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Verification of urban energy balance models Loudon, Sheila Margaret 1984

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VERIFICATION OF URBAN ENERGY BALANCE MODELS by SHEILA MARGARET LOUDON B.S c , The U n i v e r s i t y of B r i t i s h Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department bf Geography) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1984 © Sh e i l a Margaret Loudon I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e h e a d o f my d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Geography The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main M a l l V a n c o u v e r , C a n a d a V6T 1Y3 D a t e Feb. 6 , 1984 D E - 6 (3/81) ABSTRACT Three urban energy balance models have been tested against data gathered i n a suburb of Vancouver, B r i t i s h Columbia. The major in c e n t i v e behind t h i s study was that these models had not previously been v e r i f i e d i n urban areas, due mainly to the la c k of appropriate data. As the a v a i l a b l e models repres-ent a range of complexity, and u t i l i z e a v a r i e t y of d i f f e r e n t methods, i t was thought to be worthwhile to t e s t a l l three. Input f o r the models c o n s i s t s of temporal and s p a t i a l information, met-e o r o l o g i c a l data, and s i t e surface c h a r a c t e r i s t i c s . Values f o r the former two sets of inputs were r e l a t i v e l y easy to come by. Values f o r the surface c h a r a c t e r i s t i c s , however, were quite d i f f i c u l t to determine with any degree of confidence, due to the complexity of the suburban surface. This was p a r t -i c u l a r l y discouraging because i t was found that the models were very s e n s i t i v e to the values chosen f o r these inputs. The s e n s i t i v i t y analyses also showed that the simplest model (Myrup, 1969), of the three, sometimes exhibited rather u n r e a l i s t i c responses to v a r i a t i o n s i n the input parameters. The other two models (Ackerman, 1977; Carlson and Boland, 1978) u s u a l l y showed more reasonable responses. Eighteen days, from the a v a i l a b l e data set, were chosen for t e s t i n g the models. The modelled values were compared to those observed using com-parati v e s t a t i s t i c s , s c a t t e r diagrams, and time s e r i e s p l o t s . I t was found that the e r r o r s i n the modelled net all-wave r a d i a t i o n were quite small, and were due mainly to e r r o r s i n the modelling of the net long-wave radiaton. The modelled surface heat f l u x e s , on the other hand, were i n poor agreement with those observed. This was a t t r i b u t e d to the methods used for representing i i i surface moisture. The e r r o r s i n the mixed layer heights, as produced by the two more complex models, were also high. Unfortunately, the modelled surface temperatures could not be q u a n t i t a t i v e l y assessed due to the lack of approp-r i a t e measurements. However, those of the simple model seemed to be u n r e a l -i s t i c , whereas those of the other two models seemed reasonable. I t was concluded that without some improvements, p a r t i c u l a r l y i n surface moisture representation, the models would have l i t t l e p r a c t i c a l value. i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i L IST OF SYMBOLS x i i ACKNOWLEDGEMENTS x v i i CHAPTER ONE - INTRODUCTION 1 1.1 The Energy B a l a n c e 1 1.2 The Framework of Energy B a l a n c e Models 3 1.3 Model A p p l i c a t i o n s 4 1.4 The Urban Energy B a l a n c e Models: Background 5 1.5 I n c e n t i v e s and O b j e c t i v e s 9 CHAPTER TWO - THE MODELS 10 2.1 Model M 10 2.1.1 P r e l i m i n a r y C a l c u l a t i o n s 10 2.1.2 S o l a r R a d i a t i o n C a l c u l a t i o n s 14 o 2.1.3 S u r f a c e Temperature/Energy B a l a n c e C a l c u l a t i o n s 15 2.2 Model A 19 2.2.1 R a d i a t i o n C a l c u l a t i o n s 22 2.2.2 S u b s u r f a c e Temperature P r o f i l e 22 2.2.3 S u r f a c e Temperature/Energy B a l a n c e and Mixed 23 L a y e r C a l c u l a t i o n s 2.2.3.1 The U n s t a b l e Atmosphere 25 2.2.3.2 The S t a b l e Atmosphere 29 V 2.3 Model C 31 2.3.1 Preliminary Calculations 34 2.3.2 Net Radiation Calculations 36 2.3.3 The D i f f u s i v i t y Integral 39 2.3.4 Daytime Calculations 42 2.3.5 Night-time Calculations 46 2.3.6 Subsurface Temperature P r o f i l e 49 2.4 Summary 49 CHAPTER THREE - THE OBSERVATION PROGRAMME 51 3.1 The Si t e 51 3.2 Instrumentation 54 3.3 The Observation Periods 57 3.4 Surface Temperatures 60 CHAPTER FOUR - THE MODEL INPUT 65 4.1 Temporal and S p a t i a l Information 69 4.2 Meteorological Conditions 69 4.3 Site Surface C h a r a c t e r i s t i c s 75 4.4 Conclusions 82 CHAPTER FIVE - MODEL:OBSERVATION COMPARISONS 83 5.1 Net Radiation 84 5.2 Turbulent Fluxes 99 5.3 Surface Moisture 117 5.4 Subsurface Heat Flux 121 5.5 Surface Temperatures 123 5.6 Mixed Layer Heights 130 5.7 Conclusions 135 v i CHAPTER SIX - SUMMARY OF CONCLUSIONS 139 REFERENCES 141 V l l LIST OF TABLES No. Page 2.1 In p u t f o r Model M 11 2.2 In p u t f o r Model A 20 2.3 I n p u t f o r Model C 32 3.1 Summary o f m e t e o r o l o g i c a l c o n d i t i o n s f o r the days t o 59 be used i n t e s t i n g t he models 4.1 Summary of changes i n t h e maximum d a i l y v a l u e s o f t h e 66 m o d e l l e d f l u x e s , s u r f a c e t e m p e r a t u r e s , and mixed l a y e r h e i g h t s p r oduced by changes i n the g i v e n i n p u t p a r a -meters 4.1a Model M 66 4.1b Model A 67 4.Ic Model C 68 5.1 Summary of s t a t i s t i c s c omparing m o d e l l e d and o b s e r v e d 85 n e t a l l - w a v e r a d i a t i o n 5.2 Summary of s t a t i s t i c s comparing m o d e l l e d and o b s e r v e d 94 net s o l a r r a d i a t i o n and n e t long-wave r a d i a t i o n 5.3a Summary of s t a t i s t i c s comparing m o d e l l e d and o b s e r v e d . 100 s e n s i b l e heat f l u x e s 5.3b Summary o f s t a t i s t i c s comparing m o d e l l e d and o b s e r v e d 101 l a t e n t heat f l u x e s 5.4 Summary of s t a t i s t i c s comparing m o d e l l e d and o b s e r v e d 122 s u b s u r f a c e heat f l u x e s 5.5 D a i l y maximum and minimum m o d e l l e d s u r f a c e t e m p e r a t u r e s 126 and o b s e r v e d a i r t e m p e r a t u r e s 5.5a Model M 126 5.5b Model A 127 5.5c Model C 128 5.6 Summary of s t a t i s t i c s comparing m o d e l l e d and o b s e r v e d mixed l a y e r h e i g h t s 131 v i i i LIST OF FIGURES No. Page 2.1 Flow chart for Model M 12 2.2 Flow chart for Model A 21 2.3 V e r t i c a l d i s t r i b u t i o n of p o t e n t i a l temperature for an 28 unstable boundary layer capped by an inversion 2.4 Flow chart for Model C 33 2.5 Basic framework of Model C 35 3.1 The greater Vancouver area 52 3.2 Looking west from the top of the observation tower 53 3.3 Instrumentation mounted on the observation tower 55 3.4 Observed energy balance for Aug. 3, 1978 . 61 3.5 Observed energy balance for July 28, 1980 61 5.1 Modelled vs. observed net r a d i a t i o n : Model M (1977) 86 5.2 Modelled vs. observed net r a d i a t i o n : Model M (1978) 86 5.3 Modelled vs. observed net r a d i a t i o n : Model M (1980) 87 5.4 Modelled vs. observed net r a d i a t i o n : Model A '(1911) 87 5.5 Modelled vs. observed net r a d i a t i o n : Model A (1978) 88 5.6 Modelled vs. observed net r a d i a t i o n : Model A (1980) 88 5.7 Modelled vs. observed net r a d i a t i o n : Model C (1977) 89 5.8 Modelled vs. observed net r a d i a t i o n : Model C (1978) 89 5.9 Modelled vs. observed net r a d i a t i o n : Model C (1980) 90 5.10 Diurnal course of modelled and observed net r a d i a t i o n : 92 Model M 5.11 Diurnal course of modelled and observed net r a d i a t i o n : 92 Model A 5.12 Diurnal course of modelled and observed net r a d i a t i o n : 93 Model C i x 5.13 M o d e l l e d v s . o b s e r v e d n e t s o l a r r a d i a t i o n : Model M 95 (1978) 5.14 M o d e l l e d v s . o b s e r v e d n e t s o l a r r a d i a t i o n : Model C 95 (1978) 5.15 M o d e l l e d v s . o b s e r v e d n e t long-wave r a d i a t i o n : Model M 96 (1978) 5.16 M o d e l l e d v s . o b s e r v e d n e t long-wave r a d i a t i o n : Model A 96 (1978) 5.17 M o d e l l e d v s . o b s e r v e d n e t long-wave r a d i a t i o n : Model C 97 (1978) 5.18 M o d e l l e d v s . o b s e r v e d s e n s i b l e h eat f l u x e s : Model M 102 (1977) 5.19 M o d e l l e d v s . o b s e r v e d l a t e n t heat f l u x e s : Model M 102 (1977) 5.20 M o d e l l e d v s . o b s e r v e d s e n s i b l e h eat f l u x e s : Model M 103 (1978) 5.21 M o d e l l e d v s . o b s e r v e d l a t e n t heat f l u x e s : Model M 103 (1978) 5.22 M o d e l l e d v s . o b s e r v e d s e n s i b l e heat f l u x e s : Model M 104 (1980) 5.23 M o d e l l e d v s . o b s e r v e d l a t e n t h e a t f l u x e s : Model M 104 (1980) 5.24 M o d e l l e d v s . o b s e r v e d s e n s i b l e h eat f l u x e s : Model A 105 (1977) 5.25 M o d e l l e d v s . o b s e r v e d l a t e n t h e a t f l u x e s : Model A 105 (1977) 5.26 M o d e l l e d v s . o b s e r v e d s e n s i b l e h eat f l u x e s : Model A 106 (1978) 5.27 M o d e l l e d v s . o b s e r v e d l a t e n t heat f l u x e s : Model A 106 (1978) 5.28 M o d e l l e d v s . o b s e r v e d s e n s i b l e h eat f l u x e s : Model A 107 (1980) 5.29 M o d e l l e d v s . o b s e r v e d l a t e n t h eat f l u x e s : Model A 107 (1980) 5.30 M o d e l l e d v s . o b s e r v e d s e n s i b l e heat f l u x e s : Model C 108 (1977) X 5.31 Modelled vs. observed latent heat fluxes: Model C 108 (1977) 5.32 Modelled vs. observed sensible heat fluxes: Model C 109 (1978) 5.33 Modelled vs. observed latent heat fluxes: Model C 109 (1978) 5.34 Modelled vs. observed sensible heat fluxes: Model C 110 (1980) 5.35 Modelled vs. observed latent heat fluxes: Model C 110 (1980) 5.36 Diurnal course of modelled and observed sensible heat 113 fluxes: Model M 5.37 Diurnal course of modelled and observed latent heat 113 fluxes: Model M 5.38 Diurnal course of modelled and observed sensible heat 114 fluxes: Model A 5.39 Diurnal course of modelled and observed latent heat 114 fluxes: Model A 5.40 Diurnal course of modelled and observed sensible heat 115 fluxes: Model C 5.41 Diurnal course of modelled and observed latent heat 115 fluxes: Model C 5.42 Diurnal course (July 31, 1978) of modelled and observed 116 sensible heat fluxes: Model A 5.43 Diurnal course (July 31, 1978) of modelled and observed 116 latent heat fluxes: Model A 5.44 Diurnal course of modelled and observed subsurface heat 124 fluxes: Model M 5.45 Diurnal course of modelled and observed subsurface heat 124 fluxes: Model A 5.46 Diurnal course of modelled and observed subsurface heat 125 fluxes: Model C 5.47 Modelled vs. observed mixed layer heights: Model A 132 (1978) 5.48 Modelled vs. observed mixed layer heights: Model C 132 (1978) X I 5.49 D i u r n a l course of the modelled and observed mixed 134 l a y e r heights: Models A and C x i i LIST OF SYMBOLS Symbol D e f i n i t i o n SI units® a a b s o r p t i o n c o e f f i c i e n t a^ f r e e c o n v e c t i o n c o e f f i c i e n t b r a t i o o f backward t o t o t a l s c a t t e r e d r a d i a t i o n b , s u b s i d e n c e c o e f f i c i e n t s ^ sub c s p e c i f i c heat o f a i r J kg ^ -3 -1 C heat c a p a c i t y J m K C n o n - t u r b u l e n t t r a n s f e r c o e f f i c i e n t f o r he a t W m ^ ti C n o n - t u r b u l e n t t r a n s f e r c o e f f i c i e n t f o r W m "^K ^ m o i s t u r e heat t r a n s f e r c o e f f i c i e n t C T f o r n e u t r a l atmosphere 0^ f r i c t i o n c o e f f i c i e n t CT„, C T T f o r n e u t r a l atmosphere UN U -3 d a t m o s p h e r i c d u s t p a r t i c l e s cm • -2 D,, m u l t i p l y - r e f l e c t e d s o l a r r a d i a t i o n W m D -2 D d i f f u s e s o l a r r a d i a t i o n W m s e^ vapour p r e s s u r e mb "2 -1 E Q e v a p o r a t i o n kg m s • , -2 -1 E p o t e n t i a l e v a p o r a t i o n kg m s f C o r i o l i s parameter s ^ g a c c e l e r a t i o n due t o g r a v i t y m s 2 h hour a n g l e d e grees h mixed l a y e r h e i g h t m © I n t he e q u a t i o n s g i v e n i n S e c t i o n s 2.2 and 2.3 the symbols a r e i n cgs u n i t s . x i i i d i f f u s i v i t y i n t e g r a l s m -2 solar constant W m _2 d i r e c t beam solar r a d i a t i o n W m von Karman constant thermal conductivity W m ^ -2 net solar r a d i a t i o n W m Monin-Obukhov s t a b i l i t y length m -2 net long-wave r a d i a t i o n W m -2 outgoing long-wave r a d i a t i o n W m -2 incoming long-wave r a d i a t i o n W m latent heat of vaporization J kg ^ r e l a t i v e o p i t c a l a i r mass moisture a v a i l a b i l i t y factor -2 -h -1 thermal i n e r t i a J m s K surface pressure mb s p e c i f i c humidity kg kg ^ -1 s p e c i f i c humidity at z^ (Model M), s p e c i f i c kg kg humidity of the mixed layer (Model C) surface saturation s p e c i f i c humidity kg kg -2 net all-wave r a d i a t i o n W m -2 latent heat f l u x W m -2 anthropogenic heat release W m -2 subsurface heat f l u x W m -2 sensible heat flux W m radius vector gas constant J kg V ^ Richardson number x i v XV w 6 w c X X. ".1 Y 6 (SR. (ew). entrainment v e l o c i t y synoptic-scale v e r t i c a l v e l o c i t y adiabatic exchange c o e f f i c i e n t s t a b i l i t y corrected exchange c o e f f i c i e n t correction factor for multiple r e f l e c t i o n depth or height depth of uppermost subsurface layer atmospheric damping height (Model M), height of surface layer (Model C) subsurface damping depth height of t r a n s i t i o n layer surface roughness length zenith angle surface albedo Bowen r a t i o p o t e n t i a l temperature lapse rate of the inversion solar d e c l i n a t i o n r a d i a t i v e heating of the atmosphere inversion strength e f f e c t i v e atmospheric em i s s i v i t y surface emissivity s t a b i l i t y parameter po t e n t i a l temperature p o t e n t i a l temperature at (Model M), po t e n t i a l temperature of the mixed layer (Models A and C) kinematic sensible heat f l u x at the inversion base m s -1 m s -1 kg m s i - 2 - 1 kg m s m m degrees K m 1 degrees K s" 1 K K K K m s - 1 X V I -1 (8w) k i n e m a t i c s e n s i b l e heat f l u x a t the s u r f a c e K m s o 2 -1 K t h e r m a l d i f f u s i v i t y m s 2 -1 K eddy d i f f u s i v i t y f o r h e a t m s 2 -1 K eddy d i f f u s i v i t y f o r momentum m s m J J 2 -1 K eddy d i f f u s i v i t y f o r m o i s t u r e m s tij. s t a b i l i t y p arameter p a i r d e n s i t y kg m ^ -2 -4 a S t e f a n - B o l t z m a n n c o n s t a n t W m K <f> l a t i t u d e d e g rees $ s t a b i l i t y p arameter f o r h e a t H $ m s t a b i l i t y p a r a m eter f o r momentum x v i i ACKNOWLEDGEMENTS I g r e a t l y a p p r e c i a t e t he v a l u a b l e a d v i c e and t h e s u p p o r t o f my s u p e r -v i s o r , Dr. T.R. Oke. D r s . D.G. S t e y n and J.E. Hay, t h e o t h e r members o f my committee, were a l w a y s v e r y w i l l i n g t o be o f a s s i s t a n c e . I am a l s o g r a t e f u l t o D r s . T.N. C a r l s o n and T.P. Ackerman f o r s u p p l y i n g me w i t h t he c o d i n g , and i n s t r u c t i o n s f o r u s i n g , t h e i r u r b a n energy b a l a n c e models. F u r -t h e r , t h i s s t u d y would n o t have been p o s s i b l e were i t n o t f o r t h e work o f the many p e o p l e who p a r t i c i p a t e d i n the o b s e r v a t i o n programmes w h i c h p r o v i d e d the d a t a base. My f r i e n d and c o l l e a g u e , Sue Grimmond, was a l w a y s h e l p f u l , p a r t i c u l a r l y i n p r o o f r e a d i n g my f i n a l m a n u s c r i p t . The f u n d i n g f o r t h i s p r o j e c t was p r o v i d e d by the N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada t h r o u g h a p o s t g r a d u a t e s c h o l a r s h i p t o m y s e l f and r e s e a r c h g r a n t s to my s u p e r v i s o r . CHAPTER ONE INTRODUCTION M o d e l l i n g i s becoming more and more pr o m i n e n t i n c l i m a t o l o g y , n o t o n l y as an a l t e r n a t i v e t o measurement, but a l s o as an a p p r o a c h t o i n c r e a s i n g our u n d e r s t a n d i n g of a t m o s p h e r i c p r o c e s s e s and as a method f o r p r e d i c t i o n of a t m o s p h e r i c phenomena. Of i n t e r e s t h e r e a r e models d e s i g n e d t o p r e d i c t the energy b a l a n c e and t e m p e r a t u r e o f a s p e c i f i e d s u r f a c e . I n r e c e n t y e a r s , s u c h models have been a p p l i e d t o u r b a n a r e a s . I n t h i s c a s e , t h e " s u r f a c e " becomes a complex m i x t u r e o f d i f f e r e n t m a t e r i a l s and g e o m e t r i e s . T h i s c o m p l e x i t y c r e a t e s problems b o t h f o r t h e m o d e l l e r and the f i e l d o b s e r v e r , and has been p a r t l y r e s p o n s i b l e f o r the l a c k of model v e r i f i c a t i o n . The p r e s e n t s t u d y aims to h e l p overcome t h i s gap by u s i n g a r a t h e r r a r e s e t of energy b a l a n c e o b s e r v a t i o n s , g a t h e r e d i n a suburb o f Vancouver, B r i t i s h C o l u m b i a , t o t e s t the p e r f o r m a n c e of t h r e e u r b a n energy b a l a n c e / t e m p e r a t u r e models. 