergence ( B ) of 1.0 X 10~ s " 5  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 rising 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. 7.14 to 7.18 show the dependence of z - j  max  on QH » max  m e a n  Figures  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 r e a l i t y boundary conditions).  The most  sensitive being the inversion intensity which also exhibits the greatest non-linearity, 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 rationally investigated in this model as the present formulation is appropriate to a constant fetch of 8 km as described in Section 6.3.2.  A rough indication of the model sensitivity  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  £700 x o N  E 500 h  300  1  0  L  —» 100  • 200 Q  Figure 7.14:  • 300  • 400  H max < ~*> W m  Maximum Inversion Height vs Maximum Surface Sensible Heat Flux.  1 500  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" )  .04  .05  1  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  Q  340 W  u  2.0 m s -  Y  0.010 K m-  c  0.020  6  1.0 x IO" S "  x  8.00 km  u  H  Gradient rn'2  1.13 m W 3  _1  158 s  1  2.82 x 10^ m K  1  2  250 m 5  1  9.3 x 10 m s 3  1.26 x 10"  2  -1  82  8.  Conclusion This part of the study has shown that the accepted forms of  inversion rise models (typified by that of Tennekes (1973)) can be successf u l l y generalized to include the effects of advection and subsidence. The effects of advection have been modelled by including an advected heat flux term into the thermal budget equation for the mixed layer. The magnitude of this flux 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 subsidenceinduced warming of the atmosphere above the growing layer as well as imposing a subsidence velocity on the entrainment interface. This subsidence is driven by atmospheric divergence on both synoptic- and mesoscales.  The magnitude of the synoptic-scale divergence has been estimated  from observations of subsidence in potential temperature p r o f i l e s , 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 t s application to areas in which mesoscale 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 t y . 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 t e r r a i n , 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 p a r t i a l l y sufficient 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  e x p l i c i t l y urban or in any way related to the character of the underlying surface or surface-layer i t should have general applicability 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 e x p l i c i t l y 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 yield 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-stationarities, 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 s t a t i s t i c s a-j/u* over very rough surfaces (z  = 0.5 m) have adiabatic limits that agree with  those measured over much smoother surfaces. The behaviour of these s t a t i s t i c s with increasing i n s t a b i l i t y is consistent with previous results but somewhat obscured by large scatter. The integral s t a t i s t i c s are a l l related to the mixed layer variable ~ i /L and show strong increasing trends with this variable. z  Turbulent velocity spectra can be successfully measured in these highly turbulent flows with an orthogonal array of helicoid propeller 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 s t a t i s t i c a 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 relatively 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 t a t i s t i c a l theory of diffusion. 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 particular, 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 synopticand 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 sensit i v i t y being roughly 100 m for the expected range of values of surface sensible heat f l u x , mean wind, inversion strength and subsidence parameter. In addition to these conclusions, during the course of the data analysis the following techniques have been u t i l i z e d .  86  The closure of an open energy budget was achieved by distributing 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 subsidence 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 definition in the l i s t below, and w i 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  Coefficient in mixed layer.model  (Km'^s ^)  b  7  i ) Constant in mixed layer scaling  -  -  ( - )  i i ) Constant in approximate formulation for height dependence of atmospheric density and temperature c  Entrainment closure parameter  (nH) ( - )  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 t a b i l i t y length  (m)  n  Frequency  (s ^)  Q  Kinematic eddy heat flux  (K m:s )  Turbulent latent heat flux  (W m~2)  Turbulent sensible heat flux  (W m~2)  Net all-wave radiation  (W m~2)  Q  u  -  _1  hi  Q*  i ) Error ratio  r  ( - )  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  t  u  t*  (s)  Scaling time for vertical diffusion in 3  unstable conditions  (s)  Non-dimensional diffusion time  ( - )  89  u  Longitudinal component of wind velocity  (m s" )  Tj  Mean wind speed  (m s" )  Surface layer f r i c t i o n 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" )  Subsidence velocity  (m s" )  Convective velocity scale  (m s" )  u  w  *  s  w* w'  e  1  ]  T  ]  ]  ]  ]  Kinematic heat flux (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)  Inversion height  (m)  Surface roughness length  (m)  Scale height  ( - )  z. 1 z  o  a  i ) Constant in Hay-Pasquill form of  (3  Lagrangian-Eulerian transform i i ) Bowen's ratio  ( - ) ( - )  Inversion intensity (lapse rate),  Y  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)  AQ  Heat storage in urban canopy layer  (W m" )  e  Residual in energy budget closure  (W m" )  S  2  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 t a b i l i t y parameter (z/L) Mixed layer s t a b 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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A prime criterion 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  475  z  (A.l)  1 / 5 0  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 transition from a smooth-to-rough surface.  The depth of adjustment w i l l be greater for  increasing i n s t a b i l i t y , and smaller for a transition from rough-to-smooth (Peterson, 1969) or increasing s t a b i l i t y .  107 The relation (A.l) gives the required fetch which must be homogeneous 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 u t i l i z e d 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 logistical reasons i t was necessary to find a site with easily accessible electric power, reasonable security, and one at which the erection of a 30 m steel tower would not be in violation of c i t y 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 find a site t r u l y representative of urban meteorology, but i t is relatively easy to find a site surrounded by homogeneous suburban surfaces, particularly in North American c i t i e s .  Nevertheless  because of the r e s t r i c t i v e nature of the ideal requirements, i t was inevitable that some compromise would have to be made in selecting the site.  The site f i n a l l y chosen is a transformer station (operated by the  British 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  city of Vancouver has been described by Hay and Oke (1975) from a meteorologic 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 all 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 r o l l i n g " 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 direction, i t was easy to ensure that wake effects from this building did not contaminate the results of the turbulence 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 t y .  (Figure A.l is based on this analysis).  The  surface roughness length is most properly determined from wind profile 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  o  = 0.5h*-fS  v  (A.2) '  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 technique 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 landuse classification 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 classification 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 fields).  