1.1 The Energy B a l a n c e The e n e r g y b a l a n c e i s a s t a t e m e n t o f t h e p a r t i t i o n i n g o f the a v a i l a b l e r a d i a n t energy (Q*) a t the E a r t h ' s s u r f a c e i n t o the t u r b u l e n t f l u x e s of s e n s i b l e and l a t e n t h e a t (Q and Q ) , and the c o n d u c t i v e f l u x o f s e n s i b l e n E heat (Q~). The energy b a l a n c e can be w r i t t e n : Q* = Q H + Q E + Q G ( l . D D u r i n g t h e day when t h e r e i s a s u r p l u s of r a d i a n t energy the t h r e e f l u x e s on t h e r i g h t - h a n d s i d e o f eqn. (1.1) w i l l n o r m a l l y be d i r e c t e d away from the s u r f a c e . At n i g h t the o p p o s i t e i s u s u a l l y t r u e . C o n v e n t i o n a l l y , a l l f l u x e s a r e p o s i t i v e i n t h e former c a s e and n e g a t i v e i n the l a t t e r . I n u r b a n a r e a s 1 an a d d i t i o n a l term (Q_), representing the heat release due to Man's act-r i v i t i e s , i s often added to the left-hand side of eqn. (1.1). Oke (1978; _2 p.243) gives an average value of 15 W m for t h i s term for'the c i t y of Vancouver (the value may be s l i g h t l y higher i n d i f f e r e n t parts of the c i t y ) i n the summertime. As the observations used i n t h i s study were ca r r i e d out during the summer, at a l o c a t i o n possessing no obviously large heat sources, i t was decided that Q could be ignored as i t i s small i n com-parison with Q*. This decision conforms with that of others (eg. Kalanda et a l . , 1980; Oke and McCaughey, 1983). The p a r t i t i o n i n g of Q* between the three surface heat fluxes depends on the surface moisture, the substrate thermal properties, the atmospheric and substrate temperature p r o f i l e s , the atmospheric humidity, and the atmospheri d i f f u s i v i t y which depends on such things as wind speed and surface roughness The heat fluxes can be represented by the flux-gradient equations: % = - P V H H ( 1 - 2 ) 3 T where p and c are the density and s p e c i f i c heat of a i r , L i s the latent p e heat of vaporization, <^ and are the eddy d i f f u s i v i t i e s for heat and 39 3 Q water vapour, and K i s the substrate thermal conductivity. -— and -r- 1- are o Z a Z the v e r t i c a l gradients of atmospheric p o t e n t i a l temperature and s p e c i f i c 3 T humidity and -— i s that of the substrate temperature. o z A few, possibly o v e r s i m p l i f i e d , examples can be given to i l l u s t r a t e the 3 i n t e r a c t i o n s between t h e s u r f a c e c o n d i t i o n s , the s u r f a c e t e m p e r a t u r e , and the terms o f t h e energy b a l a n c e . Over a m o i s t s u r f a c e most o f t h e energy w i l l be used f o r e v a p o r a t i o n so t h a t t h e l a t e n t h e a t f l u x w i l l become the dominant term i n t h e b a l a n c e . Such a s u r f a c e w i l l have a r e l a t i v e l y low s u r f a c e t e m p e r a t u r e . On t h e o t h e r hand, i f a s u r f a c e i s d r y i t s t e m p e r a t u r e w i l l be r e l a t i v e l y h i g h and t h e t u r b u l e n t s e n s i b l e heat f l u x w i l l be dom-i n a n t i n t h e b a l a n c e . Over a v e r y rough s u r f a c e , such as a c i t y , the t u r -b u l e n t f l u x e s w i l l be enhanced due t o t h e i n c r e a s e d a b i l i t y o f the atmos-phere t o t r a n s p o r t h e a t . The r e s u l t i n g r e m oval o f h e a t from the s u r f a c e a c t s t o l o w e r i t s t e m p e r a t u r e . D u r i n g t h e n i g h t , when t h e atmosphere i s s t a b l e , b o t h t u r b u l e n t f l u x e s w i l l be r e d u c e d and the s u r f a c e t e m p e r a t u r e w i l l be l a r g e l y c o n t r o l l e d by a b a l a n c e between t h e s u b s u r f a c e heat f l u x and the n e t r a d i a t i o n . Through the energy b a l a n c e , t h e r e f o r e , the s u r f a c e c o n d i t i o n s c o n t r o l the s u r f a c e c l i m a t e . I t f o l l o w s t h a t energy b a l a n c e models can be used t o d e t e r m i n e what t h e c l i m a t e o f a s p e c i f i e d s u r f a c e w i l l be. 1.2 The Framework o f Energy B a l a n c e Models Energy b a l a n c e models a r e u s u a l l y c l a s s i f i e d as o n e - d i m e n s i o n a l , t h e o r e t i c a l models. The f o r m e r c h a r a c t e r i s t i c i s a consequence o f one o f the m ajor a s s u m p t i o n s o f t h e s e models: t h e r e i s no a d v e c t i o n o f heat o r "moisture. T h e i r b a s i s i s t h e p r i n c i p l e o f the c o n s e r v a t i o n of e n e r g y , ex-p r e s s e d by t h e energy b a l a n c e e q u a t i o n (eqn. 1.1), w h i c h r e s u l t s i n the use of t h e f o l l o w i n g p r o c e d u r e t o compute t h e s u r f a c e t e m p e r a t u r e and energy b a l a n c e components. The n e t a l l - w a v e r a d i a t i o n (Q*) i s c o m p r i s e d of t h e n e t s o l a r ( K * ) , and long-wave ( L * ) , r a d i a t i o n a t the s u r f a c e . The former can be computed based on r a d i a t i v e t r a n s f e r t h e o r y as i t r e l a t e s t o the r e c e i p t o f s o l a r r a d -i a t i o n a t t h e s u r f a c e , w h i l e t h e l a t t e r can be e x p r e s s e d : L* = o(e.T*- - e T 4 ) (1.5) A A o o where a i s t h e S t e f a n - B o l t z m a n n c o n s t a n t , e, and e a r e t h e a t m o s p h e r i c A o and s u r f a c e e m i s s i v i t i e s , and and T q t h e a t m o s p h e r i c and s u r f a c e temp-e r a t u r e s . I t i s c l e a r t h a t t h e n e t long-wave r a d i a t i o n , as w e l l as t h e t h r e e s u r f a c e h e a t f l u x e s (eqns. 1.2-1.4), a r e a l l f u n c t i o n s o f s u r f a c e t e m p e r a t u r e . (The s u r f a c e s p e c i f i c h u m i d i t y i n eqn. (1.3) can be ex-p r e s s e d as a f u n c t i o n o f s u r f a c e t e m p e r a t u r e . ) The n e t s o l a r r a d i a t i o n and a l l v a r i a b l e s i n eqns. (1.2) t h r o u g h ( 1 . 5 ) , a p a r t from the s u r f a c e t e m p e r a t u r e , can be d e t e r m i n e d based on a s e t o f s p a t i a l , t e m p o r a l , met-e o r o l o g i c a l and s u r f a c e c o n d i t i o n s w h i c h a r e i n p u t t o t h e models. The energy b a l a n c e e q u a t i o n (eqn. 1.1) can t h e n be s o l v e d i t e r a t i v e l y f o r the e q u i l i b r i u m s u r f a c e t e m p e r a t u r e . T h i s t e m p e r a t u r e , i s , i n t u r n , used t o compute t h e n e t r a d i a t i o n and s u r f a c e heat f l u x e s . 1. 3 Model A p p l i c a t i o n s Urban energy b a l a n c e models have s e v e r a l p o s s i b l e a p p l i c a t i o n s . F i r s t , t h e y have been used as t o o l s f o r i n c r e a s i n g o u r u n d e r s t a n d i n g o f t h e p r o c -e s s e s a t work i n t h e urban atmosphere (eg. Myrup, 1969). Second, t h e y c o u l d be used as a l t e r n a t i v e s t o measurement programmes wh i c h a r e o f t e n b o t h c o s t l y and time-consuming. T h i r d , t h e models might be v a l u a b l e i n urban p l a n n i n g t o p r e d i c t the e f f e c t s o f a change i n l a n d - u s e on t h e s u r f a c e c l i m a t e (eg. O u t c a l t , 1972a). F o u r t h , i t has been s u g g e s t e d t h a t urban c l i m a t e models c o u l d p r o v e t o be u s e f u l i n p u t t o m e s o s c a l e models t o d e t e r -mine t h e e f f e c t s o f t h e u r b a n s u r f a c e on l a r g e s c a l e a t m o s p h e r i c dynamics and t h u s p o s s i b l y on weather p a t t e r n s o v e r u r b a n a r e a s ( C a r l s o n e t a l . , 1981) F i n a l l y , these models might be valuable i n dealing with a i r p o l l u t i o n prob-lems. They could be used to predict the r a d i a t i v e e f f e c t s of atmospheric pollutants on the surface energy balance and temperature (eg. Ackerman, 1977). They might also provide the necessary surface boundary conditions for more complex dynamical models of the urban atmosphere, which when coupled to a i r p o l l u t i o n models, may be capable of forecasting a i r q u a l i t y i n urban regions. Knowledge of the siz e of the turbulent sensible heat fl u x i s important i n determining the extent of convective mixing and the height bf the mixed layer, both of which are important determinants of pollutant concentrations. 1.4 The Urban Energy Balance Models: Background Three urban energy balance models w i l l be tested: Myrup (1969), Ackerman (1977), and Carlson and Boland (1978). Other urban energy b a l -ance models have been developed as part of much larger urban climate models (eg. Atwater, 1972; McElroy, 1973). These are c l a s s i f i e d as d i f -f e r e n t i a l models as they are used to produce temperature and wind f i e l d s based on the equations for the conservation of energy, mass, momentum, and pollutant species. The study of these models, however, i s not within the scope of t h i s research. For brevi t y , the three models to be studied w i l l be referred to as Models M, A, and C, respectively. They u t i l i z e a v a r i e t y of d i f f e r e n t methods and increase i n complexity from Model M through to Model C. Model M requires j u s t under 1.0 s of CPU time on an Amdahl 470/V8 computer, Model A about 2.0 s, and Model C about 6.0 s. A l l three models can be applied to clear sky conditions only. Each w i l l be described i n d e t a i l i n Chapter 2, here a few background notes regarding t h e i r h i s t o r y and use w i l l be given. 6 Model M was o r i g i n a l l y d e v e l o p e d as a t o o l f o r use i n i d e n t i f y i n g t he causes o f t h e u r b a n heat i s l a n d . I t has s i n c e been m o d i f i e d s e v e r a l t i m e s , t h e most e x t e n s i v e m o d i f i c a t i o n s were t h o s e o f O u t c a l t (1972a). I n the same p a p e r , O u t c a l t d i s c u s s e s the a p p l i c a t i o n o f t h i s model t o s e v e r a l non-urban s i t u a t i o n s w h i c h , he f e l t , p r o d uced r e a s o n a b l y r e a l i s t i c r e s u l t s . He a l s o compared m o d e l l e d s u r f a c e t e m p e r a t u r e s w i t h t h o s e d e t e r m i n e d from a i r c r a f t o b s e r v a t i o n s o v e r Ann A r b o r , M i c h i g a n ( O u t c a l t , 1972b). The mod-e l l e d t e m p e r a t u r e s were w i t h i n 1.5° C o f t h e o b s e r v e d . I n O u t c a l t (1972c) Model M was used t o a n a l y z e s e a s o n a l i n f l u e n c e s on t h e f a c t o r s c a u s i n g t h e urban heat i s l a n d . Oke (1972) used t h e model t o g a i n i n s i g h t i n t o the r o l e o f e v a p o r a t i o n i n u r b a n a r e a s . Morgan e t a l . (1977) used Model M t o s i m u l a t e m i c r o - c l i m a t e s i n t h e c i t y o f Sacramento, C a l i f o r n i a . They s t a t e : "because o f t h e problems i n h e r e n t i n a p p l y i n g m i c r o m e t e o r o l o g i c a l measure-ment t e c h n i q u e s t o urban s i t e s , d e t a i l e d c o m p a r i s o n s between the c a l c u l a t e d r e s u l t s and measured v a l u e s were n o t made" ( p . 6 4 ) . Model M was used i n a s t u d y by Pease e t a l . (1980) t o produce maps o f u r b a n s u r f a c e t e m p e r a t u r e s f o r B a i t i m o r e , M a r y l a n d . Greene (1980) f u r t h e r t e s t e d t h e model, i n t h e same c i t y , a g a i n s t a i r c r a f t - d e t e r m i n e d s u r f a c e t e m p e r a t u r e s . On a v e r a g e , the model was found t o u n d e r - p r e d i c t o b s e r v e d s u r f a c e t e m p e r a t u r e s by 14° C. Tapper e t a l . (1981) compared t e m p e r a t u r e s m o d e l l e d by M t o s c r e e n - h e i g h t v a l u e s o b s e r v e d i n c a r t r a v e r s e s o f C h r i s t c h u r c h , New Z e a l a n d . They s t a t e t h a t t h e model s i m u l a t e s t h e s u r f a c e t e m p e r a t u r e f i e l d r e a s o n a b l y w e l l . Such a c o n c l u s i o n i s q u e s t i o n a b l e , however, as t h e o b s e r v e d t e m p e r a t u r e s were s c r e e n - h e i g h t t e m p e r a t u r e s , n o t s u r f a c e t e m p e r a t u r e s , as p r e d i c t e d by the model. Model M has a l s o been used e x t e n s i v e l y f o r t e a c h i n g p u r p o s e s (eg. Unwin, 1981). I n each o f the above s t u d i e s Model M was m o d i f i e d some-what from the o r i g i n a l . 7 Model A was developed for determining the e f f e c t s of atmospheric aerosols on the surface energy balance i n urban areas (Ackerman, 1976). Data from the Great Plains Meteorological Experiment, O ' N e i l l , Nebraska (Lettau and Davidson, 1957) have been used to test t h i s model. Un-fortunately, the accuracy of the values of the surface temperatures and heat fluxes from t h i s experiment are i n doubt as they were calculated, rather than d i r e c t l y measured. The net r a d i a t i o n and surface temperatures estimated by Model A agreed quite well with the observations but the ob-served and modelled heat fluxes were not nearly as well matched (Ackerman, 1977). Data from the Great Plains Meteorological Experiment were also used to test Model C (Carlson and Boland, 1978). Remarkable agreement was found between the observations and the modelled sensible and latent heat fluxes. In addition, t h i s model was tested against observations from the Wangara Experiment, Hay, A u s t r a l i a (Clarke, 1971). There was quite good agreement between observed and modelled near-surface temperatures, net r a d i a t i o n , and ground heat fluxes. F i n a l l y , the modelled sensible heat fluxes and temp-eratures at 72 m elevation, as well as the modelled mixed layer heights, were compared to two hours worth of measurements made from a i r c r a f t over Los Angeles (Carlson and Boland, 1978). The paucity of data, i n t h i s case, prev-ents any us e f u l conclusions from being made. Model C was put into use, i n the urban context, to obtain values for the substrate thermal properties and the surface moisture a v a i l a b i l i t y factor (Sec. 4.3). Normally i t i s d i f f i c u l t to determine these values for an urban area, due to the heterogeneity of the surface. The procedure, followed by Carlson et a l . (1981), to determine them involved the inversion of the 8 model such that surface temperature f i e l d s determined from s a t e l l i t e observations became model input, and the thermal properties and moisture a v a i l a b i l i t y f a c t o r s , output. From t h i s the authors were able to produce maps of these surface c h a r a c t e r i s t i c s f o r the c i t i e s of Los Angeles and St. Louis. It i s clear that although Models M, A, and C have been used i n the urban context they have not been adequately v e r i f i e d against observations i n urban areas. There are two major reasons f o r t h i s , both of which are a re s u l t of the complexity of the urban surface. F i r s t , no measurements of the energy balance had been made i n urban areas. Second, i t i s d i f f i c u l t to assign values to those model inputs which describe surface character-i s t i c s . For the purposes of th i s study the f i r s t problem has been over-come as a r e l i a b l e set of energy balance observations are av a i l a b l e for a suburb of Vancouver, B.C. These w i l l be described i n Chapter 3. Un-fortunately, the second problem remains and i t s impact w i l l be considered i n Chapter 4. A t h i r d problem which was not mentioned by any of the or i g i n a t o r s or users of the three models, apart from Carlson and Boland (1978) and Tapper et a l . (1981), also complicates t h e i r v e r i f i c a t i o n . Because of the large roughness elements i n an urban area the l o c a t i o n of the "surface" for which the temperature and energy balance components are computed w i l l be d i f f e r e n t from the actual surface. I t i s generally f e l t that the former may be located j u s t below roof l e v e l (T.R. Oke and D.G. Steyn, personal communication). This discrepancy causes obvious problems both i n assigning values to the model inputs which describe the surface, as well as i n com-paring the model output to the observations. 9 1.5 Incentives and Objectives The incentives for carrying out t h i s study are two-fold. F i r s t , and most important, the models have not been v e r i f i e d against urban obser-vations. Such te s t i n g must be done i f the models are to be applied i n urban areas. At present i t i s being i m p l i c i t l y assumed that the theory developed for simple surfaces can be applied i n the urban context. Second, the p o t e n t i a l value of these models, as outlined i n Section 1.3, cannot be r e a l i z e d u n t i l they have been v e r i f i e d . The main objective of t h i s study i s , therefore, to v e r i f y the three urban energy balance models outlined, using the a v a i l a b l e observations. In addition, the r e l a t i v e a b i l i t i e s of the three models to duplicate the obser-vations w i l l be assessed and, where possible, any apparent problem areas w i l l be i d e n t i f i e d , and a l t e r n a t i v e methods presented. CHAPTER TWO THE MODELS The general framework which i s more or les s common to each model was outlined i n Chapter 1. The purpose of the present chapter i s to give a more de t a i l e d d e s c r i p t i o n of each model, beginning with the simplest, and ending with the most complex. 2.1 Model M A l i s t of the inputs required to run t h i s model i s given i n Table 2.1 and a simple flow chart i n Figure 2.1. The model i s based on the SI system of units and i n i t i a t e s c a l c u l a t i o n s for ju s t a f t e r midnight on the s p e c i f i e d dates. The input mean d a i l y a i r temperature i s used, i n the f i r s t time step, when the surface temperature of the previous time step i s required i n the computations. The model can be divided into three major sections: the pre-liminary, the solar r a d i a t i o n , and the surface temperature/energy balance, c a l c u l a t i o n sections. 2.1.1 Preliminary Calculations In t h i s section the input data are read and several variables which thereafter remain constant throughout the model-day are calculated. F i r s t , the density and s p e c i f i c heat of a i r are calculated as functions of the input mean di u r n a l a i r temperature (T^). Second, the substrate conductivity i s determined as a product of the d i f f u s i v i t y and heat capacity. Third, the substrate damping depth (z ) i s computed as a function of the d i f -f u s i v i t y ( K ): z = ( 1 2 ^ ) ^ (2.1) 4 where t i s the length of a diurnal heat pulse (= 4.32 X 10 s ) . This depth, 10 Table 2.1 - Input f o r Model M Variable Symbol Units (SI) Start time Latitude <j> degrees Solar d e c l i n a t i o n 6 degrees Radius vector r -Mean d i u r n a l a i r temperature T^ °C Mean d i u r n a l wind speed u^ m s ^  Mean d i u r n a l vapour pressure e^ mb Station pressure P mb P r e c i p i t a b l e water w mm _3 Dust p a r t i c l e s d cm Surface albedo a Surface roughness length Z q m -3 -1 Subsurface volumetric heat capacity C J m K 2 -1 Subsurface thermal d i f f u s i v i t y K m S Moisture a v a i l a b i l i t y factor M \ 12 Preliminary Calculations Calculate K* values for e n t i r e day © Begin time step Cal c u l i ite Q* 5 Estimate T o Calculate subsurface temperature p r o f i l e Check convergence of energy balance equation No convergence ^ Convergence Calculate Q H. Q E and Q G Increment time by 15 min return to © F i g . 2.1 - Flow chart for Model M 13 which i s the base of the model's subsurface layer, i s l a t e r used as a basis f o r computing a s i x l e v e l subsurface depth p r o f i l e . The temperature at the damping depth i s the input mean d a i l y a i r temperature which remains constant. Fourth, the analogous term f o r the atmosphere, the atmospheric damping height ( z^)> I s calculated based on two e f f e c t i v e atmospheric d i f f u s i v i t i e s : bulk adiabatic d i f f u s i v i t y = (2.2) l n ( z A / z Q ) 2 disturbance penetration depth d i f f u s i v i t y = ~yit (2.3) where k i s the von Karman constant, u^ the input mean d a i l y wind speed, Z Q the roughness length, and t i s as i n eqn. (2.1). These equations are solved i t e r a t i v e l y by i n i t i a l l y s e t t i n g the damping height equal to the roughness length and subsequently increasing i t by 10 cm increments. At each step the two d i f f u s i v i t i e s (eqns. 2.2 and 2.3) are calculated. The former w i l l decrease, while the l a t t e r w i l l increase, with increasing height. The i t e r a t i v e process continues u n t i l a height i s found such that the disturbance penetration depth d i f f u s i v i t y i s greater than the bulk adiabatic d i f f u s i v i t y (Outcalt and Carlson, 1975). This height becomes the atmospheric damping height (defined as that height at which the input mean d a i l y a i r temperature and wind speed apply). F i n a l l y , the adiabatic exchange c o e f f i c i e n t (X) used i n the c a l c u l a t i o n of the turbulent fluxes, i s c a l c u l a t e d : , 2 k pu X = - (2.4) ( l n ( z A / 2 o ) ) 2 14 where p i s the density of a i r . This c o e f f i c i e n t i s s t r i c t l y a mechanical exchange term. It i s l a t e r modified, i n each time step, for buoyancy e f f e c t s using the Richardson number cor r e c t i o n factor. 2.1.2 Solar Radiation Calculations After the preliminary c a l c u l a t i o n s have been completed, quarter hourly solar r a d i a t i o n values are calculated for the s p e c i f i e d date. The method used i s from Gates (1962) and i s based on the theory of r a d i a t i v e transfer. The net solar r a d i a t i o n (K*) i s c a l c u l a t e d : K* = ( I s + D s + D B) (1 - a) (2.5) where a i s the surface albedo. In the model c a l c u l a t i o n s a shadow r a t i o i s incorporated into eqn. (2.5) which has a value of 0.0 for unshaded surfaces and a value of 1.0 for completely shaded surfaces. This factor was ignored ( i . e . shadow r a t i o = 0.0) i n the present study for two reasons. F i r s t , solar r a d i a t i o n was measured at the observation s i t e using an unobstructed instrument. Second, i f the modelling "surface" i s close to roof l e v e l , as suspected (Sec. 1.4), the shadow r a t i o w i l l probably be very small, anyway. I , D , and D_. are the d i r e c t , d i f f u s e , and m u l t i p l y - r e f l e c t e d solar rad-s s B ] i a t i o n , r e s p e c t i v e l y : (2.6) (2.7) (2.8) I i s the solar constant (= 1354.8 W m and r i s the radius vector which o i s input to the model to correct for the e l l i p t i c a l o r b i t . The cosine of I = (I /r ) cos Z exp(a + s) s o 2 D g = 0.5(1 / r ) cos Z (1 - exp s) •DB = 0.5a(I + D J (1 - exp s ) . 