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  Eif i  s=  s  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 *  th  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 . the absolute value of z  0  While  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 f a 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  houses  garages  larger  open space  (m)  o  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 p r o f i l e s , 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 r e a l i s t i c 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 l a t t i c e free-standing tower constructed by the LeBlanc and Royle company to their LR324 series S/S specifications., The tower consisted of six l a t t i c e sections (A,A,B,C,D and E of their specifications), 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 e d 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 Mainwaring 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 r a i l e r at the base of the tower housed the recording and logging  equipment (Figure B.l). The surrounding houses are b u 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  20-1 m  Level U  18-5  Level 3  13-4m -  Level 2  11-6 m -  Level 1  9-4 m -  Surface layer datum  0-0 m  m  9-0m Zo+d  W////////// 85m a.m.s.I.  - 0-0m 1:200  Figure B . 1 :  The  Tower and  Embankments  119  Figure B.2: 1. 2. 3. 4. 5. 6.  Upper sections of the tower showing surface layer instrumentation.  U.V.W. Anemometer. Microvane and cup anemometer. Differential psychrometer, upper sensor. Net pyrradiometer. Yaw sphere-thermometer. 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 pitch, 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 ( a /u)) encountered in the study w _1  should be 1.3 m for the horizontal sensors and 1.5 m for the v e r t i c a l . The starting speed of the sensors is in the region of 0.15 m s (McBean, 1972).  _ 1  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 u t i l i z e d . 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 s p i r i t level.  The  levelling required an operator at the top of the tower, and the flexing of the tower would certainly alter the levelling; 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 r a i l e 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 tenfold amplification, 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 i g i t a 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 directly measured by determining the correlation of temperature and vertical velocity fluctuations.  Any of a wide range of velocity and temperature sensors can  be used to achieve this end.  A particularly 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 utilised 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 l u x . 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 t a t i s t i c a l stationarity in signals with large variances), and the need for short averaging times (to satisfy the assumption of temporal stationarity in the signals).  The problem of s t a t i s t i c a 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). energy budget is described in Appendix E.  The estimation of a detailed 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 ratio of turbulent sensible-  to turbulent latent-heat  flux in the surface layer may be estimated by measuring simultaneously the vertical gradients of atmospheric temperature and humidity.  The  method, as implemented in this study u t i l i s e s a pair of v e r t i c a l l y separated 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 r a t i o .  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 ratio of 0.44.  The analogue signals from these sensors were conditioned  and integrated by a Campbell Scientific 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 flux 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 r a c t i o n , and  though there are no installations in the site with temperatures s i g n 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 ± 0.1 C and a time constant G  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 l o 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 e d with an output lineariser and chart recorder which provided a temperature-time plot.  In order to trans-  form to a temperature-height p l o t , 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 g h 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 digitized 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 Hall, 1974).  The instrument in i t s most basic form trans-  mits a pulse of sound v e r t i c a l l y 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 continuous record of the height of thermal turbulence structure above the site.  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 r a i l e 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 atmosphere 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 (particularly t r a f f i c ) noise.  With  the above settings this interference was at an acceptable minimum, producing a l i g h t 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 digitized 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 t s 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 s a t e l l i t e 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 dominated 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 until the 26th/27th when a deepening closed vortex over the Pacific Ocean began to dominate the flow at a l l levels.  In the  transition between these two regimes, large-scale motions without frontal origin realised potential i n s t a b i l i t y , producing wide-spread convective a c t i v i t y 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 bringing further clear skies and continued subsidence of warm air to the Southwestern British 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 i n a 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 earlier days. Mean hourly averaged wind speed and direction at level 4 (Figure B.l) from July 16th to August 8th. Potential temperature profiles of the planetary boundary layer taken intermittently throughout each of the days with at least three f l i g h t s on each day. Hourly blocks of three-dimensional turbulence s t a t i s t i c s 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 l e s in The University of British 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 l u x , Bowen r a t i o and net all-wave radiant heat flux 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  + AQ  (E.I)  S  Q* is the net all-wave radiant flux density,  the turbulent sensible heat  flux density, Q the turbulent latent heat flux density and A Q the canopy s  E  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 measurements of the turbulent fluxes are subject to relatively 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: AQ  when Q* * 5.5 Wm  2  S  = 0.24(Q* - 17.0)  (E.2)  130 and  AQ  S  = 0.70  (E.3)  Q*  otherwise. Given the three measured quantities and this parameterization, there are five different energy budgets that can be constructed (three closed and two open).  The following (unfortunately somewhat complicated)  notation is introduced in order to i l l u s t r a t e the five budgets. Q* AQ  s  -  net radiation as measured.  -  canopy layer heat storage, parameter!'sed as eqns (E.2) and (E.3).  g  -  turbulent sensible heat flux as measured by the YST system.  -  Bowen ratio as measured by the differential psychrometer system.  -  turbulent sensible heat f l u x , calculated as jj^y  E  -  turbulent latent heat f l u x , calculated as  Q^'  -  turbulent latent heat f l u x , calculated as Q^/g.  Q ''  -  turbulent latent heat f l u x , calculated as Q* -  -  canopy layer heat storage calculated as Q -  Q  I  E  S  AQ )  -  S  1 ^ (i+g) ( Q " Q ^ A  S  - AQ . S  *  1  AQ  (Q*  ii -  .  The three closed budgets are: Q*  = Q|!) + Q  E  + AQ  S  (E.4)  This budget is independent of the YST system. Q*  = Q  H  + Q ' E  +  A  QJ  < - ) E  5  131  This budget uses data from both the differential psychrometer and YST systems and does not use the parameter!'zations for canopy layer heat storage. Q* = Q  H  + Q" E  + AQ  (E.6)  S  This budget is independent of the differential psychrometer system.  The two open budgets are:  and  Q* = Q  H  + Q  Q* = Q  H  + Q'  E  E  + Q  + e-,  S  + Q  S  (E-7)  + ^  ( E  -  8 )  Both open budgets u t i l i z e data from both the differential 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 i n t e r v a l , 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 differential 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 right hand terms to within a (small) error, here taken to be W  nr . 2  0.125Q  +  10.0  132 ii)  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 flux greater than net radiation.  c)  turbulent latent heat flux 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  r a t i o 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 measurement errors rather than advective e f f e c t s .  