15 the zenith angle (cos Z) i s calculated using the solar d e c l i n a t i o n (6) and l a t i t u d e (((>) which are also input to the model, and the solar hour angle (h) which i s computed for each time step: cos Z = sin<}> sin<5 + cose(> cosS cos h. (2.9) The absorbing and sc a t t e r i n g c o e f f i c i e n t s (a and s) are given by Brooks (1959): a = -0.174(wm/20)°" 6 (2.10) s = -0.089(Pm/1013)°' 7 5 - 0.083(dm)° - 9 (2.11) where w i s the p r e c i p i t a b l e water, P the s t a t i o n pressure, and d the number of dust p a r t i c l e s per unit volume of a i r . A value for each i s input to the model, m i s the r e l a t i v e o p t i c a l . a i r mass which i s the r a t i o of the actual path length of the Sun's rays through the atmosphere, to the zenith path length. It i s given by an approximation formula devised by Kasten (1966) : m = (P/1013) (cos Z. + 0.15(93.885 - Z) J ) (2.12) which includes a pressure correction. 2.1.3 Surface Temperature/Energy Balance Calculations The energy balance components and the surface temperature are computed using an i n t e r v a l - h a l v i n g i t e r a t i o n algorithm. This i s a very simple tech-nique which begins, i n t h i s case, with two extreme temperature values (T. - 20 and T + 25) and averages them to obtain an i n i t i a l surface temp-erature estimate. As the input mean d a i l y a i r temperature (T ) does not /A vary through the day each time step begins with the same i n i t i a l temperature estimate. If t h i s temperature does not provide a s o l u t i o n to the balance 16 equation the i n t e r v a l between the two extremes i s halved and a new surface temperature estimate i s obtained by averaging over t h i s smaller i n t e r v a l . This procedure continues u n t i l a temperature i s found such that the con-vergence c r i t e r i o n of the model can be met. In t h i s model the sum of the -2 components of the energy balance equation must not exceed 1 W m . Id e a l l y , of course, t h i s sum should equal zero. The following outlines the procedure followed at f i f t e e n minute i n t e r v a l s , the length of a time step, to obtain the surface temperature and energy balance components using the i n t e r v a l -halving technique. To begin with the net long-wave r a d i a t i o n (L*) i s computed as a func-ti o n of the surface temperature of the previous time step, using Stefan-Boltzmann's law. With the assumption that the surface e m i s s i v i t y i s unity ( i . e . e Q = 1.0), the long-wave r a d i a t i o n emitted by the surface (L+) i s : L+ = e aT 4 (2.13) o o where a i s the Stefan-Boltzmann constant. The incoming long-wave r a d i a t i o n (L+) i s calculated using the formula of Idso and Jackson (1969) for the emissivity of the atmosphere (e.): L+ = {1 - 0.261 exp(-7.77 X 10 _ 4(273 - T ) 2)} aT 4 (2.14) o o Because the surface temperature of the previous time step i s used the net long-wave r a d i a t i o n w i l l remain constant throughout the i t e r a t i v e sequence. The net long-wave r a d i a t i o n i s added to the computed net solar r a d i a t i o n for the time step to give the net all-wave r a d i a t i o n (Q*). Thus the value of the f i r s t component of the energy balance equation i s found based on the surface temperature of the previous time step. 17 Using the current surface temperature estimate, each of the three remaining terms i n the balance i s computed, beginning with the subsurface heat f l u x . For each surface temperature estimate a new subsurface temp-perature p r o f i l e i s determined using the f i n i t e difference s o l u t i o n of the d i f f u s i o n equation: 9T. 9 2T. • i - F ^ T T 1 (2.15) dZ th where T\ i s the temperature of the i l e v e l . For the f i r s t time step t h i s procedure cannot be used because a p r o f i l e from the previous time step i s required i n the computations. The temperature p r o f i l e of the f i r s t time step i s , therefore, set using the subsurface damping depth, the input mean d a i l y a i r temperature, and the surface temperature estimate. Depths are geometrically-spaced between the surface and the damping depth, and temp-eratures are geometrically-spaced between the surface temperature estimate and the mean d a i l y a i r temperature. This r e s u l t s i n a l i n e a r subsurface temperature p r o f i l e i n which the depths of the layers increase with depth i n the substrate. Because a l i n e a r temperature p r o f i l e at midnight, the f i r s t time step, i s u n r e a l i s t i c the model i s run through two diu r n a l cycles. The output of the second cycle i s used i n the analyses (Ch. 5), as the temperature p r o f i l e w i l l no longer be l i n e a r at the second midnight (Outcalt, 1972a). Using the temperature (T^) of the highest l e v e l (z^) the subsurface heat f l u x (CO i s computed: Q G = K(T X - T o ) / Z l (2.16) where K i s the substrate conductivity. (In th i s model, as well as i n Model A, the usual sign convention for the three surface heat fluxes (Sec. 1.1) 18 has been reversed for the purposes of the computations only.) Next the turbulent fluxes (Q and Q ) are determined. For each surface H E temperature estimate the adiabatic exchange c o e f f i c i e n t , calculated as i n eqn. (2.4), i s adjusted for s t a b i l i t y . For t h i s purpose, the Richardson number (Ri), a measure of atmospheric s t a b i l i t y , i s calc u l a t e d : Ri = g/T { ( 9 A - T q ) /U 2 } U n ( z A / z o ) } . (2.17) (As used i n t h i s model, the Richardson number i s not dimensionless, as i t should be.) In eqn. (2.17), 9^ i s the atmospheric temperature (T^) converted to p o t e n t i a l temperature, f i s the average of the surface and atmospheric temperatures, and g i s the acceleration due to gravity. The exchange co e f f -i c i e n t can then be adjusted using a correction f a c t o r : correction factor = | l - 32Ri|'5. (2.18.) The s t a b i l i t y corrected exchange c o e f f i c i e n t (X ) i s used to cal c u l a t e the sensible and latent heat fluxes: % = W 9 A " V <2-19> where c^ i s the s p e c i f i c heat of a i r , L g the latent heat of vaporization, and M the moisture a v a i l a b i l i t y f actor. The atmospheric s p e c i f i c humidity (q^) i s calculated using: q A = 0.622 e A/P. (2.21) The atmospheric vapour pressure (e A) i s equal to the product of the saturation vapour pressure at the atmospheric temperature and the r e l a t i v e humidity of 19 the time step. The r e l a t i v e humidity i s calculated for each time step as a function of the surface temperature at the previous time step and the input vapour pressure, which remains constant throughout the model-day. The surface saturation s p e c i f i c humidity ( c l o s ) i s calculated using the saturation vapour pressure at the surface temperature. The moisture a v a i l a b i l i t y factor i s used to account for the fa c t that the surface i s not always saturated. This factor w i l l be discussed further i n Chapter 4. If the s p e c i f i c humidity gradient i s p o s i t i v e , i n d i c a t i n g dewfall, then the moisture a v a i l a b i l i t y factor i s not used i n the c a l c u l a t i o n of the latent heat f l u x . At t h i s point values have been obtained for each of the components of the energy balance equation based on a t r i a l surface temperature. If these _2 values s a t i s f y the energy balance equation to within 1 W m the t r i a l sur-face temperature becomes the equilibrium surface temperature for the time step, and the cycle begins again for the next time step. If the equation i s not s a t i s f i e d , a new surface temperature estimate i s determined, as des-cribed e a r l i e r , and the cycle begins again for the same time step. ' -2.2 Model A A l i s t of inputs for Model A i s given i n Table 2.2 and a simple flow chart i n Figure 2.2. The computations of t h i s model are based on the cgs system of units but the output has been converted to SI u n i t s . The approp-r i a t e inputs are used, i n the f i r s t time step, when values from a previous time step are required i n the computations. In addition, the atmospheric l e v e l s i n the model w i l l be those of the input atmospheric temperature p r o f i l e . This model i s set to i n i t i a t e c a l c u l a t i o n s at about 0400 LST (Local Solar Time) on the s p e c i f i e d dates. As with Model M, i t i s run through two d i u r n a l cycles and the output from the second cycle i s used i n the analyses. Model A can Table 2.2 - Input for Model A Variable Symbol Units (cgs) -2 -1 Solar r a d i a t i o n K* ergs cm s Atmospheric temperature sounding height z cm p o t e n t i a l temperature 9 K I n i t i a l surface temperature T Q K Surface temperature one time step before the s t a r t T K o I n i t i a l mixed layer temperature 6^ K I n i t i a l mixed layer height h cm Climatic mean temperature T^ K Mean diu r n a l wind speed u^ cm s ^  -2 Station pressure P g cm Pr e c i p i t a b l e water w cm Subsidence c o e f f i c i e n t b , s ^" sub Surface roughness length z^ cm -3 -1 Subsurface volumetric heat capacity C ergs cm K 2 -1 Subsurface thermal d i f f u s i v i t y K cm s Bowen r a t i o (3 © Begin time step Determin e K* + L4-21 Calculate subsurface temperature p r o f i l e Compare T and 9. to o • A determine atmospheric s t a b i l i t y T < 0 , o 1 Stable 4-T > 9. o A Unstable Calculate T and mixed o layer height 7F sc. Calculate T ~7f Check convergence of energy balance and mixed layer height equations Check convergence of energy balance equation When convergence i s obtained calculate atmospheric temperature p r o f i l e _sk_ When convergence i s obtained ca l c u l a t e mixed layer height and atmospheric temperature p r o f i l e Calculate Q*, Increment time by 15 min return to © Fig . 2.2 - Flow chart for Model A (after Ackerman, 1976) 22 be d i v i d e d i n t o t h r e e major s e c t i o n s : t h e r a d i a t i o n , t h e s u b s u r f a c e temp-e r a t u r e p r o f i l e , and t h e s u r f a c e t e m p e r a t u r e / e n e r g y b a l a n c e and mixed l a y e r , c a l c u l a t i o n s e c t i o n s . Each s e c t i o n i s c o n t a i n e d i n a l a r g e l o o p w h i c h i s c y c l e d t h r o u g h once f o r e v e r y f i f t e e n m i n u t e time s t e p . The t h i r d s e c t i o n can be f u r t h e r s u b d i v i d e d i n t o a s e t o f c a l c u l a t i o n s f o r t h e s t a b l e , and a n o t h e r f o r t h e u n s t a b l e , atmosphere. 2.2.1 R a d i a t i o n C a l c u l a t i o n s A time s t e p b e g i n s w i t h t h e d e t e r m i n a t i o n o f t h e n e t s o l a r r a d i a t i o n and the i n c o m i n g long-wave r a d i a t i o n . Due t o t h e c o m p l e x i t y of the o r i g i n a l r a d i a t i o n c a l c u l a t i o n s , and the l a c k o f a p p r o p r i a t e i n p u t v a r i a b l e s t o r u n them i n t h e p r e s e n t s t u d y , t h e s e were not used. I n s t e a d , b o t h the n e t s o l a r r a d i a t i o n v a l u e s computed by Model C, and the i n c o m i n g long-wave r a d i a t i o n e q u a t i o n s o f Model C, were used i n Model A. I t was r e a l i z e d t h a t q u a r t e r h o u r l y s o l a r r a d i a t i o n v a l u e s from Model C c o u l d s i m p l y be r e a d i n t o Model A, w h i l e t h e i n c o m i n g long-wave r a d i a t i o n had t o be computed w i t h i n the model because, t h e c a l c u l a t i o n s r e q u i r e m o d e l l e d s u r f a c e and mixed l a y e r t e m p e r a t u r e s . Model C's i n c o m i n g long-wave r a d i a t i o n e q u a t i o n s (Sec. 2.3.2) were, t h e r e f o r e , i n c o r p o r a t e d i n t o Model A. The s u r f a c e and mixed l a y e r t e m p e r a t u r e s used i n t h e s e c o m p u t a t i o n s a r e from t h e p r e v i o u s time s t e p so t h a t t h e i n c o m i n g long-wave r a d i a t i o n r e m a i n s c o n s t a n t t h r o u g h o u t the i t -e r a t i v e sequence. The o u t g o i n g long-wave r a d i a t i o n , however, i s c a l c u l a t e d as p a r t of the i t e r a t i v e sequence u s i n g eqn. (2.13) and, a g a i n , the assump-t i o n i s made t h a t the s u r f a c e e m i s s i v i t y i s u n i t y . 2.2.2 S u b s u r f a c e Temperature P r o f i l e I n t h e f i r s t t i m e s t e p t h e d e p t h p r o f i l e i s c a l c u l a t e d u s i n g : z = R 2 (2.22) 23 where R i s i n i t i a l l y equal to the square root of two and i s subsequently incremented by t h i s value to produce a power law depth p r o f i l e i n which the depths of the layers increase with depth i n the substrate. The temperatures corresponding to these l e v e l s are computed for each time step using the f i n i t e d i f f e r e n c e form of the d i f f u s i o n equation (eqn. 2.15). To compute the p r o f i l e for the f i r s t time step an i n t i t i a l p r o f i l e i s required. For th i s study the equations used i n Model M to set up an i n i t i a l p r o f i l e were inserted into Model A. This i n i t i a l p r o f i l e i s interpolated so as to cor-respond with the calculated depth p r o f i l e and i s then used to compute the temperature p r o f i l e f o r the f i r s t time step. 2.2.3 Surface,Temperature/Energy Balance and Mixed Layer Calculations Once the net solar r a d i a t i o n , the incoming long-wave r a d i a t i o n , and the s o i l temperature p r o f i l e have been determined for the time step, the surface and mixed layer temperatures of the previous time step are compared to decide whether or not to follow the stable, or unstable, atmosphere c a l c u l a t i o n sequence. It i s within these two sequences that the i t e r a t i o n s used to de-termine the surface temperature and energy balance components are c a r r i e d out. In addition, each contains i t s own mixed layer height and atmospheric temperature p r o f i l e c a l c u l a t i o n s . Before the unstable and stable routines are described i n d i v i d u a l l y a b r i e f o u t l i n e of the theory common to both w i l l be presented. Since t h i s model i s based on the use of the Bowen r a t i o to describe the surface moisture status, the energy balance equation takes the following, s l i g h t l y d i f f e r e n t form, to that described previously: Q* + Q G + BQ R = 0. (2.23) 24 In eqn. (2.23), B = 1 + 3 where 3 , the Bowen r a t i o , i s the r a t i o of the sensible, to the l a t e n t , heat f l u x . The t h i r d term i n the balance i s , therefore, equivalent to the sum of the lat e n t and sensible heat fluxes, as would be expected. The sensible heat f l u x i s calculated based on a heat transfer coef-f i c i e n t (C^) and a f r i c t i o n c o e f f i c i e n t (C^) which are both dependent upon s t a b i l i t y : QH = p C p C T C U U A ( 9 A " V (2'24) where 6^ and u^ are the p o t e n t i a l temperature, and wind speed, of the mixed layer. The neutral atmosphere equivalents of Cy and C T ( C ^ and C^) are functions of the roughness length and the height of the mixed layer (h): C"* = 1/k {ln(0.025h/z Q)} + 8.4 (2.25) el}. = 0.74/k {ln(0.025h/z )} + 7.3. (2.26) IN O C _ and C are used, along with the bulk Richardson number ( R i _ ) : UN IN D R i B = gh/e A { ( e A-T o)/u 2} (2.27) to c a l c u l a t e C y and C T (Deardorff, 1972). For the unstable atmosphere: C" 1 = C"* - 25 exp(0.26? - 0.03c2) (2.28) CT1 - CTN + " cm (2-29) where X, = l o g ( - R i B ) - 3.5. (2.30) 25 For the stable atmosphere: CU = CUN ( 1 " R i B / R 1 C ) ( 2 ' 3 1 ) C T = C T N ( 1 - R i B / R i c ) (2.32) where R i ^ j the c r i t i c a l Richardson number, i s set equal to a constant value ,of 2.0 in the model. The latent heat f l u x i s not calculated using flux-gradient r e l a t i o n s h i p s i n t h i s model. It i s simply determined using the calculated sensible heat fl u x and the input Bowen r a t i o . The subsurface heat f l u x i s calculated as i n Model M (eqn. 2.16). The equations for the net r a d i a t i o n , the subsurface heat f l u x , and the turbulent sensible heat f l u x are substituted into eqn. (2.23) to obtain: K* + L+ - aT 4 + ( K / z 1 ) ( T 1 - T Q) + Bpc ^ C ^ O A - T Q ) = 0. (2.33) Since K*, L+, and T^ w i l l have been determined for the time step the remaining variables i n t h i s equation are h, found i n C T T and C_, 9., and T . The method U T A o of handling these v a r i a b l e s , so that eqn. (2.33) can be solved for the surface temperature, depends on the s t a b i l i t y regime of the atmosphere. 2.2.3.1 The Unstable Atmosphere Using both the mixed layer temperature and height from the previous time step the surface temperature becomes the only v a r i a b l e i n eqn. (2.33). The equation i s then solved i t e r a t i v e l y f o r the surface temperature using the secant method. This method requires two i n i t i a l temperature estimates for which the model uses the equilibrium surface temperatures from the two previous time steps. These temperatures, T (I - 1) and T Q ( I - 2 ) , are each 26 used i n eqn. (2.33) t o g i v e two r e s i d u a l s , F ( T Q ( I - 1)) and F(T ( I - 2 ) ) . Eqn. (2.34) i s t h e n used t o compute a new s u r f a c e t e m p e r a t u r e e s t i m a t e f o r t h e t i m e s t e p : T ( I - 2) F(T ( I - 1)) - T ( I - 1) F(T ( I - 2)) T ( I ) = -9 ^ 2 _o ^ ( 2 > 3 4 ) ° F ( T Q ( I - 1)) - F ( T Q ( I - 2)) The above p r o c e d u r e i s r e p e a t e d u n t i l a s u r f a c e t e m p e r a t u r e i s found w h i c h _2 when s u b s t i t u t e d i n t o eqn. (2.33) g i v e s a r e s i d u a l (F) o f l e s s t h a n 0.5 W m T h i s t e m p e r a t u r e i s used t o compute the net r a d i a t i o n and en e r g y b a l a n c e components f o r the ti m e s t e p . Once t h e s u r f a c e t e m p e r a t u r e has been d e t e r m i n e d t h e h e i g h t o f the mixed l a y e r and t h e a t m o s p h e r i c t e m p e r a t u r e p r o f i l e can be c a l c u l a t e d . T h i s i s done f o l l o w i n g S t u l l ( 1 9 7 3 ) . F i r s t , t h e e n t r a i n m e n t v e l o c i t y (w ) and t h e s y n o p t i c - s c a l e v e r t i c a l v e l o c i t y (w g) a r e used t o compute t h e new mixed l a y e r h e i g h t : 4£ = w + w (2.35) d t e s where a (8w) w = (2.36) e A and w = -b ,h. (2.37) s sub (9w) i s t h e k i n e m a t i c s e n s i b l e heat f l u x a t the s u r f a c e , w h i c h i s c a l -o c u l a t e d u s i n g t h e new s u r f a c e t e m p e r a t u r e , and i s e q u i v a l e n t t o Q^/pc . a p J 27 A, the inversion strength (Fig. 2.3), i s calculated using the temperature p r o f i l e of the previous time step, a^ i s a free convection c o e f f i c i e n t set equal to 0.2 and 1>SU^ i s the input subsidence c o e f f i c i e n t . F i n a l l y , a new atmospheric temperature p r o f i l e i s computed. To s t a r t , for a l l l e v e l s up to the height of the mixed layer the temperatures are set equal to the mean temperature of the mixed layer: 39 1.2(8w) sr s-2- • <2-38) Eqn. (2.38) contains the assumption that (9w)_^ = -0.2(8w) Q where (6w)_^ i s the kinematic sensible heat f l u x at the inversion base. The temperatures for each l e v e l above the mixed layer are calculated using: | f=-w g Y (2.39) where y, the p o t e n t i a l temperature lapse rate of the inversion, i s computed using the temperature p r o f i l e of the previous time step. wg i s calculated for each height (z), above the mixed layer, as: w = -b ,z. (2.40) s sub O r i g i n a l l y , a term for the r a d i a t i v e heating of the atmosphere was i n -cluded i n eqns. (2.38) and (2.39), and an advective term was included"in eqn. (2.38). The former was set equal to zero when the o r i g i n a l r a d i a t i o n routine was removed (T.P. Ackerman, personal communication), while the up-wind temperature and wind p r o f i l e s required to cal c u l a t e the l a t t e r (Ackerman, 1976) were not av a i l a b l e for t h i s study. (Eqn. (2.48) for the stable case (Sec. 2.2.3.2), also included these terms.) 