This i s consistent with our  assumption of e f f e c t i v e surface homogeneity (see Appendix A) and i s supported by Figure  E.2  which shows the residual E-| to be independent  of wind d i r e c t i o n {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 i n 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 e p t o k u r t i c ,  e-j being the more extreme.  For t h i s reason ( E . 7 ) was chosen as the open  budget, and e-j was d i s t r i b u t e d between  and Q according to the f o l l o w i n g E  scheme. Consider the f o l l o w i n g open energy budget: Q* = Q  + Q  H  + AQ  E  s  +  e  (the complex notation has been dropped f o r c l a r i t y ) .  The residual e is  assumed to be composed of two portions a r i s i n g from measurement errors i n the two t u r b u l e n t terms only v i z : Q* = ( Q  where e  u  n  + e  r  h  e )  + H  H  + (Q  £  + e ) £  + AQ  s  = e , and our estimated turbulent fluxes are:  %  Q  = %  E  = Q  +  E  £  +  H  E  e  .  134 No  Q,  Budgets 1, 2 and 3 agree to within 6Q  Q*  = Q  Use  Q*  =  + Q  H  Qu  +  H  +  E  Qu  (68)  S  Q'r  Qp + +  H  0  AQ  E  E  0  + —  AQ  S  ^  +  AQ' ^>  (8)  1  If  Q*  Q  >  H u  >  0 and  •Use Q* ='Q|!| + Q + A Q E  Q*  I f 0* > 0 and Q  Use  > 1.25  E  If AQ'  Q* S  If AQ'  S  <  Q*  0 and  >0  Q*  >  <  0  0 and  Use  Q*  = Q  Use  Q*  = Q  u  +  Q  E  •Use Q * = Q  H  +Q  E  + Q '  +  1  E  U  +  (41)  S  Q  A  AQ  + AQ  S  $  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 right. Notes:  1. 2.  SQ = 0.125 Q * + 10.0 W m 6Q'  = 0.7  Q*  2  (19)  (1)  (14)  (62)  (35)  360  OO O  O O  o  O O CD O O OOi O O OOO O O O O0 D O O O OODOO O O(3D QDOOO OOO O OO OOO O © O CD (BD CD) O CD O O OO O O O O O O O  O  °S  240  O  0  c o •j: u  z I  O  OO 60  GOO o O  o oo oc&xr) o OO  CD O O O O ® OO  oo  C5DOO  o o O  -150  Figure E.2:  O ©  cn  o o o o oo o o oo o  o o  -250  OOO GEO OOO O  o o  O  oCD O  QD  -50 Residual  Residual  O 50 (Wm" ) a  vs Wind Direction  150  250  136  -600  -400  -200  <?,  -600  -400  400  600  200  400  600  [ W m ' )  -200  ej [ W  Figure E.3:  200  m  J  ]  Frequency Distribution of e1 and e . 2  137 A s t a t i s t i c a l treatment of experimental data yields probable errors <5Q and <$Q in the two fluxes (Fuchs and Tanner, 1970; Bailey, 1977; H  E  Kalanda, 1979).  We require that: e  Solving for  u  and e e  / e  =  E  6Q /6Q = ± r  ±  H  E  leads to  E  = / ( l ± r) e  u  =  e£  e / ( l ± 1/r)  where the positive sign is for <5Q^ and 6Q having the same sign and the E  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 w i 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*  Q  R  Q  E  Q  s  e  149.1 = 95.0 + 67.8 + 34.3 - 48.0;  6Q  U  6Q  E  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 r e a l i s t i c one, and both fluxes had been overestimated by the instrumentation. the hour ending 0900 LST on the 5th of August.  A second example is from  138  Q*  Q  H  Q  Q  E  s  e  6Q  382.5 = 313.0 + 46.1 + 87.7 - 64.3; Positive r:  382.5 = 269.3 + 25.5 + 87.7  Negative r:  382.5 = 191.7 + 103.1 + 87.7.  6Q  H  E  17.1; 36.3  The second case is taken to be the best estimate as the Bowen's ratio, in the f i r s t case is unrealistically high. ated  and underestimated E.2  The instrumentation overestim-  in this case.  Examples of an Urban Surface Energy Budget A previous energy budget study at this site (Kalanda, 1979;  Kalanda et a l . , 1980) u t i l i z e d only the differential psychrometer system for estimating the turbulent fluxes (the budget is as eqn. (E.4)). Energy budgets measured in that study are not significantly different from those of the present study and exhibit the usual temporal variation of fluxes and the relative magnitudes of Q„, Q and AC) now known to r  n  typify urban surfaces (Oke, 1978).  t  S  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 e x p l i c i t l y utilized in this  study, but the surface turbulent sensible heat flux is a crucial parameter in both Parts One and Two.  The extra e f f o r t involved in determining  the entire budget was deemed j u s t i f i e d by the increased confidence in Q when the other three terms were determined and shown to be reasonable. u  CO  Local solar time (h)  Figure E.4:  Suburban Surface Energy Budget.  140 F.  Spectral Analysis The magnetic tapes containing the Gill UVW signals, collected,  f i l t e r e d and recorded as in Section B.2.1 were transferred to The University of British 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 l e t s .  Each raw data block was s l i g h t l y 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 r a i l i n g 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 discont i n u i t i e s 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 s p 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 t a t i s t i c s 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 t a b 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 t a b i l i t y class.  This  was achieved by averaging the spectral amplitudes from a l l eight data blocks in each z/L class into bands, each of width 0.1 units of non-dimensional 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 frequency end.  8 ) , thus resulting in some scatter at the low  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 l u s t r a t i n g the low frequency scatter.  The slight 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 replotting, and indicates that spectral forms are largely independent of s t a b i l i t y .  No systematic ordering of  curves with s t a b i l i t y could be discerned, presumably due to purely s t a t i s t i c a l fluctuation masking the weak trends with s t a b 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 digitized for use in determining the dispersion functions. Figure 3.4.  These are the spectra shown in  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  Figure F.2b:  -1-0  log (f)  00  1 0  Construction of Composite Spectrum. Crosswind Component.  20  3-0  -2 0  -1-0  00 l o g (f)  Figure F.2c:  Construction of Composite Spectrum. Vertical component.  1 0  2-0  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 DIMENSION F ( 1 0 0 ) , Y ( 1 0 0 ) ,FN<100) , 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 READ<7,103)A(1) ,A<2),A(3) 103 FORMAT(3 A4) 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 I N LOG SPACE 66 READC7,100,END=11)DX,DY CCCCCCC 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 CCCCCCC LOOP THROUGH TRAVEL T I M E 0 TO 1 5 0 I N 3S DO 22 J T = 1 , 5 1 TI(JT)=FLOAT(JT-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 . 283185 F2=SIN(F1) I F ( F 1 . E Q . O . O ) G O TO 5 5 F3=F2/Fl GO TO 3 3 55 F3=1.0 CCCCCCC COMPUTE INTEGRAND 33 FN(JK)=Y(JK)*F3*F3 CCCCCCC I N T E G R A T E FUNCTION FI(JT)=SQRT(QINT4P(F,FN,N,1,NH 22 CONTINUE CCCCCCC CHECK OUTPUT S P E C T R A L I N T E G R A L AINT=FI(It WRITE(6,101)AINT 101 F 0 R M A T ( 1 X , » T* F(T*)•,3X,F7.5) CCCCCCC OUTPUT T A B L E AND P L O T F I T * ) DO 4 4 J K = 1 , 5 1 FI(JK)=FI(JKl/AINT WRITE ( 6 , 1 0 2 ) T I ( J K ) , F I < J K ) TICJK)=TI(JKl/25.0 44 FI(JK>=FI(JK)*5.C 102 F 0 R M A T ( 2 X , F 6 . 0 , 3 X , F 6 . 4 ) CALL 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) CALL L I N E ( T I , F I , 5 1 , U ) CALL 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 i n g 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  a  = 0.50 m s  _ 1  u  a  = 0.38 m s  _ 1  v  z  _ 1  = 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* =  =1.50  ta/z  The dispersion function (from Equation 4.1) i s :  so  S = 0.57 y a = S t/a y  y  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 r a t i f i e d atmosphere or one with high wind speed (> 10 m s ; Pasquill (1974)) _ 1  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 < z  Q  < 1.0 m, take Z = 0.6 m. q  From Counihan's (1975) Equation 4 = 2.28 m s  _ 1  a = 1.71 m s  - 1  a  and  u  v  for z = 20 m from Counihan's (1975) Equation 3.  The values of TJ, a , a and z can now be applied as in Example 1 to v  provide an estimate of a  (the result f o r t = 60 s is a = 9.7 m). #  y  Example 3. In an unstable atmosphere, in the absence of measured wind variances, some estimate of the turbulent sensible heat flux must be available, either through direct measurement or parameterization.  The  former method is demanding in terms of instrumentation and operational requirements and the latter is at best rough with the currently available schemes. Given a measured value of Q^, a value for u*/u may be e s t i mated from Pasquill's (1974) Figures 6.3 and 6.4, using a value of z  Q  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 G  v  which w i l l allow estimation of a  speed.  