28 P o t e n t i a l Temperature F i 8 - 2-3 - V e r t i c a l d i s t r i b u t i o n of p o t e n t i a l temperature f o r an unstable boundary l a y e r cappped by an i n v e r s i o n ( a f t e r Tennekes, 1973) 2.2.3.2 The Stable Atmosphere In the case of a stable atmosphere i t s equivalent of the unstable mixed layer i s the r e s u l t of f r i c t i o n , rather than heating, producing a small turbulent layer at the surface. The height of t h i s layer i s c a l -culated using the approach of Businger and Arya (1974): 0.72u^ h r r - 1 (2.41) where v * = fL ( 2 - 4 2 ) and A * L = _ . (2.43) kg(6w) o u j V i s the f r i c t i o n v e l o c i t y , a s t a b i l i t y parameter, L the Monin-Obukhov length, and f the C o r i o l i s parameter. Eqn. (2.41) i s a diagnostic equation, while the mixed layer height equation for the unstable atmosphere (eqn. 2.35) i s p r e d i c t i v e . Since the diagnostic equation gives le s s r e l i a b l e r e s u l t s than the p r e d i c t i v e equation (Nieuwstadt, 1981), the mixed layer height and temperature of the previous time step are not used i n eqn. (2.33) to deter-mine the surface temperature as they were i n the unstable case. Instead, eqn. (2.33), and eqn. (2.41) with the appropriate substitutions (eqn. 2.44), are used simultaneously to solve for both the surface temperature and the mixed layer height: 30 h - 0.72 (6 - T ) 2 = 0. (2.44) This pair of equations i s solved, i t e r a t i v e l y using the•.Newton-Raphson method. This requires estimates of both a temperature and mixed layer height, for which the values computed i n the previous time step are used. These are 3F 1 3F 1 3F 2 9F 2 used to compute the d e r i v a t i v e s , r-^—, r r — , -rzr~, and -——, where F. i s the d 1 dn dT dn I o o r e s i d u a l for eqn. (2.33) and that for eqn. (2.44). Eqns. (2.45) and (2.46) are then used to c a l c u l a t e new values for the surface temperature and mixed layer height: T o ( I ) = T o ( I - 1) + -3F 1 3 F 9 ) 2 3h - F 1 3h (2.45) h(I) = h(I - 1) + 3F, , i 1 3T 3F, - F 2 3T (2.46) where D = 3F X 3F 2 3 F 1 8 F 2 3T 3h 3h 3T ' o o (2.47) Two convergence c r i t e r i a were necessary i n t h i s case. F i r s t , the equilibrium surface temperature i s that which when substituted back into eqn. (2.33) produces a r e s i d u a l of le s s than 0.05 W m . Second, the change i n the height of the mixed layer from the l a s t i t e r a t i o n must be less than 50 cm. If the convergence c r i t e r i a are not met the values computed by eqns. (2.45) and (2.46) are used as new estimates and the procedure i s repeated. The new temperature and mixed layer height are used to compute the net r a d i a t i o n and energy balance components for the time step. The next step i s to compute a new atmospheric temperature p r o f i l e . 31 The mechanisms at work e f f e c t i n g the temperature changes d i f f e r depending upon the height i n the atmosphere. Up to the height ( z g ) where the sen-s i b l e heat f l u x becomes zero, eqn. (2.48) i s used to c a l c u l a t e the temp-eratures : 3T = - r — . z < z s (2.48) s Above z g , eqn. (2.39), as i n the unstable case, i s used. At the s t a r t of the stable regime z^ i s set equal to the depth of the lowest model layer. As the night goes on i t i s allowed to r i s e as a r e s u l t of shear-generated turbulence which works to gradually increase the depth of the very stable layer at the surface. This r i s e i s accomplished i n the model by c a l c u l a t i n g the l o c a l Richardson number at z g for each time step. When the Richardson number reaches a c r i t i c a l value of 0.75, z g jumps to the height of the next model layer. As a r e s u l t of t h i s z g may r i s e by three or four model layer depths by morning. This procedure prevents the lowest layer from becoming unreasonably cold during the night (Ackerman, 1976; pp.73-74). 2.3 Model C A l i s t of the inputs required to run Model C i s given i n Table 2.3 and a flow chart i s outlined i n Figure 2.4. As with Model A, i t i s based on the cgs system but the output has been converted to SI u n i t s . This model i n i t i a t e s c a l c u l a t i o n s at about 0600 LST on the s p e c i f i e d dates and, a l -though the time step i s only 3 min, the model output i s quarter hourly, as i n Models M and A. It begins with a sequence of preliminary c a l c u l a t i o n s which set the i n i t i a l conditions. It then proceeds through a s e r i e s of Table 2.3 - Input for Model C 32 Variable Symbol Units (cgs) Date Start time Latitude Atmospheric temperature/pressure sounding pressure l e v e l temperature dew point depression Climatic mean temperature Wind sounding d i r e c t i o n height speed Horizontal pressure gradients Horizontal temperature gradients P r e c i p i t a b l e water Surface albedo Surface roughness length Thermal i n e r t i a Moisture a v a i l a b i l i t y factor w degrees mb o„ K P M f t knots mb K cm cm -2 -1 -h c a l cm K s Preliminary c a l c u l a t i o n s , includes estimate of i n i t i a l T © Begin time step Calculate Q< Check sign of Q Q H < 0 Night-time tr Q H > o Daytime Compute: Q atmospheric n temperature and wind p r o f i l e s , d i f f u s i v i t y i n t e g r a l , Q and T E o Compute: d i f f u s i v i t y i n t e g r a l , mixed layer height, wind p r o f i l e , T G, Q E and Q H Calculate subsurface temperature p r o f i l e Increment time by 3 min return to © Fig . 2.4 - Flow chart for Model C 34 computations for every 3 min time step. F i r s t , the net r a d i a t i o n i s deter-mined. Second, the energy balance components and surface temperature are calculated. This l a t t e r set of c a l c u l a t i o n s d i f f e r s depending upon whether i t i s day- or night-time, that i s , depending upon whether the sensible heat f l u x of the previous time step i s p o s i t i v e or negative, r e s p e c t i v e l y . F i n a l l y , the s o i l temperature p r o f i l e for the next time step i s computed. 2.3.1 Preliminary Calculations Atmospheric temperature and wind p r o f i l e s are input to the model to set up most of the i n i t i a l meteorological conditions. F i r s t , p o t e n t i a l temperature and s p e c i f i c humidity lapse rates are determined. Second, the temperature (9 ) and s p e c i f i c humidity (q.) at z are determined by A A A l i n e a r i n t e r p o l a t i o n , z^, the top of the surface or constant f l u x layer (Fig. 2.5), remains at a constant height of 50 m for a l l model c a l c u l a t i o n s . This i s also the i n i t i a l height of the mixed layer. Third, i n i t i a l estimates of the surface temperature and s p e c i f i c humidity are made. The surface temperature i s simply set at one degree l e s s than the temperature at the lowest l e v e l of the sounding, the screen-level temperature, while the surface s p e c i f i c humidity i s calculated as a function of the vapour pressure corresponding to the surface temperature estimate. Fourth, a cubic spline i s used to int e r p o l a t e the i n i t i a l wind sounding so that i t corresponds with the model l e v e l s . The model l e v e l s are 250 m apart extending from a height of 50 m up to a height of 2300 m. The f r i c t i o n v e l o c i t y for the f i r s t time step i s computed using the interpolated wind speed at 50 m (u ): A k u A u* . (2.49) ln ( z . / z ) - $ A o m Because n e u t r a l i t y i s assumed for the f i r s t time step the s t a b i l i t y para-35 Mixed Layer Surface Layer T r a n s i t i o n Layer 50 m Subsurface Layer F i g . 2.5 - Basic framework of Model C (after Carlson and Boland, 1978) meter (<J> ) i s zero, m Geostrophic wind p r o f i l e s are computed using the input horizontal pressure and temperature gradients. Two geostrophic wind p r o f i l e s are com-puted; one for use i n the night-time computations and the other, which covers a considerably deeper atmospheric layer, for use i n the daytime computations. The atmospheric layers for the nocturnal c a l c u l a t i o n s begin at a height of 50 m and are 50 m apart up to a height of 500 m. <. F i n a l l y , the subsurface temperature p r o f i l e of the f i r s t time step i s calculated. The depth p r o f i l e i s set using: z = ( e R - l ) s (2.50) where R i s i n i t i a l l y set equal to zero and subsequently incremented by one to set up f i v e s o i l l a y e r s . s i s set equal to 2 cm unless the d i f f u s i v i t y , calculated as a function of the input thermal i n e r t i a , i s greater than about —6 2—1 1.6 X 10 m s , i n which case s i s calculated as a function of the d i f -f u s i v i t y . The temperatures corresponding to these layers are determined by uniformly subdividing the temperature range between the surface temp-erature estimate and the input c l i m a t i c mean temperature. These computations r e s u l t i n an exponential temperature p r o f i l e i n which the depths of the layers increase with depth i n the substrate. 2.3.2 Net Radiation Calculations The solar r a d i a t i o n c a l c u l a t i o n s i n t h i s model are based on r a d i a t i v e transfer theory and u t i l i z e a v a r i e t y of parameterizations to account for absorption and sca t t e r i n g by atmospheric constituents (Boland, 1977.; pp.25-27 and Augustine, 1978; pp.113-116). The net solar r a d i a t i o n for each time 37 step i s calculated using: I T (1 - a) cos Z K * = " 1 - a Y ' <2'51> _2 where I q i s the solar constant (= 1353 W m ), T a transmission term, and Y a cor r e c t i o n factor f o r multiple r e f l e c t i o n between the surface and the atmosphere. The l a t t e r two terms, as well as cos Z, are calculated within the model for each time step. cos Z i s given by eqn. (2.9). As i n Model M, the l a t i t u d e i s input to the model and the hour angle i s recomputed at each time step. Unlike Model M, however, the solar d e c l i n a t i o n i s computed within the model. The equation used includes a correction for the e l l i p t i c a l o r b i t . The zenith angle i s also used to compute the o p t i c a l a i r mass for each time step. For thi s the equation of Kasten (1966) i s used, as i n Model M, but without the pressure c o r r e c t i o n . The o p t i c a l a i r mass i s required to determine the Sun's path length through the various absorbing and sc a t t e r i n g media, as required i n the computation of the transmission term and the cor r e c t i o n factor for multiple r e f l e c t i o n . These l a t t e r are calculated using: -T = T T T {T DT + (1 - T^T )(1 - b)} (2.52) oz w aa R as R as Y = b'T' T'T' (1 - TlT' ) (2.53) oz w aa R as where T , T , and T are the transmission functions due to absorption by oz w aa r ozone, water vapour, and aerosols, r e s p e c t i v e l y , and T R and T^ s are those due to Rayleigh scattering and aerosol s c a t t e r i n g , r e s p e c t i v e l y . T and '' 3.3. T are determined by c a l c u l a t i n g T , which i s the o v e r a l l transmission due 3S 3 38 to aerosols, and then apportioning one quarter of T to absorption and the remainder to s c a t t e r i n g . The primed quantities r e f e r to transmission of d i f f u s e r a d i a t i o n which i s characterized by a constant o p t i c a l a i r mass of 1 . 7 . b i s the r a t i o of backward scattered r a d i a t i o n to t o t a l scattered r a d i a t i o n and i s calculated using: 0 . 2 ( 1 - T a s ) + 0 .5 (1 - T R) " " (1 - T „ ) + (1 - T R) • < 2- 5 4> This equation i s based on the assumption that Rayleigh s c a t t e r i n g i s i s o t r o p i c and that aerosol scatters 80% forward and 20% backwards. The transmission functions are calculated using a v a r i e t y of par-ameterizations chosen from several a v a i l a b l e i n the solar r a d i a t i o n l i t -erature. T q z and T w are computed as simple empirical functions of ozone and water vapour path lengths, r e s p e c t i v e l y . T_ and T , on the other hand, are determined by in t e g r a t i n g t h e i r respective o p t i c a l depths, calculated as functions of wavelength, over the e n t i r e solar spectrum. Because these integrations add s i g n i f i c a n t l y to the t o t a l time required to run the model, transmission "look-up" tables were created (T. Carlson, personal com-munication) . These eliminate the need to redo the integrations every time the model i s run. For the c a l c u l a t i o n of T and T the depths of ozone and water vapour oz w , i n an atmospheric column must be s p e c i f i e d . The depth of ozone i s set, by the table values, to a constant value of 0 . 3 cm. The p r e c i p i t a b l e water, however, can be s p e c i f i e d as an input. In the c a l c u l a t i o n of the o p t i c a l depth due to aerosol attenuation the aerosol content of the atmosphere i s s p e c i f i e d by a t u r b i d i t y c o e f f i c i e n t which i s the o p t i c a l depth of aerosols 3 9 at 0.5 urn. This i s set, again by the table values, to a value of 0.35 which i s f a i r l y t y p i c a l for urban areas (Toon and Pollack, 1976). The tables, therefore, are entered, at each time step, with a value for p r e c i p i t a b l e water and a value for the o p t i c a l a i r mass to obtain values for the trans-mission a f t e r absorption, the transmission a f t e r s c a t t e r i n g , and the back-scattered f r a c t i o n . These are substituted into eqns. (2.52) and (2.53) to obtain T and Y required i n eqn. (2.51) to ca l c u l a t e the net solar r a d i a t i o n . Next, the net long-wave r a d i a t i o n i s determined. The outgoing component i s calculated using eqn. (2.13) i n which the surface emissivity i s set to unity, as i n Models M and A. The incoming long-wave r a d i a t i o n i s calculated using: i n which the atmospheric emissivity i s determined as a function of precip-i t a b l e water (Monteith, 1961): The net long-wave r a d i a t i o n i s then added to the net solar r a d i a t i o n to give the net all-wave r a d i a t i o n for the time step. 2.3.3 The D i f f u s i v i t y Integral Atmospheric d i f f u s i v i t y i s handled i n a unique way i n t h i s model through the incorporation of a d i f f u s i v i t y i n t e g r a l . The theory behind t h i s i n t e g r a l w i l l be described b r i e f l y before i t i s discussed i n the context of the model ca l c u l a t i o n s . This i n t e g r a l not only allows the d i f f u s i v i t y to vary with height i n the atmosphere but also allows heat transfer i n the t r a n s i t i o n layer to be handled separately from that i n the surface or constant flux L+ = e A ° U e A + T q ) / 2 } 4 (2.55) e A = 0.725 + 0.17 ln(w). (2.56) 40 layer (Fig. 2.5). Within the t r a n s i t i o n layer heat i s transferred by a combination of both turbulent and non-turbulent processes, whereas i n the surface layer heat i s transferred by turbulent processes alone. The d i f -f u s i v i t y i n t e g r a l a r i s e s as a consequence of the integ r a t i o n of eqns. (2.57) and (2.58) for the sensible and latent heat fluxes: % M -< PVH + C H > ^ < 2 - 5 7 > % - -(pVq + V t- ' (2-58) CTT and C are heat transfer c o e f f i c i e n t s which account for the non-turbulent H q processes of the t r a n s i t i o n layer while and are eddy d i f f u s i v i t i e s which account for the turbulent heat transfer processes of both the tran-s i t i o n and surface layers. Integration of these equations, from z = 0 to z = z^ gives equations for the surface temperature and s p e c i f i c humidity, res p e c t i v e l y : Q H i T = 9 + (2.59) o A pc % = <A + pir- <2-60> The d i f f u s i v i t y i n t e g r a l (I) i n eqn. (2.59) i s expressed: I = A 3z KH + C H / p c p (2.61) A s i m i l a r equation can be written i n terms of C and K for the water vapour q q d i f f u s i v i t y . Because heat i s transported d i f f e r e n t l y i n the t r a n s i t i o n and surface 41 layers the d i f f u s i v i t y i n t e g r a l must be solved i n two parts. and are given d i f f e r e n t values, while and are always equal. The d i f f u s i v i t y i n t e g r a l s for heat and water vapour w i l l , therefore, d i f f e r i n the tran-s i t i o n layer but they w i l l be equal i n the surface layer. For s i m p l i c i t y , that for heat only w i l l be considered i n further discussion. For the tran-s i t i o n layer the i n t e g r a l i s solved from z = 0 to z = z , where z i s set •Li J-J equal to 1.0 cm. The non-turbulent heat transfer c o e f f i c i e n t s (CT, and C ) H q are set equal to 3.0 X 10 4 and 7.11 X 10 ^ c a l cm *s \ r e s p e c t i v e l y , and: K r = ku Az. (2.62) For t h i s case i n t e g r a t i o n of eqn. (2.61) y i e l d s : h " ku" l n z Tpc ku, + L P * (2.63) For the surface layer the i n t e g r a l i s solved from z = z to z = z . For t h i s layer and are set equal to zero and: ku^ .z K H = — . ' (2.64) rt <S> i s a s t a b i l i t y parameter whose value depends on the Monin-Obukhov length H (L): 3 ~ u * c p L = (2.65) kgR gQ R(l + 0.7/|6|) where R i s the gas constant. L w i l l be p o s i t i v e for a stable atmosphere and negative for an unstable atmosphere. The form of the r e l a t i o n s h i p between the Monin-Obukhov length and the s t a b i l i t y parameter depends upon 42 the s t a b i l i t y regime. The upper part of the d i f f u s i v i t y i n t e g r a l must, therefore, be solved separately for each of the three d i f f e r e n t s t a b i l i t y conditions which can occur i n the surface layer. Under neutral s t a b i l i t y , $„ = 1.0 and the i n t e g r a l from z T to z, i s H b L A simply: ln ( z /z ) T2 " leu, • ( 2 ' 6 6 ) This formulation i s used only i n the f i r s t time step when the Monin-Obukhov length cannot be calculated because the sensible heat f l u x i s s t i l l equal to i t s i n i t i a l value of zero. For stable conditions: * R =•1 + 4.7(z/L) (2.67) and the i n t e g r a l from z to z. i s : ]_i /A l n ( z A / z L ) + 4.7(z A/L) - 4.7(z L/L) I 0 = : . (2.68) 2 ku.. For unstable conditions: * = {1 - 15(z/L)} - 1 5. (2.69) In t h i s case the d i f f u s i v i t y i n t e g r a l from z to z i s considerably more ij A complex than i n the other two cases and i t i s solved using a method proposed by Benoit (1977). F i n a l l y , 1^ and 1^ are added together to give I. 2.3.4 Daytime Calculations If the sensible heat f l u x of the previous time step i s p o s i t i v e the following set of c a l c u l a t i o n s are executed for the current time step. A 43 p o s i t i v e sensible heat f l u x indicates an unstable atmosphere, as i s normally the case during the day. The upward heat f l u x during the day causes the mixed layer to r i s e above i t s i n i t i a l depth of 50 m. The methods of Tennekes (1973) and t h e i r mod-i f i c a t i o n s ( Z i l i t i n k e v i c h , 1975; Tennekes, 1975) are used to determine the height, and mean temperature, of the mixed layer at each time step. The rate of change of the mixed layer height i s dependent on the size of the downward heat f l u x at the inversion base and on the strength of the i n -version: dh dt - (8w). (2.70) where: (9w) . = -0.5(9w) x o 2.6(6w) 2/3 1 + ( g h / T o ) 1 / 3 A -1 (2.71) and: dA dh (9w) (9w). dt Y d t + (2.72) The main purpose of incorporating the mixed layer formulation into the model was to allow the mean temperature of the mixed layer to vary over the day. The rate of change of 0 can be calculated from: A d9 (9w) (9w). dt h A (2.73) where <SR i s the r a d i a t i v e heating i n the layer (= 1.67 X 10 ^ °C s ^) A 44 Because of the dependence of the va r i a b l e s , h, (9w)., A, and 9 , upon each dA d 9 A X — other the equations used to cal c u l a t e , and (0w)^ are s i m p l i f i e d for the f i r s t time step i n which the mixed layer formulation i s implemented. The change i n the mixed layer height through the day also a f f e c t s the s p e c i f i c humidity of the mixed layer. This change i s a r e s u l t of the entrain-ment of d r i e r a i r from above the mixed layer and the increase i n moisture due to evaporation at the ground surface (Carlson and Boland, 1978). The new s p e c i f i c humidity i s determined using: q'(ph + p 6h) - 0.5 Y n(6h) 2p + t/L q = -A h a _ h E e. ( 2 > 7 4 ) A ph where p and are the de n s i t i e s at the middle and top of the mixed layer, respectively, q^ i s the s p e c i f i c humidity of the previous time step, At the length of the time step, 5h the change i n the mixed layer height over At, and the s p e c i f i c humidity lapse rate above the mixed layer. In order for the f r i c t i o n v e l o c i t y , used i n the c a l c u l a t i o n of the Monin-Obukhov length and calculated as a function of the wind at 50 m, to vary throughout the day the wind p r o f i l e i s recalculated at each time step. For these c a l c u l a t i o n s a p r o f i l e of momentum d i f f u s i v i t i e s (K ) must be m computed. The momentum d i f f u s i v i t y p r o f i l e i s used, along with the computed geostrophic wind p r o f i l e , and the wind p r o f i l e from the previous time step, to determine the new wind p r o f i l e . The wind p r o f i l e s computed are the r e s u l t of a balance between the acc e l e r a t i o n term which i s attempting to force the winds into geostrophic balance, and the mixing term which i s tr y i n g to force the winds out of geostrophic balance. This l a t t e r i s a function of the momentum d i f f u s i v i t y at each l e v e l (T. Carlson, personal communication). 