y  and  as in example 1. for a given wind  149 I.  Theodolite-Tracked, Balloon-Borne Temperature Soundings Data from the theodolite sightings (altitude 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 profiles 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 prohibitively large, as indicated by Schaefer and Doswell (1978) and Netterville 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 baseline) 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 line"  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 profiles from these soundings. In order to produce detailed potential temperature profiles the temperature-time output of the radiosonde receiver was digitized 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 g h 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 linearly interpolated 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 p r o f i l e s , 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 profiles 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 profile 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 profile 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 profiles in determining the subsi-  dence rate is i l l u s t r a t e d 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 profiles 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. regression equation being  The linear  151 TJ. = -0.32 + 1.02 u tower balloon L  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 profiles 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 profiles 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 profiles.  The diagonal line represents agreement.  The data show no  obvious trend and indicate that s t a t i s t i c a l l y the best estimate of inversion 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 z  Figure J . l :  0-4 i  sounder  0-5 , k m  0-6  '  Inversion height from acoustic sounder and potential temperature p r o f i l e .  0-7  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 profiles above the inversion base.  A pair of;such profiles  is shown in Figure K.l, where a "kink" in the potential profile 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 (K.l)  ^JT = w ( e , Y , a , 3 , z ) . 0  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 j u s t i f i e d here since the conditions of the approximation are met (see Section 6.3.1). e b + 2 (1 Y  Q  b(e  0  + ( l - a)z) Y  Integrating this leads to: b ( e b + 2 (1 - a ) ) ( t Q  Y  ( e l n ( z / z ) + ( l - ct)(z o  1  - t ) 2  2  1  Y  2  2 ]  ))  (K.2)  e (K) Figure K.1:  Subsidence in potential temperature profiles on August 8th.  The subsiding "kink" is indicated by an arrow, in each case, with i t s height in metres. Relevant parameters are as follows Time kink z  Y  ? z  i  (LST) (m)  (K m- ) (K) (m) 1  1000 850 0.0176 300.1 315  1200 696 0.0171 303.1 370  157 Substituting the mean values of e , y and a and the values for t-| and  z^,  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 j u s t i f i a b l e ) approximation (Equation 6.14) is taken, a similar analysis results in 3 = 2.78 x I O '  5  s"  1  .  Similar analyses were performed for as many pairs of f l i g h t s 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 rise.  DIMENSION DIMENSION  Q H Q 4 ) ,A< 7) , Z I ( THE(14)  140), TH(140)  ,DT(140),VP(3),TV(6)  DIMENSION T(140),S(2),GAM(14),UB(14),V(3),QHI( COMMON A , P 2 , T I S , B EXTERNAL F CCCCCCC CCCCCCC CCCCCCC CCCCCCC CCCCCCC CCCCCCC •CCCCCCC  '  B O Y L A Y SIM PI^SY , BACK READ BOUNDARY CONDITIONS LE HEAT FLUX, LAPSE RATE, N , AND I N I T I A L INVERSION TE, TWO P A R A M E T E R S , S T A R T SJoS I D E N C - E , A N D M E A S U R E D LAYER.  1 0 ) , THV(  6)  PLOT , WITH SUBSIDENCE HTN3. AS H O U R L Y A V E R A G E D V A L U E S OF SENSIB MEAN WIND S P E E D , MEAN WIND DIRECTI HEIGHT, MIXED LAYER TEMPERATURE, DA TIME AND EXPONENT FOR M E S 3 SCALE T I M E AND M E A N T E M P E R A T U R E OF MIXED  R'€AD<5, 1 3 0 ) Q I , ( Q H ( I ) , I = 1 , 1 4 ) , 3 F , ( GAM( 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 B , G 0 2 , ( T V ( I ) , T H V ( I ) , 1 = 1 , 6 ) CCCCCCWRITE(6,103)S(1),S{2) C=0.20 N=3 REL=1.0E-10 AbS=1.0E-10 DTO=Q.1 f  CCCCCCC DO  LOOP THROUGH 22 J = l , 1 4  FOURTEEN  CCCCCCC INTERPOLATE QH AT DO 7 7 IK=1,10 I F ( J . £ Q . 1 ) G 0 TO 99 Qi=QH<J-1) GO TO 5 5 99 Q1=QI 55 Q2=QH(J) IF(J.EQ.14JG0  44 222  TO  44  IF(IK.GT.5JG0 TO DQ=(Q2-Q1)/10.0  88  TEN  WITHIN  THIS  HOUR.  IK+5)  UQ=1Q3-Q2)/10.0 QrlI(IO=Q2 +DQ*FL0AT(  77  POINTS  Q3=QH(J + l ) GO TO 2 2 2 Q3 = QF  QHI(I.<J=Q1+DQ*FL0AT( GO TO 77 88  HOURS  IK-5)  CONTINUE  CCCCL.C DO  LOOP 11  THROUGH  TEN  SIX  MINUTE  INTERVALS  K = l , 1 0  JK=iO*(J-1)+K JK1=JK-1 I F ( Q H H K ) IF(QriKK) i FLAG=1  . L T . 0 . 0 . AND. J . E Q . 1JG0 , L T . 0 . 0 ) G 0 TO 8 7 8  CCCCLCC COMPUTE FETCH RS=7.U0 THES=0.3937  OF  I F ( T H E ( J ) . L T . 9 0 . 0 )  URBAN  GO  TO  TO  SURFACE  901  66  FOR  ELLIPTICAL  CITY  159 PHI=(THE(J)-90.0)*1.7 GO  901 *02  453E-02  T O 902  P H I = ( 2 7 0 . 0 + T H E ( J ) ) * 1 . 7 453E-03 CS=CGS(PHI)**2 SI=SIN(PHI)**2 RP=SiRT(7056.0/(196.0*SI+36.0*CS)) DX=1000.0*( ( R P * C O S ( P H I )+RS*COS {THES ) ) * * 2 ++(RP*SiN(PHI) +RS*SIN(THES))**2)**0.5  :CCCCCC  Al  S E T CDE F F I C I E N T S  TO  A7  A(1)=3HI(K)*(1 .0+ 0 / 1 2 1 2 . 0 A ( 2 ) = 0.7i71*SQRT(UtJ{ J ) * Q H I ( K ) * P l * G A M ( J ) / ( 1 2 1 2 . 0 * D X ) ) A(3)=C*QHI(K)/1212.0 IF(JK.GE.ITSJGO TO 6 A(4)=B GO TO 7 b A(4)=B*EXP(F8*0.1*FL0AT(JK-ITS)) 7 At 5 ) = S c 3 R T ( U B ( 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)=GAM{J)*P1 A(7)=B*G0 TIS=36Q.0*FLOAT(JK) TI=0.0 TF=360.0 ;cc:ccc S E T ARRAY VALUES O F I N V E R S I O N HEIGHT, MEAN TEMPERATURE A N D CCCCCCC  TEMPERATURE  STEP  IF(JK1.GT.OJGO  67  TO  TO I N I T I A L  OR COMPUTED  VALUES.  67  V(1)=TH0 V(2)=ZI0 V(3)=DT0 GO TO 1 V(1)=TH(JK1) V< 2 ) = Z I ( J K 1 ) V(3)=DT(JK1)  CCCCCCC  COMPUTE  D E R I V A T I V E S  FOR  OUTPUT  VP(2)=A(3)/V(3)-A(4)*V(2)-A(5) VP(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 )  *RITE(6,104)J,K,(V(IJ,1=1,3),(VP(I),1=1,3),(A(I),1=1,7} CCCCCCC  U S E NUMERICAL  SOLUTION  CCCCCCC  NErt  T H E THREE  I  CALL  3 4 , 66  VALUES  OF  OF D I F F E R E N T I A L V A R I A B L E S .  D E ( F , N , V , T I , T F , R E L , A B S , I F L A G )  GO TQ<3,4,1,1,1,3),IFLAG WRITE(6,101)J,K,IFLAG,TI,TF,REL,ABS,N GO TO 33 TH(JK)=V(1) ZI(JK)=V(2) DT(JK)=V(3) GO TO 11 DT(JK)=DTO Z I ( J K ) = Z I O T H ( J K ) = T H O  II  CONTINUE  22  CONTINUE  CCCCCCC  878  LOOP  THROUGH  D O 111 M=1,JK1 THS=TH0-2.0 THF=TH0+18.0  ARRAYS  FOR  PLDTTIMG.  EQUATIONS  TO  COMPUTE  160 T(M)=1.0+FLOAT(M)/20.0 I F ( T H ( M ) o L T . T H S ) T H ( M ) = THS IF(TH(M).GT.THF)TH(M)=THF TH(M)=6.0+1TH<M)-THS)/10.0 111 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)•,20,5.,90.,0.,200.) CALL AXIS< 1 . 0 , 1 . 0 S O L A R TIME (H)*,-14,7.0,0.0,5.0,2.0) 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 ) CALL  PL0T(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 P L O T T I N G . 4 4 4 READ( 5 » 1 0 5 , E N D = 3 3 3 ) T D , Z D , N P E N TD=(TD-5.0)*0.5+1.0 ZD=ZD/200.0+1.0 CALL P L O T ( T D , Z D , N P E N ) GO TO 4 4 4 3 3 3 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 8 8 8 TVT=(TV(I)-5.0)*0.5+1.0 THT=6.0+(THV(I)-THS)/10.0 CALL S Y M B O L ( T V T , T H T , 0 . 0 7 , 1 , 0 . 0 , - 1 ) GO TO 7 7 7 8 8 8 CALL L I N E ( T , T H , J K 1 , + 1 ) CALL SYMBOL(1.2,7.6,0.14,S,0.0,8) CALL PLOTND 33 STOP 100 F0KMAT(10F6.1,/,6F6.1,/,3(10FS.1,/,4F6.1,/),2(1X,F5.1),1X, 1E7.1,2A4,2(1X,F4.2),1X,I3,1X,F4.2,2X,F6.1,/,6(F6.2,F6.1)) 1 3 1 F O R M A T ( 1 0 X , 2 ( 1 2 , 2 X ) , « E R R O R I N DE - CODE •,12,2(2X,F6.1) 1,2(2X,E14.4),2X,I3) 1 0 3 F O R M A T ( 2 X , • I N V E R S I O N RISE FOR »,2A4) 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(1X,E14.7)) 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 D E R I V A T I V E S (CALLED BY D E ) . COMMON A , P 2 , T I S , B DIMENSION V(3),VP(3),A(7) VP(1)=A(1)/V(2)-A(2) VP(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 67.4 104.6 369.3 261.8 161.0 113.7 .0200 .0320 .0440 .0368 .0175 .0185 .0195 .0196 1.4 2.4 2.6 2.8 4.3 4.2 4.6 3.9 134.0 145.0 162.0 158.0 150.0 149.0 146.0 138.0 35.0 290.0 5.2E-06 AUG 7.70 291.4 10.53 294.4 5.081 45.841 3 5.127 64.558 4  170.9 170.4 210.3 213.7 147.7 239.2 10.8 -14.3 .0248 .0128 .0134 .0144 .0154 .0165 2.4  2.6  2.6  2.7  3.3  4.9  166.0 182.0 179.0 2C4.0 224.0 152.0 01 0.70 1.00 15.62 296.4  65 0.45  0.0124  Sample input data for inversion rise simulation.  


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