45 F i n a l l y , the surface temperature and the latent and sensible heat fluxes are calculated. The surface temperature and latent heat f l u x are determined using eqns. (2.59) and (2.75), re s p e c t i v e l y , with the d i f f u s i v i t y i n t e g r a l solved f o r unstable conditions i n both cases: pL M = - T - ^ o s ~ «A> * ( 2 - 7 5 ) In eqn. (2.75) M, the moisture a v a i l a b i l i t y f a c t o r , i s the same as that used i n Model M and Q o s> the surface saturation s p e c i f i c humidity, i s computed as a function of surface temperature: = 1 0{6.1989 - ( 2 3 5 3 / T ^ Mos The sensible heat f l u x i s calculated using eqn. (2.77) which was derived by s u b s t i t u t i n g eqn. (2.59) for the surface temperature and eqn. (2.16) for the subsurface heat f l u x into the energy balance equation: Q* - Q - K/z (8 - T ) ' Qu = ~ — -.. (2.77) 1 + I(K/pc z.) P 1 For each of the surface temperature, the latent heat f l u x , and the sensible heat f l u x the value for the time step i s the average of that computed i n the previous time step, and that computed i n the current time step. The subsurface heat f l u x i s simply computed as a r e s i d u a l . The sequence described above i s followed for every time step i n which the sensible heat f l u x of the previous time step i s p o s i t i v e . Convergence i s achieved when there are n e g l i g i b l e changes i n the sensible heat f l u x and i n the f r i c t i o n v e l o c i t y between cycles. It was found that only one cycle 46 t h r o u g h the sequence was r e q u i r e d f o r convergence ( C a r l s o n and B o l a n d , 1978) 2.3.5 N i g h t - t i m e C a l c u l a t i o n s The n i g h t - t i m e c a l c u l a t i o n sequence i s f o l l o w e d f o r e v e r y time s t e p i n w h i c h t h e s e n s i b l e heat f l u x i s n e g a t i v e . A l t h o u g h a n e g a t i v e s e n s i b l e h eat f l u x i n d i c a t e s a s t a b l e atmosphere, the model does a l l o w f o r u n s t a b l e c o n d i t i o n s t o o c c u r a t n i g h t . The methods used f o r ,the n i g h t - t i m e r o u t i n e a r e based on a model d e v e l o p e d by B l a c k a d a r (1979). At n i g h t t h e s e n s i b l e heat f l u x i s c a l c u l a t e d u s i n g : Q = - k pu,T,. (2.78) H p " " u A i s d e t e r m i n e d u s i n g eqn. (2.49), w h i l e T.,. i s d e t e r m i n e d u s i n g : 9 A " 9 1 T;V = - (2.79) l n ( z A / z ) - <J>R where 0 ^ ,. t h e t e m p e r a t u r e a t the h e i g h t z (= 1 m), i s d e t e r m i n e d u s i n g an e m p i r i c a l r e l a t i o n s h i p w h i c h i n c o r p o r a t e s the e f f e c t s of r a d i a t i v e f l u x d i v e r g e n c e and t u r b u l e n t f l u x c o n v e r g e n c e . $ and <5> , i n eqns. (2.49) and m rl (2.79), a r e f u n c t i o n s o f the Monin-Obukhov l e n g t h where the form of the r e l a t i o n s h i p depends on t h e a t m o s p h e r i c s t a b i l i t y as g i v e n by the b u l k R i c h a r d s o n number: g Z A R i B = ~2 { ( 9 A " V + T - l n ( z / z c ) } - (2.80) 9 U A 9 i s t h e mean t e m p e r a t u r e o f t h e s u r f a c e l a y e r . I f R i i s p o s i t i v e but l e s s B t h a n the c r i t i c a l R i c h a r d s o n number (eqn. 2.81) the atmosphere i s s t a b l e ; i f R i i s p o s i t i v e and g r e a t e r t h a n R i the atmosphere i s n o n - t u r b u l e n t or 47 extremely stable; and i f Ri i s negative the atmosphere i s unstable. The c r i t i c a l Richardson number i s defined: Ri_ = 0.5512 exp{-0.2129(u 2 .+ v 2 ) ^ + 0.2} C ^ g g (2.81) where u and v are the geostrophic wind components. For the stable case the Monin-Obukhov length i s computed as a function of the Richardson numbers: L * = l n ( z . / z ) — A o z. R i T 5 ( R i c - R i B ) (2.82) With m^ = *H = " 5 ( Z A / L ) (2.83) u.,. and Tj. can be determined. For the unstable case the Monin-Obukhov length, from the previous time step, i s used to compute x, where: x = (1 - 16(z A /L)} \ (2.84) Now, = 2 l n 1 + x (2.85) and + TT + 2 In 1 + x - 2 tan ^x. (2.86) These values are used i n eqns. (2.49) and (2.79) to compute u.,. and T.,.. For the non-turbulent case the momentum and heat fluxes at the surface 48 are set equal to zero. The l a t t e r two cases r a r e l y occur. A new temperature and wind p r o f i l e are calculated for each time step during the night by inte g r a t i o n of the u and v momentum equations, and the thermodynamic equation, according to a scheme devised by Blackadar (1979). The latent heat f l u x i s calculated using eqn. (2.75), as i n the daytime routine, with the d i f f u s i v i t y i n t e g r a l solved for the appropriate s t a b i l i t y conditions. When the latent heat f l u x becomes negative i t i s set to zero. Although the d i f f u s i v i t y i n t e g r a l i s used to determine the latent heat f l u x for a short time at night the d i f f u s i v i t y i n t e g r a l s are, i n f a c t , un-defined once turbulence ceases, that i s , once the sensible heat flux becomes negative. Because of t h i s the surface temperature cannot be determined as i t was for the daytime routine and i s , instead, determined as an equilibrium so l u t i o n to the energy balance equation. Solving t h i s equation for the surface temperature gives the following quartic equation: AT 4 + BT + C = 0 (2.87) o o where A = z a o B = K/z . C = KT 1/z 1 + Q R + Q E - £ A o T A + K*. This i s solved for the surface temperature, for each time step during the night, using Newton's i t e r a t i v e technique for fi n d i n g the roots of a poly-nomial . 49 As w i t h t h e da y t i m e r o u t i n e i t was found t h a t one c y c l e t h r o u g h t h e above s e t o f c a l c u l a t i o n s was s u f f i c i e n t f o r conv e r g e n c e . The t i m e s t e p a t n i g h t i s h a l f t h a t d u r i n g the day. 2.3.6 S u b s u r f a c e Temperature P r o f i l e Once t h e energy b a l a n c e components and the s u r f a c e t e m p e r a t u r e have been computed f o r a t i m e s t e p t h e s u b s t r a t e t e m p e r a t u r e p r o f i l e f o r the next t i m e s t e p i s computed u s i n g the f i n i t e d i f f e r e n c e form o f t h e d i f f u s i o n e q u a t i o n (eqn. 2.15) as i n the o t h e r two models. T h i s p r o c e d u r e p r o v i d e s a v a l u e f o r T^ f o r use i n eqns. (2.77) and ( 2 . 8 7 ) . 2.4 Summary The most o b v i o u s d i f f e r e n c e between the t h r e e models i s i n t h e i r l e v e l o f d e t a i l . As t h e c o m p l e x i t y i n c r e a s e s from Model M t o Model C problems a r e b e i n g d e a l t w i t h i n a more p h y s i c a l l y r e a l i s t i c f a s h i o n w i t h l e s s simp-l i f y i n g a s s u m p t i o n s . Model M r e p r e s e n t s the b a s i c s e t of c a l c u l a t i o n s ne-c e s s a r y t o s o l v e t h e pr o b l e m . To t h i s a dynamic mixed l a y e r , w h i c h a l l o w s the upper l e v e l t e m p e r a t u r e t o v a r y , as w e l l as a more d e t a i l e d c o n s i d e r a t i o n o f a t m o s p h e r i c d i f f u s i v i t y , have been added i n Model A. I n Model C the s o l a r r a d i a t i o n c a l c u l a t i o n s a r e much more s o p h i s t i c a t e d t h a n t h o s e o f Model M, the wind speed p r o f i l e i s r e c a l c u l a t e d a t e v e r y t i m e s t e p , and the atmos-p h e r i c d i f f u s i v i t y c a l c u l a t i o n s a r e more d e t a i l e d t h a n t h e y a r e i n Model A. I n a d d i t i o n t o the o v e r a l l d i f f e r e n c e i n d e t a i l a v a r i e t y o f d i f f e r e n t methods have been used i n the models. F i r s t , t h e b a s i s f o r the a t m o s p h e r i c d i f f u s i v i t y c a l c u l a t i o n s i s d i f f e r e n t i n each model. Model M uses a s i m p l e R i c h a r d s o n number c o r r e c t i o n f a c t o r , Model A uses heat t r a n s f e r and f r i c t i o n c o e f f i c i e n t s , and Model C uses a d i f f u s i v i t y i n t e g r a l d u r i n g t h e day and 50 a scheme based on the bulk Richardson number at night. In addition, Models A and C each contain separate c a l c u l a t i o n sequences for stable and unstable regimes, while Model M does not. Second, i n Model C the mixed layer depth ca l c u l a t i o n s of Tennekes (1973) are used whereas i n Model A those of S t u l l (1973) are used for unstable conditions and those of Businger and Arya (1974) are used for stable conditions. F i n a l l y , the i t e r a t i v e methods used to a r r i v e at the equilibrium surface temperature are d i f f e r e n t i n each model The above are major differences there are, i n addition, many smaller and less s i g n i f i c a n t ones. CHAPTER THREE THE OBSERVATION PROGRAMME This chapter contains a discussion of the measurement programme from which the observations used to test the models were obtained. I t includes a d e s c r i p t i o n of the s i t e , the observation periods, and the instrumentation. More de t a i l e d descriptions of both the s i t e and the instrumentation can be found i n both Kalanda (1979) and Steyn (1980). The nature of urban surface temperatures i s also explored. 3.1 The Site The measurements were conducted i n the c i t y of Vancouver, B.C: a c i t y bounded by the P a c i f i c Ocean to the west, mountains to the north, and farm-land to the south and east. Figure 3.1 i s a map of the c i t y showing a l l locations relevant to t h i s study. The observation tower was located at the B.C. Hydro Mainwaring Substation i n the suburb of Sunset. The surrounding area i s v i r t u a l l y f l a t with a s l i g h t southwestward slope. The predominant land-use i s one- and two-storey single family dwellings (Fig. 3.2). A deta i l e d analysis of surface cover was c a r r i e d out for a c i r c u l a r area of radius 2 km surrounding the s i t e . 64% of t h i s area i s greenspace, 14% i s houses, 11% i s commercial and i n s t i t u t i o n a l (mainly stores and schools), and 11% i s pavement (Kalanda et a l . , 1980). Solar r a d i a t i o n measurements were used to determine that the area has a surface albedo ranging between 0.12 and 0.14 (Steyn and Oke, 1981). A d e t a i l e d s e c t o r i a l analysis of the rough-ness elements around the s i t e and the equation of Lettau (1969) gave a mean surface roughness length for the area of 0.5 m. Various methods were used to a r r i v e at an aerodynamic displacement height of 3.5 m (Steyn, 1980). In the determination of the energy balance, from observations, i t was 51 F i g . 3.1 - The greater Vancouver area F i g . 3.2 - Looking west from the top of the o b s e r v a t i o n tower 54 assumed that advective heat fluxes were n e g l i g i b l e . The models contain a sim i l a r assumption. Given the roughness length of 0.5 m and an observation tower of height 30 m, Steyn (1980) determined that the fetch must be homo-geneous for at least 1.5 km from the observation s i t e , i n order to s a t i s f y the above assumption. Therefore, the Sunset s i t e i s "ideal because of the uniformity of the surrounding land-uses for at least 2 km i n a l l d i r e c t i o n s . An analysis of the measured fluxes and t h e i r v a r i a t i o n with wind d i r e c t i o n further supports the assumption that the former were probably not affected by advection (Steyn, 1980). 3.2 Instrumentation In order to obtain energy balance observations representative of a com-posite suburban surface, rather than of a single surface (such as a lawn or roo f ) , the measurements must be made well above the height of the roughness elements so that turbulent mixing can provide an homogenizing r o l e . For t h i s reason the instruments were mounted on a 30 m tower erected at the obser-vation s i t e (Fig. 3.3). No sensors were located lower than a height of 20 m. It i s being assumed that the observed components of the energy balance apply to the same "surface" as do t h e i r modelled equivalents. This i s probably a reasonable assumption considering the height from which the measurements were made. The observations a v a i l a b l e for tes t i n g the models are the net solar r a d i a t i o n (K-c) , the net all-wave r a d i a t i o n (Q*) , the three components of the surface energy balance (Q„, Q„, and Q ), and the mixed layer height (h). Unfortunately, no surface temperature observations are av a i l a b l e . To accom-odate the fac t that these observations, apart from the mixed layer heights, were produced as hourly averages, the 15 min model output was converted to 55 F i g . 3 . 3 - I n s t r u m e n t a t i o n mounted on the o b s e r v a t i o n tower 56 hourly averages. Observations, as well as the model output, are i n solar time. The net short-wave r a d i a t i o n was measured using upward- and downward-facing pyranometers. The net all-wave r a d i a t i o n was measured using a net pyrradiometer mounted about 22 m from the base of the tower. The error i n -volved i n measuring these two terms i s about 5% (Suckling and Hay, 1976; Kalanda, 1979). The net long-wave r a d i a t i o n (L*) was calculated as: L* = Q* - K*. ' (3.1) The subsurface heat f l u x was calculated using parameterizations dev-eloped s p e c i f i c a l l y for the observation area (Oke et a l . , 1981). The day-and night-time parameterizations are: Q G = 0.25(Q* - 27.0) Day (3.2) Q G = 0.67Q*. Night (3.3) Parameterization i s necessary i n a suburban area because d i r e c t measurement of t h i s flux i s a l l but impossible due to the v a r i e t y of surface types and configurations. On most occasions the two turbulent fluxes were obtained i n d i r e c t l y , by determining the Bowen r a t i o (B = Q„/Q_) from temperature and humidity gradients, measured using a reversing d i f f e r e n t i a l psychrometer system. This method has been recommended for use i n determining evaporation i n a forest (Spittlehouse and Black, 1980) which, i n some respects, represents an environment s i m i l a r to, though not as complex as, a suburban area. Given the a v a i l a b l e energy (Q* -• Q_), the turbulent fluxes are calculated as: 57 Q H = 8(Q* - Q G)/(1 + B) (3.4) Q E = (Q* - Q G)/(1 + B). (3.5) Detailed c a l c u l a t i o n s by Kalanda (1979) indicate that the errors i n deter-mining the turbulent fluxes using t h i s method vary from about 10 to 20% during the day, and are normally greater than 25% at night. During the 1978 period (Sec. 3.3) the turbulent sensible heat flux was also determined d i r e c t l y , using the eddy c o r r e l a t i o n technique v i a a yaw sphere-thermometer system (Yap et a l . , 1974). The measurement error i n t h i s "case i s estimated to be about 5 to 15% (Steyn, 1980). During t h i s period the subsurface heat fl u x could be determined as a r e s i d u a l when d i r e c t sensible heat f l u x measurements, as well as Bowen r a t i o measurements, were a v a i l a b l e . The values determined i n t h i s way f i t the parameterized r e l a t i o n s h i p s reasonably well (Oke et a l . , 1981). Other methods a v a i l a b l e for measuring the energy balance are summarized i n Kalanda (1979). From his summary i t i s clear that the two methods used are among the most r e l i a b l e and p r a c t i c a l a v a i l a b l e . Observations of the mixed layer height were made during the 1978 period using both an acoustic sounder, and theodolite-tracked balloons carrying miniature radio transmitting temperature sensors (Steyn, 1980). These ob-servations are not av a i l a b l e for a l l hours. 3.3 The Observation Periods The data used to test the models were obtained from three separate ob-servation periods. A l l observations were made during the summer. The f i r s t observation period took place during the month of September, 1977. This period followed a summer of f a i r l y average r a i n f a l l with a wetter than normal period near the end of August. September, i t s e l f , was also wetter 58 than normal. The second observation period extended over a period of approximately three weeks, from July 18 to August 8, 1978. This period was characterized by a stable a n t i c y c l o n i c regime which resulted i n very warm, dry weather. In addition, the month and a half preceding the ob-servation period was d r i e r than normal. The t h i r d observation period ex-tended over much of July and -August, 1980. Both June and July of t h i s year were much wetter than normal, while August was s l i g h t l y d r i e r . The average Bowen r a t i o s for the three observation periods were 0.8, 1.2, and 0.3, respe c t i v e l y (T.R. Oke, personal communication). To s a t i s f y the requirements of the solar r a d i a t i o n c a l c u l a t i o n s , i n the three models, only cloudless days could be chosen from the data sets. Eliminating days with more than two hours of missing data, as well as days with any cloud cover, l e f t eighteen days appropriate to t h i s study. These days are l i s t e d i n Table 3.1, along with t h e i r mean d a i l y meteorological c h a r a c t e r i s t i c s . The measurements were taken by the Atmospheric Environment Service at the Vancouver International A i r p o r t , which i s located about 8 km southwest of the Sunset tower s i t e . Table 3.1 shows that the data set i n -cludes days for which the mean d a i l y a i r temperature ranged from about 11 to 23° C, and the average wind speed from about 1 to 6 m s ^. The values for the accumulated p r e c i p i t a t i o n of the previous t h i r t y days, as well as for the mean d a i l y vapour pressure d e f i c i t , c l e a r l y show that 1980 was a much wetter summer than was 1978. The very d i f f e r e n t moisture conditions of the 1978 and 1980 observation periods s i g n i f i c a n t l y affected the observed daytime energy balances of the days chosen for study. On s i x of the days from the 1978 period the sensible heat f l u x was s l i g h t l y higher than the latent heat flux ( i . e . (3 > 1, Fig. 59 Table 3.1 - Summary of meteorological conditions for the days to be used i n t e s t i n g the models Date Mean d a i l y a i r temperature Mean d a i l y wind speed Accumulated 30 day p r e c i p i t a t i o n Mean d a i l y vapour pressure d e f i c i t (°C) (m s ' ) (mm) (kPa) Sept, . 10, 1977 13.5 4.5 104.8 0.297 Sept, . 25, 1977 11.1 > 2.5 106.5 0.324 July 20, 1978 15.3 2.2 1.4 2.491 July 29, 1978 19.0 2.4 7.8 0.961 July 30, 1978 19.8 2.8 7.8 1.045 July 31, 1978 17.8 3.4 7.8 0.693 Aug. 1, 1978 17.9 3.5 7.8 0.683 Aug. 2, 1978 18.6 2.7 7.8 0.916 Aug. 3, 1978 19.9 2.7 7.8 0.887 Aug. 4, 1978 19.0 3.9 7.8 0.820 Aug. 8, 1978 23.3 3.8 7.8 2.141 July 24, 1980 17.3 2.2 97.8 0.655 July 26, 1980 18.1 2.1 71.0 0.352 July 27, 1980 18.9 2.4 68.8 0.595 July 28, 1980 17.9 5.5 67.6 0.394 July 30, 1980 16.5 1.5 67.6 0.735 Aug. 9, 1980 18.5 2.0 45.6 0.777 Aug. 10, 1980 20.0 1.8 38.0 1.420 60 3.4). On the remaining three days the Bowen r a t i o was as high as 2.0. These observations r e f l e c t the dry conditions prevalent during t h i s part-i c u l a r year. The energy budgets of the seven days chosen from the 1980 data set were a l l very s i m i l a r . They were characterized by very high latent heat fluxes and negative sensible heat fluxes (eg. F i g . 3.5). This seemingly unusual s i t u a t i o n has been a t t r i b u t e d to the wet conditions of t h i s summer, coupled with the i r r i g a t i o n of greenspace, and the jux t a p o s i t i o n of pervious and impervious surfaces i n suburban areas (Oke and McCaughey, 1983). I t i s u n l i k e l y that the models w i l l be able to reproduce these energy balances. It would have been misleading to include 1977 i n the above comparison because t h i s observation period took place l a t e r i n the year than the other two. Of the two days chosen from the 1977 data set the e a r l i e r was charact-erized by an average daytime Bowen r a t i o of almost 2.0 while that of the l a t t e r was about 1.0. Even though the accumulated p r e c i p i t a t i o n was s l i g h t l y higher during t h i s period than i t was i n 1980 (Table 3.1) the Bowen r a t i o was not as low. This i s probably due to there being less a v a i l a b l e energy to drive the "oasis" e f f e c t , and less i r r i g a t i o n to supplement the s o i l moisture, at t h i s time of year. 3.4 Surface Temperatures As noted, no surface temperature observations are a v a i l a b l e . It i s v i r t u a l l y impossible to make d i r e c t observations of the surface temperature as the surface i t s e l f i s nothing more than an i n f i n i t e l y thin plane. It i s often possible, however, to extrapolate measured s o i l temperature p r o f i l e s to estimate surface temperatures. This technique cannot be used to obtain a representative temperature of an "urban surface", however, because of i t s heterogeneity. In addition, there are obvious p r a c t i c a l problems involved 61 AUG 3 . J 9 1 8 F i g . 3.4 - Observed energy balance for Aug. 3, 1978 JULY 2 8 . 1980 • OX O OH A OE O OG F i g . 3.5 - Observed energy balance for July 28, 1980 62 with determining substrate temperature p r o f i l e s i n urban areas. Such ob-servations might be of l i t t l e use, anyway, as the "surface" to which the modelled temperatures apply i s probably quite d i f f e r e n t from the actual surface (Sec. 1.4). The problems involved i n obtaining urban surface temperatures have been p a r t i a l l y solved by the use of remote sensing techniques. It i s f e l t that temperatures obtained i n t h i s way w i l l be representative of the i n t e r -action of the various geometric configurations and thermal properties of which the urban surface i s composed (Greene, 1980).and as such that they might correspond with modelled surface temperatures. However, remote sensing i s also l i m i t e d because v e r t i c a l surfaces w i l l not be "seen". This method involves the estimation, of the long-wave r a d i a t i o n emitted by the surface, from a i r c r a f t or s a t e l l i t e s . Using Stefan-Boltzmann's law, and an assumption of the surface emissivity, the corresponding surface temp-erature can be calculated. S i m i l a r i l y , the outgoing long-wave r a d i a t i o n measured from the Sunset tower (as was done during the 1977 and. 1978 ob-servation periods), might have been used to estimate surface temperatures. Unfortunately, these measurements appear to be u n r e l i a b l e (Kwasnicky, 1980) and so w i l l not be used. Surface temperatures obtained by remote sensing techniques i n other studies can provide ins i g h t into the nature of urban surface temperatures under clear sky conditions and are reviewed here. Lewis et a l . (1976) show that urban surface temperatures determined from a i r c r a f t observations of Baltimore, Maryland i n the month of May ranged from 25 to 30° C i n veg-etated areas, to 45 to 55° C i n built-up areas. Goward (1981) used a i r -c r a f t observations to analyse surface temperatures of roofs and pavements 63 i n Indianapolis, Indiana i n August. He found that the temperatures of these surfaces.varied between 30 and 55° C depending on t h e i r albedos. In com-parison, the surface temperatures of nearby vegetated areas were only about 20° C. In both studies, the quoted temperatures were observed near midday, therefore, they are probably s l i g h t l y l e s s than the maximum d a i l y values because maximum d a i l y surface temperatures are expected to occur i n mid-afternoon (Oke, 1978; p.32). Similar studies give an i n d i c a t i o n of the observed d i u r n a l range i n surface temperatures. Using surface temperatures determined from a i r c r a f t observations over Ann Arbor, Michigan i n August, Outcalt (1972b) found that the surface temperatures i n the c i t y centre ranged from 21° C i n the very early morning to 31° C at about noon. Carlson et a l . (1981) report surface temperatures determined from s a t e l l i t e data for both Los Angeles and St. Louis i n springtime. Surface temperatures i n Los Angeles i n the early a f t e r -noon (1330 LSI) ranged from 24° C i n vegetated areas to 34° C i n i n d u s t r i a l areas. In the very early morning (0230 LST) the temperatures ranged from 10 to 13° C. In St. Louis the early afternoon range was from 23 to 34° C, and the early morning range from 3.5 to 6.5° C. For the above cases the actual d a i l y range i s probably s l i g h t l y larger than the quoted temperatures suggest because the observations were taken an hour or two before the times of the normal maximum and minimum temperatures. Minimum d a i l y surface temperatures are expected to occur j u s t a f t e r sunrise (Oke, 1978; p.32). It i s also worthwhile to add that the surface temperature i s normally les s than the a i r temperature at night, and greater during the day ( S e l l e r s , 1965; p.62). I t follows that the d a i l y surface temperature range i s greater than that of the a i r temperature. 64 The above shows that surface p r o p e r t i e s are a major determinant of the surface temperature. At a given time surface temperatures can vary w i t h i n a c i t y by as much as 20° C or more depending on the surface type. Given a m i d - l a t i t u d e c i t y on a c l e a r summer day i t i s , t h e r e f o r e , expected that the maximum surface temperature of a suburban area, such as that i n the present study, might w e l l f a l l between 25 and 40° C. This i s a f a i r l y wide range, but i t can be used as a b a s i s f o r d e c i d i n g whether modelled surface temperatures are w i t h i n reason. In a d d i t i o n , the modelled temperatures should peak i n mid-afternoon and e x h i b i t a greater d i u r n a l range than the observed a i r temperatures. CHAPTER FOUR THE MODEL INPUT L i s t s of the inputs required for each model were given i n Chapter 2. These inputs f a l l into three major categories: s p a t i a l and temporal i n -formation, meteorological conditions, and s i t e surface c h a r a c t e r i s t i c s . This chapter i s a discussion of how the values for each of the inputs was obtained and the s e n s i t i v i t y of the models to them. The s e n s i t i v i t y analyses were done by varying a s p e c i f i c input while a l l other inputs were held constant at t h e i r base values. The base values used were the inputs for July 30, 1978 as these were f e l t to be f a i r l y representative of the data set as a whole. For s i m p l i c i t y , the e f f e c t of these changes on the computed fluxes, surface temperatures, and mixed layer heights was only considered i n terms of t h e i r maximum d a i l y values. Ackerman (1977) and Carlson and Boland (1978) have shown, i n t h e i r own s e n s i t i v i t y analyses, that i t i s usually the maximum d a i l y values that are affected most by changes i n the input parameters. There are two major reasons for conducting s e n s i t i v i t y analyses. F i r s t , since there i s uncertainty about the values to be assigned to c e r t a i n of the inputs i t i s useful to know how much these p a r t i c u l a r v a r i a b l e s a f f e c t the model output. Second, the models can be assessed i n terms of how p h y s i c a l l y r e a l -i s t i c the r e s u l t s are. This i s not completely straightforward, however, because the complexity of the i n t e r r e l a t i o n s h i p s between the surface temp-erature, energy balance components, and mixed layer height often makes i t d i f f i c u l t to assess exactly what e f f e c t the change i n a p a r t i c u l a r input should have. Table 4.1 i s a summary of the r e s u l t s of the s e n s i t i v i t y analyses. The values l i s t e d are the changes produced by varying the inputs continuously, 65 Table 4.1 - Summary of changes i n the maximum d a i l y values of _2 the modelled fluxes (W m ), surface temperatures (°C), and mixed layer heights (m) produced by changes (as s p e c i f i e d i n the text) i n the given input parameters Table 4.la - Model M Q* T 0 Maximum Q d a i l y values 656.5 86.0 553.8 24.3 21.5 A i r temperature +9+ -245 +260 -10 +8 Wind speed -1 -27 +26 -.3 -.6 Vapour pressure +2 +110 -106 +6 +2 Pr e c i p i t a b l e water -45 -12 -33 -1 -.2 Dust p a r t i c l e s -10 -3 -7 -.2 -.1 Surface albedo -75 -20 -55 . -2 -.3 Surface roughness -2 -55 +60 -10 -1 Heat capacity 0 -5 -13 +24 0 Thermal d i f f u s i v i t y 0 -2 -5 +9 0 Moisture a v a i l a b i l i t y -7 -385 +395 -25 -4 These are the maximum d a i l y values as computed using the base values (July 30, 1978) for a l l inputs. For changes of more than one unit, the values have been rounded off to the nearest u n i t . 67 Table 4.1b - Model A Maximum d a i l y values % % % T o 659.9 238.9 238.9 194.4 32.1 919.8 Climatic mean temperature Wind speed Subsidence c o e f f i c i e n t Surface roughness Heat capacity Thermal d i f f u s i v i t y Bowen r a t i o -6 +5 +5 -30 +2 -15 -.6 -1 -1 +3 +.1 +10 +6 -6 -6 -19 -.5 -1110 +8 +8 +8 -8 -1 .-4 -3 -60 -60 +120 -.3 -12 +.3 -28 -28 +48 -.4 -10 -19 +130 -232 +67 +9 +258 68 Table 4.Ic - Model C °* % ' QG T o Maximum d a i l y values 635.7 153.7 363.6 165.4 32.6 960.2 Climatic mean temperature Surface albedo Surface roughness Thermal i n e r t i a Moisture a v a i l a b i l i t y -2 +7 +20 -35 +1 +70 Pr e c i p i t a b l e + 3 + 2 water -75 -18 -53 -12 -2 -115 -1 -1 -.1 +3 +.2 -5 +17 -50 -145 +225 -5 -320 +40 -140 +255 -95 -12 -700 69 over the ranges given i n the text. The meteorological conditions were varied over the approximate ranges observed i n Vancouver during the observation periods. The surface c h a r a c t e r i s t i c s were, i n general, varied over ranges thought to be representative of urban areas. 4.1 Temporal and S p a t i a l Information This set of var i a b l e s i s used i n the c a l c u l a t i o n of the incoming solar r a d i a t i o n . In Model M i t includes a s t a r t time, the l a t i t u d e of the s i t e , the solar d e c l i n a t i o n , and the radius vector. The l a t t e r two were obtained from the Smithsonian Meteorological Tables ( L i s t , 1965). Model C also r e -quires the input of l a t i t u d e and s t a r t time but the solar d e c l i n a t i o n i s calculated i n the model, given the date. Because the equation' used corrects for the e l l i p t i c a l o r b i t there i s no need to input a value for the radius vector. Since Model A simply obtains, as input, quarter hourly solar rad-i a t i o n values calculated by Model C the inputs noted above are not required. 4.2 Meteorological Conditions This set of inputs represents a v a r i e t y of meteorological conditions most of which vary from day-to-day. These are used to e s t a b l i s h i n i t i a l values for the s t a b i l i t y parameters, the upper l e v e l wind, temperature and humidity, and the subsurface temperature p r o f i l e . They are also used as a basis for making i n i t i a l surface temperature estimates and for estab-l i s h i n g atmospheric t u r b i d i t y conditions. The input requirement's i n t h i s category are b a s i c a l l y the same for each model although they d i f f e r i n the l e v e l of d e t a i l , required. The mean d a i l y a i r temperature required i n Model M was obtained from Atmospheric Environment Service observations made at the Vancouver Inter-70 nationa l A i r p o r t . It was found that the modelled turbulent fluxes were extremely s e n s i t i v e to v a r i a t i o n s i n t h i s temperature. For an increase i n mean d a i l y a i r temperature of from 10 to 20° C the sensible heat f l u x _2 decreased about 250 W m , while the latent heat f l u x increased by a. s i m i l a r -2 amount. In addition, the subsurface heat f l u x decreased about 10 W m Although the magnitude of, p a r t i c u l a r l y , the former change i s unreasonably large the nature of the changes seem p h y s i c a l l y p l a u s i b l e . They a r i s e because the input mean d a i l y a i r temperature represents the temperature at the atmospheric damping height and the subsurface damping depth. Increasing the input temperature causes a decrease i n the atmospheric and subsurface temperature gradients, and thus the sensible heat fluxes (Q and Q ), because the modelled surface temperature does not increase i n proportion with the input temperature. In Models A and C an upper a i r temperature sounding, rather than a singl e screen-level temperature, i s required. The nearest s t a t i o n from which upper a i r soundings are av a i l a b l e i s Quillayute, Washington situated 171 km southwest of Vancouver. As these soundings are to represent i n i t i a l conditions the observations recorded at 0400 PST were used. For Model A the sounding temperatures had to be input as p o t e n t i a l temperatures, while for Model C they were input as actual temperatures and converted to p o t e n t i a l values i n the model. S e n s i t i v i t y analyses were not conducted, i n th i s case, as the soundings are only used to set i n i t i a l conditions. In a ddition to the upper a i r temperature sounding, Model A requires the input of surface temperature, and mixed layer temperature and height, estimates to set i n i t i a l conditions. The following scheme was devised (based on T.P. Ackerman's example run of the model). The i n i t i a l mixed layer temperature 71 was simply set equal to the temperature at the lowest l e v e l of the upper a i r sounding. The i n i t i a l surface temperature was set approximately 5° C less than t h i s , and the surface temperature one time step before the s t a r t of the model was set at 4° C l e s s . The i n i t i a l mixed layer height was set equal to 50 m. Models A and C require a c l i m a t i c mean temperature as input to represent the temperature at,the lowest l e v e l of the substrate temperature p r o f i l e . A value of 10° C was used based on Atmospheric Environment Service climat-o l o g i c a l summary data. The s e n s i t i v i t y of the two models to t h i s input was determined by varying i t from 5 to 20° C. This change caused a decrease of - 2 30 W m i n the subsurface heat flux modelled by A, while that modelled by C _2 decreased by 35 W m . Because an increase i n the temperature at the lowest substrate l e v e l causes a decrease i n the subsurface temperature gradient, the observed decreases i n the subsurface heat fluxes are expected. The corres-ponding increase i n the turbulent sensible heat f l u x appears to be rela t e d to the increase, of 70 m, i n the mixed layer height modelled by C. The average d a i l y wind speed input to both Models M and A was also obtained from the Atmospheric Environment Service observations at the Van-couver International A i r p o r t . Over the range of from 2 to 4 m s ^, Model M's latent and sensible heat fluxes increased and decreased, re s p e c t i v e l y , - 2 by less than 30 W m . These r e s u l t s appear to be p h y s i c a l l y u n r e a l i s t i c as one would expect both turbulent fluxes to increase with an increase i n wind speed because of the associated increase i n the atmosphere's a b i l i t y to transport energy. Instead, the latent heat flux increased at the expense of the sensible heat flux . Model A was r e l a t i v e l y i n s e n s i t i v e to changes in wind speed over t h i s range. Model C requires a wind sounding rather than a single wind speed, as in Models M and A. This information was included on the upper a i r obser-vations recorded at Quillayute, Washington. This sounding was required to set i n i t i a l conditions only. To e s t a b l i s h upper l e v e l humidities Model M requires the input of a mean d a i l y screen height vapour pressure, while Model C u t i l i z e s the dew point depression values of the upper a i r soundings. While the former remains constant through the model-day, the l a t t e r i s only used to set i n i t i a l con-d i t i o n s . Model A does not require an equivalent input because the latent heat f l u x i s not e x p l i c i t l y calculated i n t h i s case. The vapour pressures, required i n Model M, were again obtained from the Atmospheric Environment Service observat ions. For vapour pressures ranging from 10 to 20 mb the sensible and lat e n t heat fluxes increased and decreased, r e s p e c t i v e l y , by -2 jus t over 100 W m . This trend i s expected because as the upper l e v e l vapour pressure increases the atmospheric humidity gradient and, therefore, the latent heat flux,, decreases. The associated increase i n surface temp-erature of 2° C also seems reasonable. Surface pressure maps were used to obtain the s t a t i o n pressures r e -quired i n Models M and A and the hori z o n t a l pressure and temperature grad-ients required i n Model C. The l a t t e r are used to determine the geostrophic wind p r o f i l e s . The gradients were determined by taking differences between the pressures and temperatures located 2° of l a t i t u d e to the north and south, and 2° of longitude to the east and west, of the c i t y . This was a r e l a t i v e l y crude procedure and the errors i n the values obtained are probably quite large. Maps for 0400 PST were a v a i l a b l e for 1980 only, so that e i t h e r 1000 PST or 1600 PST maps had to be used for 1977 and 1978. The 0400 PST maps 73 were preferable because t h i s i s the time at which the upper a i r soundings, used here, are taken. These discrepancies i n the accuracy of the values extracted, and i n the timing of the maps used, are probably not serious because the model i s f a i r l y i n s e n s i t i v e to changes i n the pressure and temperature gradients. P r e c i p i t a b l e water was required as an input to a l l three models. No observations were a v a i l a b l e so the equation of Smith (1966), for estimating p r e c i p i t a b l e water from dew point temperatures (T^), was used: l n w = {0.1133 - ln(X + 1)} + 0.0393T_,. (4.1) d Smith s p e c i f i e s a value for X of 2.77 for summer i n the l a t i t u d e zone 40-50°. Dew point temperatures were obtained from observations made at the University of B r i t i s h Columbia climate s t a t i o n situated about 12 km west of the ob-servation s i t e . For the s e n s i t i v i t y analyses, p r e c i p i t a b l e water was varied from 2 to 3.5 cm which covers the range of values calculated for the ob-servation periods. Over t h i s range the net r a d i a t i o n modelled by C i n --2 creased only a few W m . For the same range of values the net r a d i a t i o n _2 modelled by M dropped about 45 W m . This r e l a t i v e l y high s e n s i t i v i t y indicates that, i n t h i s case, there may be some concern that the p r e c i p i t a b l e water values input to the model are only estimates. The greater s e n s i t i v i t y to p r e c i p i t a b l e water of Model M may be a r e s u l t of the fact that precip-i t a b l e water i s only used to compute the solar r a d i a t i o n i n t h i s model, whereas i n Model C i t i s used to compute both the solar r a d i a t i o n and the incoming long-wave r a d i a t i o n . Since an increase i n p r e c i p i t a b l e water de-creases the solar r a d i a t i o n but increases the incoming long-wave r a d i a t i o n the e f f e c t of a change i n p r e c i p i t a b l e water on the net r a d i a t i o n of Model 74 C w i l l be smaller. It would have been misleading to do a s i m i l a r analysis for Model A because even though p r e c i p i t a b l e water i s input to t h i s model to compute the incoming long-wave r a d i a t i o n the solar r a d i a t i o n , which i s also a function of p r e c i p i t a b l e water, was not calculated within the model. For the solar r a d i a t i o n c a l c u l a t i o n s Model M also requires a value for the concentration of atmospheric dust p a r t i c l e s . The choice of a value for t h i s parameter was based on the work of Gates (1962). He suggests values _3 ranging between 1.4 and 2.0 p a r t i c l e s cm for urban areas. Since Vancouver - 3 i s a medium-sized, f a i r l y n o n - i n d u s t r i a l c i t y a value of 1.7 p a r t i c l e s cm was chosen. Over the range of values suggested for urban areas the net -2 r a d i a t i o n only decreased by 10 W m The f i n a l input v a r i a b l e to be considered i n t h i s section i s the sub-sidence c o e f f i c i e n t required i n Model A for the mixed layer c a l c u l a t i o n s . During the 1978 observation period atmospheric temperature soundings were p e r i o d i c a l l y taken at the s i t e . From these, subsidence c o e f f i c i e n t s were estimated (D.G. Steyn, personal communication). For 1977 and 1980 a L —6 —1 constant value of 1.0 X 10 s was used. This i s the mean of those values determined i n 1978. Model s e n s i t i v i t y to t h i s input was not s t r a i g h t -forward. The subsidence c o e f f i c i e n t was given the values: 0, 1.0 X 10 7 , —6 — 5 — 1 1.0 X 10 , and 1.0 X 10 s , which covers the range of values calculated from the 1978 observations. For the f i r s t three of these values the fluxes, surface temperature, and mixed layer height remained more or le s s constant. For the value 1.0 X 10 s \ however, the changes i n the fluxes were more s i g n i f i c a n t and the mixed layer height dropped by about 1100 m. Although the nature of the l a s t change i s p h y s i c a l l y reasonable, i t s magnitude appears to be rather extreme. 75 4.3 Site Surface C h a r a c t e r i s t i c s The f i n a l set of inputs i s comprised of those variables that describe the surface f or which the fluxes are to be modelled. These include surface albedo, surface roughness, substrate thermal c h a r a c t e r i s t i c s , and surface moisture. In general, these parameters remain constant from day-to-day for a s p e c i f i c s i t e and they are the most d i f f i c u l t of the inputs to assign. This i s due mainly to the complexity of the suburban surface. Based on the discussion of Chapter 3, i t was decided that a value of 0.13 would be used for the surface albedo. S e n s i t i v i t y analyses were done by varying the albedo from 0.10 to 0.20. The e f f e c t of the albedo v a r i a t i o n , on modelled net r a d i a t i o n , was v i r t u a l l y the same for both Models M and C: -2 i t dropped by about 75 W m . The surface temperature of Model M was more or l e s s unaffected, while that of Model C dropped by 2° C. The height of the mixed layer dropped by j u s t over 100 m due to the decrease i n surface heating. Given the s e n s i t i v i t y of the modelled net r a d i a t i o n to the value of the albedo i t i s fortunate that i t was determined from measurements i n th i s case. Where no observations are a v a i l a b l e , however, the albedo would have to be estimated. This may be d i f f i c u l t for a suburban or urban surface, but Oke (1983) shows these values to be f a i r l y conservative. A value of 0.5 m, again as noted i n Chapter 3, was used for the surface roughness. It should be mentioned that t h i s i s not expected to be much more than an order of.magnitude estimate (Steyn, 1980). A s e n s i t i v i t y analysis was c a r r i e d out for each of the three models by varying the roughness length from 0.2 to 1.0 m. Strangely, Model C was almost completely i n s e n s i t i v e to changes i n roughness over t h i s range. S i m i l a r l y , Carlson (1981) found that changing the roughness length by a factor of 10 produced a change of less than 1 C i n the modelled surface temperature. This s i t u a t i o n may indicate inconsistencies i n the atmospheric d i f f u s i v i t y c a l c u l a t i o n s of t h i s model. Model A, however, was only s l i g h t l y more s e n s i t i v e to these -2 changes. In t h i s case, the turbulent fluxes both increased about 8 W m due to the increased atmospheric turbulence associated with greater surface roughness, while the surface temperature dropped about 1° C due to the increased removal of surface heat. Ackerman (1976, p. 148) also found that model s e n s i t i v i t y to roughness length was f a i r l y i n s i g n i f i c a n t . Model M was the most s e n s i t i v e to changes i n roughness length; the la t e n t and sen-_2 s i b l e heat fluxes increased and decreased, re s p e c t i v e l y , by about 60 W m Similar unexpected changes were found when examining the s e n s i t i v i t y of Model M to wind speed. ,In the case of Model M there should be some concern about the uncertainty involved i n the determination of the roughness length. The next set of inputs are the substrate thermal c h a r a c t e r i s t i c s , or those properties which govern the transfer of heat through the substrate. Models M and A require values for both the substrate heat capacity and the thermal d i f f u s i v i t y from which a value for conductivity i s calculated i n the models. Model C requires- a value for the thermal i n e r t i a (p) which i s a function of the heat capacity (C) and thermal d i f f u s i v i t y (K): p = C i A (4.2) In t h i s model the thermal i n e r t i a i s used i n a second-order regression equation to determine the substrate conductivity and, from t h i s , the thermal d i f f u s i v i t y . For a simple surface, such as a bare s o i l , i t would be an easy matter to assign values to these inputs because well-established thermal properties have been determined for s p e c i f i c surface types (eg. Goward, 1981). However, an urban surface i s a complex mixture of many d i f f e r e n t materials. In addition, because the modelling "surface" i s probably located at the height of the roughness elements (Sec. 1.4), and should include the "honeycomb" nature of bu i l d i n g s , the thermal properties of a i r may also be s i g n i f i c a n t . These features obviously make i t very d i f f i c u l t to assign values to the thermal properties of an urban surface. Nevertheless, a rather wide range of possible values have been reported i n the l i t e r a t u r e . A search produced a range of values f o r the thermal i n e r t i a of urban areas of from 800 to 3000 J m~2s~\~l, (eg. Oke, 1981). From these a value of 1260 J m~2s~\~l was chosen as appropriate for the thermal i n e r t i a i n Model C. This value i s the most commonly c i t e d i n the l i t e r a t u r e for suburban areas, while higher values are normal for urban areas. Using the second-order regression equation i n Model C t h i s value r e s u l t s i n a conductivity of 1.0 W m ^ -7 2 -1 and a d i f f u s i v i t y of 7.0 X 10 m s . The conductivity value agrees well with those given for suburban areas but the d i f f u s i v i t y seems s l i g h t l y low. It was decided that these values would be used as a basis for Models M and -7 2 -1 A so that the values input were 7.0 X 10 m s for the d i f f u s i v i t y and 6 —3—1 1.5 X 10 J m K for heat capacity. The heat capacity value corresponds well with those given i n the l i t e r a t u r e for suburban areas. In each model i t i s only the conductivity and d i f f u s i v i t y which are e x p l i c i t l y used i n the computations. The conductivity i s used i n determining the subsurface heat f l u x , while the d i f f u s i v i t y i s used i n the substrate temperature p r o f i l e c a l c u l a t i o n s . S e n s i t i v i t y analyses were c a r r i e d out on the above parameters using the ranges of values given i n the l i t e r a t u r e f o r urban and suburban areas. 78 For Models M and A the heat capacity and d i f f u s i v i t y were varied separately, as w e l l as together. F i r s t , the heat capacity was varied from 1.0 X 10 6 —3 — 1 through 2.2 X 10 J m, K . In Model M the subsurface heat f l u x increased -2 by almost 25 W m , at the expense of the turbulent heat fluxes. Model A _2 was more s e n s i t i v e : the subsurface heat f l u x increased by about 120 W m —6 —6 2 —1 The thermal d i f f u s i v i t y was varied from 1.0 X 10 to 2.0 X 10 m s . In _2 Model M the substrate heat f l u x increased by almost 10 W m , while i n Model -2 A i t increased by almost 50 W m . The above r e s u l t s appear to be p h y s i c a l l y r e a l i s t i c because an increase i n e i t h e r the heat capacity or d i f f u s i v i t y increases the a b i l i t y of the substrate to transport heat. S e n s i t i v i t y analyses were also done on combinations of values of heat capacity and thermal d i f f u s i v i t y . The combinations ranged from 1.0 X 10 J m - 3K - 1 and 1.0 X 10~ 6 m 2s _ 1 to 2.2 X 10 6 J nfV"1 and 2.0 X 1 0 _ 6 m 2s _ 1 for heat capacity and d i f f u s i v i t y , r e s p e c t i v e l y . I t was found, i n Model _2 M, that the subsurface heat f l u x increased by almost 40 W m , while the surface temperature remained constant. In Model A, for the same combinations, -2 the subsurface heat f l u x increased by 185 W m , the turbulent heat fluxes _2 decreased by 90 W m , the surface temperature dropped by 2 C, and the mixed layer height dropped by almost 260 m. The l a t t e r two changes are related to the decrease i n the turbulent sensible heat f l u x . _2 - l i For Model C thermal i n e r t i a was varied from 800 to 3000 J m s K This range i s almost equivalent to the range represented by the combinations of heat capacity and d i f f u s i v i t y used above. Over t h i s range, the net -2 r a d i a t i o n increased by almost 20 W m and the subsurface heat f l u x increased 2 by 225 W m . The surface temperature dropped by 5 C and the mixed layer height by over 300 m. The r e s u l t s of increasing the thermal i n e r t i a i n 79 Model C a r e as would be e x p e c t e d : an i n c r e a s e i n the t h e r m a l i n e r t i a i n -c r e a s e s t h e a b i l i t y o f the s u b s u r f a c e t o t r a n s p o r t h e a t . A l t h o u g h i t i s d i f f i c u l t t o make c o m p a r i s o n s between the t h r e e models as t o t h e i r s e n s i t i v i t i e s t o t h e t h e r m a l s u b s t r a t e p a r a m e t e r s i t seems t h a t Model M i s d e f i n i t e l y t h e l e a s t s e n s i t i v e t o them. F u r t h e r , i t i s s u p r i s i n g t h a t Model M's s u r f a c e t e m p e r a t u r e s do n o t r e s p o n d a t a l l t o changes i n t h e s e i n p u t s . The f i n a l i n p u t d e s c r i b e s the s u r f a c e m o i s t u r e . I n Models M and C a m o i s t u r e a v a i l a b i l i t y p arameter (M) i s used w h i l e , i n Model A t h e Bowen r a t i o (B) i s used. Of a l l the i n p u t p a r a m e t e r s t h e s e a r e the most d i f f i c u l t t o a s s i g n v a l u e s . I n t h e o r i g i n a l v e r s i o n o f Model M t h e s u r f a c e r e l a t i v e h u m i d i t y (RH) was u s e d , r a t h e r t h a n M, i n d e t e r m i n i n g t h e s u r f a c e e v a p o r a t i o n . I t was m u l t i p l i e d by the s u r f a c e s a t u r a t i o n s p e c i f i c h u m i d i t y : Q E = X A L e ( « A " R H % s } ' < 4- 3 ) and, f o r u r b a n a r e a s , was assumed e q u i v a l e n t t o " t h e f r a c t i o n o f t o t a l a r e a o c c u p i e d by t r a n s p i r i n g p l a n t s " (Myrup, 1969). I n eqn. ( 2 . 2 0 ) , as used i n t h e p r e s e n t v e r s i o n o f Model M, and eqn. ( 2 . 7 5 ) , as used i n Model C, M i s m u l t i p l i e d by the g r a d i e n t , r a t h e r t h a n the s u r f a c e s a t u r a t i o n v a l u e , of s p e c i f i c h u m i d i t y . M i s d e f i n e d as (Nappo, 1975): E = ME (4.4) o op where E i s the a c t u a l , and E the p o t e n t i a l , e v a p o r a t i o n . I n words, M o op ^ > r i s " t h e f r a c t i o n of maximum p o s s i b l e e v a p o r a t i o n f o r a s a t u r a t e d s u r f a c e " 80 (Carlson and Boland, 1978). Unfortunately, t h i s d e f i n i t i o n does not suggest a basis for assigning a value to M. Nappo (1975) c a r r i e d out a study of the two methods ( i . e . eqn. (4.3) compared to eqns. (2.20) and (2.75)) for c a l c u l a t i n g evaporation i n energy balance models. He concluded'that i n urban areas with small evaporative area (the present study area i s not one of small evaporative area) M should be m u l t i p l i e d by the s p e c i f i c humidity gradient, as i n eqns. (2.20) and (2.75), but interpreted as the f r a c t i o n of t o t a l area covered by t r a n s p i r i n g vegetation, as was done for RH i n eqn. (4.3). He had stated previously, however, that i n t e r p r e t i n g M i n t h i s way was questionable. In t h e i r c r i t i c i s m of Model M, M i l l e r et a l . (1972) i n -dependently argued that the above method (that suggested by Nappo) should be used. In addition, T. Carlson (personal communication) believes that i t i s possible that M i s c l o s e l y related to the f r a c t i o n of vegetation, a l -though he did add that t h i s term i s the "big unknown" i n the model. The moisture a v a i l a b i l i t y f actor has, therefore, been given a value of 0.64 which i s the f r a c t i o n of greenspace i n the 2 km radius c i r c l e surrounding the ob-servation s i t e . A drawback of using t h i s p a r t i c u l a r i n t e r p r e t a t i o n of the moisture a v a i l a b i l i t y i s that i t implies that the value w i l l be c h a r a c t e r i s t i c of a s p e c i f i c s i t e and, therefore, that i t should not vary from day-to-day with varying moisture content. In both Models M and C i t was found that the s e n s i t i v i t y to moisture a v a i l a b i l i t y decreased as t h i s term was increased from 0.10 to 0.99. The sensible and latent heat fluxes modelled by M decreased and increased, res-_2 p e c t i v e l y , by j u s t under 400 W m , while the surface temperature decreased by 4° C. In Model C the net r a d i a t i o n increased by 40 W m 2 , the sensible -2 heat f l u x decreased by 140 W m , and the latent heat f l u x increased by over 81 250 W m . The surface temperature dropped 12 C and the mixed layer height dropped by almost 700 m. These r e s u l t s are as expected. An increase i n surface moisture should cause an increase i n the latent heat f l u x thereby leaving l e s s energy a v a i l a b l e for the sensible heat fluxes (Q u and CO. This, i n turn, causes a drop i n the surface temperature and mixed layer height. Problems were also encountered i n deciding on the value for the input Bowen r a t i o for Model A. Ackerman (1976, p.122) states that the choice of an input Bowen r a t i o has to be made f a i r l y a r b i t r a r i l y because l i t t l e i s known about i t s behaviour i n urban areas. He chose a value of 2.0 for the Bowen r a t i o for h i s work with the model i n Los Angeles. A value of 1.0 was chosen f o r the current study as the observations have shown that t h i s i s a f a i r l y representative value for the s i t e (Sec. 3.3). Placing a single value on the Bowen r a t i o , however, i s u n r e a l i s t i c as i t has been observed to vary from hour-to-hour and from day-to-day with varying moisture content (eg. Kalanda et a l . , 1980). Ackerman (personal communication) states that the Bowen r a t i o can be s p e c i f i e d as a function of time, but i t was d i f f i c u l t to know how t h i s could be done beyond using the observed hourly values for each day, as input. The l a t t e r would c l e a r l y defeat the purpose of energy balance modelling. The s e n s i t i v i t y of Model A to changes i n the Bowen r a t i o was investigated by varying t h i s parameter from 0.5 through 2.5. As the Bowen r a t i o increased the s e n s i t i v i t y of the model to i t decreased. The net r a d i a t i o n decreased -2 -2 almost 20 W m , the sensible heat f l u x increased by 130 W m , and the latent -2 heat f l u x decreased about 230 W m . The surface temperature increased by 9° C, while the mixed layer height increased about 260 m. Again these r e s u l t s appear to be p h y s i c a l l y r e a l i s t i c . 82 I t i s c l e a r t h a t a l l three models are very s e n s i t i v e to the input para-meter d e s c r i b i n g s u r f a c e moisture which i s , u n f o r t u n a t e l y , probably the i n -put c o n t a i n i n g the most u n c e r t a i n t y . 4 . 4 Conclusions The c a r e f u l choice of val u e s f o r the model i n p u t s i s very important because the accuracy of the model output depends to a great extent on them. This i s where one of the reasons f o r doing the s e n s i t i v i t y analyses becomes most apparent; they g i v e some i d e a as to which i n p u t s a f f e c t the output to the g r e a t e s t e x t e n t . U n f o r t u n a t e l y , i t was found t h a t the two s e t s of i n p u t s to which the models were most s e n s i t i v e (the s u b s t r a t e thermal p r o p e r t i e s and the sur f a c e moisture c h a r a c t e r i s t i c s ) were a l s o those whose val u e s c o n t a i n the g r e a t e s t u n c e r t a i n t y . These v a l u e s , and those of the other i n p u t s , were chosen as c l o s e l y as p o s s i b l e to the s p e c i f i c a t i o n s of the o r i g i n a t i n g mod-e l l e r s . The f a c t t h a t some i n p u t s were d i f f i c u l t to set i s , of course, a weakness of the models: i f inp u t v a l u e s cannot be e a s i l y assigned f o r a given s i t u a t i o n the models have l i t t l e u s e f u l n e s s . CHAPTER FIVE  MODEL:OBSERVATION COMPARISONS Using the input data discussed i n Chapter 4, the three models were run f o r the eighteen days l i s t e d i n Chapter 3. The output from the models was compared to the observations using s c a t t e r diagrams, comparative s t a t i s t i c s , and time s e r i e s p l o t s . A l l analyses were conducted f o r the hours from approximately 0600 LST to sunset. Hourly, as w e l l as t o t a l daytime, energy values were used. In a d d i t i o n , each of the three observation years was i n i t i a l l y considered separately because of the d i f f e r e n c e s i n moisture con-d i t i o n s between them. The s t a t i s t i c s (summarized i n tables throughout t h i s chapter) include the means (X) and standard deviations (a) of the observed and modelled values, and the model o b s e r v a t i o n root mean square (RMS) e r r o r and c o e f f i c i e n t of 2 2 determination (r ). I t should be noted that r can be a misleading s t a t i s i t c : a high value i n d i c a t e s nothing more than a constant r e l a t i o n s h i p , not neces-s a r i l y a 1:1 r e l a t i o n s h i p , between two v a r i a b l e s . This means that a very consistent b i a s , as frequently occurs here, could r e s u l t i n a high c o e f f i c i e n t of determination. August 3, 1978 ( F i g . 3.4) was chosen as the day to be used f o r com-parison of observed and modelled time s e r i e s . This day was selected be-cause the observed fluxes showed l e s s hour-to-hour v a r i a b i l i t y than d i d those of any other day i n the data set, and because i t was reasonably representative of the energy balances observed i n the 1977 and 1978 observation periods. In a d d i t i o n , i t would hardly be " f a i r " to compare modelled time s e r i e s p l o t s with any of those observed i n 1980. The i m p l i c a t i o n of t h i s choice i s , 83 84 therefore, that i f the models cannot d u p l i c a t e the observations of August 3, 1978 i t i s u n l i k e l y that they w i l l be able to d u p l i c a t e those of any other day. 5.1 Net Radiation Before considering the net r a d i a t i o n r e s u l t s i t should be remembered that Model A does not contain i t s o r i g i n a l net r a d i a t i o n c a l c u l a t i o n s (Sec. 2.2.1). Instead, i t s inputs include s o l a r r a d i a t i o n values c a l c u l a t e d by Model C, and the method used i n Model C f o r c a l c u l a t i n g incoming long-wave r a d i a t i o n was incorporated into A. Net r a d i a t i o n values c a l c u l a t e d by these two models w i l l , therefore, d i f f e r only because of d i f f e r e n c e s i n the mod-e l l e d surface and mixed layer temperatures used i n computing the net long-wave r a d i a t i o n . The s c a t t e r diagrams (Figs. 5.1-5.9) and the s t a t i s t i c s (Table 5.1) show that the hourly values of modelled net r a d i a t i o n are reasonably s i m i l a r to those observed i n a l l cases. Of the three years, the RMS e r r o r s are highest for 1980 and, of the three models, t h i s s t a t i s t i c i s lowest for Model C. On the average, the net r a d i a t i o n i s overestimated by a l l three models. In a d d i t i o n , the observed hourly v a r i a t i o n of net r a d i a t i o n i s overestimated by the models, as shown by the standard d e v i a t i o n s . S t a t i s t i c s f o r the daytime t o t a l s of net r a d i a t i o n are also included i n Table 5.1. These were again most c l o s e l y predicted by Model C and con-s i s t e n t l y overestimated. The day-to-day v a r i a t i o n of net r a d i a t i o n i s w e l l duplicated by the models as shown by the standard deviations for the energy t o t a l s . The time s e r i e s p l o t s of modelled and observed net r a d i a t i o n (Figs. 85 Table 5.1 - Summary of s t a t i s t i c s comparing modelled and observed net all-wave r a d i a t i o n (X, o, and RMSE have u n i t s of -2 -2 -1 W m f o r h o u r l y v a l u e s and MJ m d f o r t o t a l v a l u e s ; r 2 i s dimensionless) Observed Model M Model A Model C 1977 -X 238.6 253.7 251.9 239.1 a 192.8 200.5 201.7 191.2 RMSE 20.7 17.9 8.3 r 2 0.99 0.99 0.99 1978 X 325.0 364.9 361.5 351.4 a 206.9 229.6 238.4 226.2 RMSE 48.1 49.6 35.8 r 2 0.99 0.99 0.99 1980 X 295.8 340.8 345.8 328.5 a 201.0 240.9 251.0 236.2 RMSE 63.8 75.3 54.3 r 2 0.99 0.99 0.99 A l l X 305.1 344.6 344.8 331.5 years a 204.9 233.8 242.3 229.4 RMSE 53.2 59.5 42.7 r 2 0.99 0.99 0.99 Daytime X 14.9 16.6 16.6 16.0 t o t a l s a 2.6 2.8 2.8 2.7 RMSE 1.9 2.0 1.4 r 2 0.92 0.88 '0.91 86 -J0.0 0.0 10.0 20.0 30.0 40.0 , 50.0 OBSERVED FLUX (W/M2) CX10J ) 60.0 70.0 F i g . 5.1 - Modelled v s . observed net r a d i a t i o n : • Model M (1977) -10.0 0.0 10.0 20.0 30.0 40.0 . SO.O OBSERVED FLUX (W/M2) (X101 ) 60.0 70.0 F i g . 5.2 - Modelled v s . observed net r a d i a t i o n : Model M (1978) 87 F i g . 5.3 - Modelled v s . observed net r a d i a t i o n : Model M (1980) F i g . 5.4 - Modelled v s . observed net r a d i a t i o n : Model A (1977) 88 , 1 1 1 1 r— T i i ) -JO.3 0.3 10.3 ?a 0 .30.0 40.0 SO.O &Q.Q "JO.O OBSERVED FLUX (V7M2 ) (X101 J F i g . 5.5 - Modelled v s . observed net r a d i a t i o n : Model A (1978) OX-OflT-80 4* / / ^ + + > + y f t * A* , ~\ 1 1 1 [ I 1 I -10 0 0 0 10.0 20.0 30.0 0 , 0 50.0 60,0 70.0 OBSERVED FLUX [W/M21 (X101 ) F i g . 5.6 - Modelled v s . observed net r a d i a t i o n : Model A (1980) 89 F i g . 5.8 - M o d e l l e d v s . o b s e r v e d n e t r a d i a t i o n : M o d e l C (1978) 90 F i g . 5.9 - Modelled v s . observed net r a d i a t i o n : Model C (1980) 91 5.10-5.12) show c l e a r l y that f o r each model i t i s approximately the s i x hours around noon i n which t h i s f l u x i s overestimated. This overestimation i s c l e a r l y greater f o r Models M and A than i t i s f o r C. Net all-wave r a d i a t i o n can be separated i n t o i t s two components: the so l a r and long-wave r a d i a t i o n . The modelling of these terms was assessed f o r 1978 only (Table 5.2 and F i g s . 5.13-5.17). Solar r a d i a t i o n was s l i g h t l y underestimated by Model M and almost p e r f e c t l y predicted by Model C. This i s not p a r t i c u l a r l y s u r p r i s i n g because c l e a r sky s o l a r r a d i a t i o n can be accurately estimated (eg. Suckling and Hay, 1976). Most of the e r r o r i n the modelling of net r a d i a t i o n must, therefore, l i e i n the long-wave comp-onent- The RMS e r r o r s involved i n modelling t h i s term are, indeed, l a r g e r than those f o r the s o l a r r a d i a t i o n . The e r r o r i s lowest f o r Model C and highest f o r Model M. The overestimation of the long-wave r a d i a t i o n , by a l l three models, p a r t i a l l y compensates f o r the underestimation i n modelled so l a r r a d i a t i o n r e s u l t i n g i n the good o v e r a l l agreement between modelled and observed net all-wave r a d i a t i o n . Hourly values of long-wave r a d i a t i o n modelled by M show very l i t t l e v a r i a t i o n : f o r an observed standard d e v i a t i o n -2 -2 of 36.6 W m , the v a r i a t i o n i n the modelled values i s as low as 2.8 W m -2 That f o r Model A i s 13.4 W m , while values modelled by C show the most -2 r e a l i s t i c v a r i a t i o n with a standard d e v i a t i o n of j u s t over 20 W m Unfortunately, i t i s not p o s s i b l e to d i r e c t l y assess the modelling of long-wave r a d i a t i o n i n any d e t a i l because no r e l i a b l e observations of i t s incoming arid outgoing components are a v a i l a b l e f o r the observation periods. However, con s i d e r a t i o n of the methods used i n the models, along with some speculation, make i t p o s s i b l e to i d e n t i f y p o s s i b l e problem areas, and to formulate some recommendations. A l l three models use Stefan-Boltzmann's 92 RUG. 3-. \31S. IM) 2 4 . 0 SOLAR TIME (HOURS) F i g . 5.10 - Diurnal course of modelled and observed net r a d i a t i o n : Model M SOLAR TIME (HOURS) F i g . 5.11 - Diurnal course of modelled and observed net r a d i a t i o n : Model A 93 -a—a—a 22.0 24.0 F i g . 5.12 - D i u r n a l course of modelled and observed net r a d i a t i o n : Model C Table 5.2 - Summary of s t a t i s t i c s comparing modelled and observed net s o l a r r a d i a t i o n and net long-wave -2 r a d i a t i o n (X, a, and RMSE have u n i t s of W m ; r 2 i s dimensionless) Observed Model M Model A Model C K* X 467.3 449.0 — 466.6 a 241.0 228.3 - 244.9 RMSE 23.2 - 9.4 2 r 0.99 — 0.99 L* X -139.2 -80.5 -101.2 -111.4 0 36.6 2.8 13.4 21.2 RMSE 70.0 47.6 34.9 2 r 0.23 0.50 0.75 95 3?H KX - 1978 0.0 10.0 20.0 30.0 40.0 SO.O 60.0 OBSERVED FLUX IW/M2) DUO 1 J 7o.o ao.o F i g . 5.13 - Modelled v s . observed net s o l a r r a d i a t i o n : Model M (1978) 0.0 10.0 70.0 30.0 «.0 50.0 , 60.0 OBSERVED FLUX IU/M2) (XlO 1 J F i g . 5.14 - Modelled v s . observed net s o l a r r a d i a t i o n : Model C (1978) 96 -335.0 -200.0 -J15.0 -JSO.0 -J25.0 -)00.0 -15.0 -SO.O -25.0 OBSERVED FLUX IV/M2) F i g . 5.15'- Modelled vs. observed net long-wave r a d i a t i o n : Model M (1978) F i g . 5.16 - Modelled vs. observed net long-wave r a d i a t i o n : Model A (1978) 97 r F i g . 5.17 - Modelled v s . observed net long-wave r a d i a t i o n : Model C (1978) 98 law as a bas i s f o r computing both the incoming and outgoing components of long-wave r a d i a t i o n . I f both the e m i s s i v i t y and temperature of the r a d i a t i n g medium are known the emitted long-wave r a d i a t i o n can be ex a c t l y c a l c u l a t e d using t h i s law. The outgoing component of long-wave r a d i a t i o n i s c a l c u l a t e d using the modelled surface temperature and the assumption that the surface e m i s s i v i t y i s u n i t y . There are e r r o r s i n the modelled surface temperatures which w i l l a f f e c t the c a l c u l a t i o n of outgoing long-wave r a d i a t i o n but t h e i r magnitude i s not known (Sec. 5.5). I t i s p o s s i b l e , however, that the out-going component of long-wave r a d i a t i o n could be i n e r r o r by as much as 5% due to the assumption that the surface e m i s s i v i t y i s unity. The e m i s s i v i t y of a composite suburban surface i s probably c l o s e r to 0.95 as reported by A r n f i e l d (1982). The modelling of incoming long-wave r a d i a t i o n i s l e s s straightforward. In t h i s case an estimate must be made of the r a d i a t i v e temperature of the atmosphere. In Model M the modelled surface temperature i s used for t h i s purpose, while i n Models A and C the average of the modelled surface, and mixed l a y e r , temperatures i s used. I t i s recommended that a i r temperature alone be used for t h i s purpose (eg. Monteith, 1961; Idso and Jackson, 1969). (Recently, T. Carlson (personal communication) made a change i n Model C so that the incoming long-wave r a d i a t i o n i s c a l c u l a t e d as a function of (1.89^ + 0.2T Q)/2.) In the case of Model M incoming long-wave r a d i a t i o n would, therefore, be constant throughout the model-day, while i n Model C i t would simply be l e s s v a r i a b l e than i t i s at present. I t i s quite r e a l i s t i c for t h i s component of long-wave r a d i a t i o n to vary very l i t t l e over the course of a day (eg. Oke, 1978; p.127). The apparent atmospheric e m i s s i v i t y was c a l c u l a t e d using the method of Idso and Jackson (1969) i n Model M (eqn. 2.14) 99 and the method of Monteith (1961) i n Models A and C (eqn. 2.56). The l a t t e r may be a poor choice as Monteith (1961) developed h i s method for the B r i t i s h I s l e s . I t i s suggested that the method r e c e n t l y developed by Idso (1981), f o r c a l c u l a t i n g atmospheric e m i s s i v i t y , might be an improvement over those used. This method requires values f o r a i r temperature and vapour pressure and could be e a s i l y incorporated i n t o Models M and C. I t would be l e s s a t -t r a c t i v e to incorporate i t i n t o Model A because atmospheric humidity values are not otherwise required i n t h i s model. In conclusion, the e r r o r s i n modelling net all-wave r a d i a t i o n are due mainly to e r r o r s i n modelling net long-wave r a d i a t i o n . These, i n turn, are probably due to e r r o r s i n both the modelled temperatures and the values used f o r surface and atmospheric e m i s s i v i t i e s . 5.2 Turbulent Fluxes It i s i n the modelling of the turbulent heat fluxes that studying the three years of observations separately i s c l e a r l y valuable. The models overestimated the r e l a t i v e l y small l a t e n t heat fluxes observed i n 1978 and, with the exception of Model M, underestimated the high l a t e n t heat fluxes observed i n 1980. The s t a t i s t i c s (Table 5.3) and the s c a t t e r p l o t s (Figs. 5.18-5.35) show that the observed fluxes are, i n general, poorly estimated by the models. As with the net r a d i a t i o n , the RMS e r r o r s are highest f o r 1980. In 1977 and 1978 t h i s s t a t i s t i c i s lowest for Model A and highest f o r Model M, while i n 1980 the reverse i s the case. When the three years are combined the RMS e r r o r s f or a l l three models, except that for the l a t e n t heat fluxes modelled by M, are s i m i l a r . The 1977 and 1978 s c a t t e r diagrams for Model A show l e s s of a tendency toward bias than do those of e i t h e r Models M or C. The s c a t t e r diagrams f o r 1980 show that the l a t e n t 100 Table 5.3a - Summary of s t a t i s t i c s comparing modelled and ob-served s e n s i b l e heat f l u x e s (X, a, and RMSE have -2 -2 -1 u n i t s of W m f o r h o u r l y v a l u e s and MJ m d 2 f o r t o t a l v a l u e s ; r i s dimensionless) Observed Model M Model A Model C 1977 X 111.6 50.8 89.7 63.9 a 86.8 68.0 64.5 50.7 RMSE 74.0 44.8 66.2 r 2 0.77 0.83 0.83 1978 X 149.8 30.2 126.2 74.9 a 96.5 56.2 83.5 56.2 RMSE 136.0 59.4 96.1 r 2 0.58 0.68 0.66 1980 X -13.5 30.1 128.6 77.9 a 96.0 70.7 92.6 58.1 RMSE 129.7 195.3 146.9 r 2 0.0 0.0 0.0 A l l X 81.1 32.1 123.7 75.1 years a 123.0 63.8 86.4 56.6 RMSE 128.8 131.0 117.0 r 2 0.10 0.12 0.11 Daytime X 4.1 1.5 6.0 3.6 t o t a l s a 4.2 1.1 1.1 0.7 RMSE 5.2 4.8 4.4 r 2 0.02 0.0 0.02 101 Table 5.3b - Summary of s t a t i s t i c s comparing modelled and ob-served l a t e n t heat f l u x e s (X, cr, and RMSE have -2 -2 -1 u n i t s of W m f o r h o u r l y v a l u e s and MJ m d 2 f o r t o t a l v a l u e s ; r i s dimensionless) Observed Model M Model A Model C 1977 X 77.3 197.2 89.7 116.4 a 69.0 122.3 64.5 79.5 RMSE 140.6 35.5 56.9 2 0.72 0.77 0.73 1978 X 102.9 327.9 126.2 190.4 a 87.0 165.7 83.5 125.7 RMSE 251.4 58.7 128.8 r 2 0.61 0.64 0.44 1980 X 244.4 303.8 128.6 166.9 a 182.0 169.2 92.6 122.4 RMSE 119.6 166.7 135.0 r 2 0.69 0.66 0.65 A l l X 156.8 305.8 123.7 174.0 years a 150.4 167.6 86.4 122.6 RMSE 198.9 113.7 126.4 r 2 0.44 0.49 0.36 Daytime t o t a l s X a RMSE 2 7.8 3.8 14.8 3.1 8.2 0.08 6.0 1.1 3.9 0.20 8.4 1.8 4.1 0.01 R. 5 .19 - Modelled vs. observed l a t e n t heat f l u x e s : Model M (1977) 103 0H-Dfir-18 ++ + - 1 0 . 0 0.1 10.0 70 .0 30.9 40.0 SO.O OBSERVED FLUX (W/M2) (XlO 1 ) F i g . 5.20 - Modelled vs. observed s e n s i b l e heat f l u x e s : Model M (1978) . 8 -. 9 -QE-ORT-18 + + + + * + + + - 1 0 . 0 0 . 0 10.0 20.0 30.0 40.0 , SO.O OBSERVED FLUX (W/M2) (XlO 1 ) F i g . 5.21 - Modelled vs. observed l a t e n t heat f l u x e s : Model M (1978) 104 F i g . 5.22 - Modelled vs. observed sensib l e heat f l u x e s : Model M (1980) F i g . 5.23 - Modelled vs. observed l a t e n t heat f l u x e s : Model M (1980) 105 F i g . 5.25 - Modelled v s . observed l a t e n t heat f l u x e s : Model A (1977) 106 -10.0 0.0 10.0 23.0 30.0 40.0 SO .0 60.0 70.0 OBSERVED FLUX (V/M21 (X]0J 1 F i g . 5.26 - Modelled vs. observed s e n s i b l e heat f l u x e s : Model A (1978) -10.0 0.0 tO.O 20.0 30.0 40.0 50.0 60.0 70 0 OBSERVED FLUX (U/M2) (X101 ) F i g . 5.27 - Modelled vs. observed l a t e n t heat f l u x e s : Model A (1978) 107 F i g . 5.29 - Modelled v s . observed l a t e n t heat f l u x e s : Model A (1980) 108 i i i 1 1 1 1 1 1 1 -10.0 0.0 10.0 20.0 30.0 «.0 , 50.0 " 60.0 70.0 OBSERVED FLUX (W/M2) (X10 1 ) F i g . 5.31 - Modelled v s . observed l a t e n t heat f l u x e s : Model C (1977) 109 F i g . 5.33 - Modelled v s . observed l a t e n t heat f l u x e s : Model C (1978) I l l heat fluxes p r e d i c t e d by Models A and C are consistent underestimates of the observed and that, f o r a l l three models, there i s v i r t u a l l y no r e l a t i o n -ship between the modelled and observed sensib l e heat f l u x e s . The f a c t that the RMS e r r o r s f o r the turbulent fluxes modelled by A were lowest of the three models f o r 1977 and 1978 and those of M were lowest for 1980 may be somewhat misleading. In the case of Model A the value of the Bowen r a t i o input to t,he model was observed to be reasonably represent-a t i v e of the 1977 and 1978 observation periods. This i s c l e a r l y a major reason why the turbulent fluxes modelled by A were c l o s e r to those observed i n these two years than were those modelled by e i t h e r M or C. I t i s s i g -n i f i c a n t that such an appropriate choice for the Bowen r a t i o could probably not have been made without the a i d of p r i o r observations. In the case of Model M the l a t e n t heat fluxes are j u s t as high f o r 1978 as they are f o r 1980. In a d d i t i o n , on average,.the l a t e n t heat fluxes mod-e l l e d by M are always much l a r g e r than are those of Models A and C, and the average sensib l e heat fluxes are always smaller (Table 5.3). The f a c t that Model M has produced l a t e n t heat f l u x e s , which are c l o s e r to those observed, than those produced by Models A or C, does not, therefore, n e c e s s a r i l y imply that t h i s model i s better able to simulate the extreme conditions of 1980. Instead, the r e l a t i v e l y good correspondence seems to be merely c o i n c i d e n t a l . One might think that the r e l a t i v e l y high l a t e n t heat fluxes of Model M i n -di c a t e s that t h i s model i s a f f e c t e d more than Model C by the high moisture a v a i l a b i l i t y f a c t o r . This may be true, but i t was shown i n Chapter 4 that the s i z e of the l a t e n t heat f l u x modelled by M i s also very s e n s i t i v e to the s i z e of the input mean d a i l y a i r temperature (due to the f a c t that as the l a t t e r increased, the atmospheric temperature gradient, and thus the 112 sensibl e heat f l u x , decreased). The model output also seems to show a d i r e c t r e l a t i o n s h i p between the s i z e of the input temperature and the modelled l a t e n t heat f l u x . For example, the highest mean d a i l y a i r temp-erature was input f o r August 8, 1978 (23.3° C). This was also the date when -2 Model M produced a maximum l a t e n t heat f l u x of 577 W m , a maximum sensible -2 heat f l u x of only 42 W m , and when the sensible heat f l u x was p o s i t i v e f o r only s i x hours. i Because of the great hour-to-hour v a r i a b i l i t y e xhibited by the observed turbulent fluxes i t was expected that the RMS e r r o r s of the daytime t o t a l s may have been smaller, i n r e l a t i v e terms, than those of the hourly values. In general, t h i s was not the case (Table 5.3). In a d d i t i o n , the standard deviations of the modelled daytime t o t a l s are much smaller than those ob-served i n d i c a t i n g that the models do not d u p l i c a t e the observed day-to-day v a r i a b i l i t y . Figures 5.36 to 5.41 are time s e r i e s p l o t s of the observed and modelled turbulent fluxes f o r August 3, 1978. Those f o r Models M and C c l e a r l y show the overestimation of the l a t e n t heat f l u x and the underestimation of the sensible heat f l u x . Those f o r Model A show remarkable agreement between the observed and modelled turbulent f l u x e s . This i s due to the f a c t that the average observed Bowen r a t i o f o r t h i s p a r t i c u l a r day was very close to 1.0, the input value f o r Model A. For days with average observed Bowen r a t i o s d i f f e r e n t from 1.0 (for example, J u l y 31, 1978) the r e s u l t s are not quite as good (Figs. 5.42-5.43). The curves of the turbulent flu x e s modelled by M are more symmetric about s o l a r noon than are those of Models A and C. I t i s d i f f i c u l t to assess 11 R U G . 3 . \31B- (MJ O OBSERVED OH O MODELLED OH o—e—es—s-- e—o 0-0- 2.0 4.0 . o.o e.o 10.0 12-0 |4.0 SOLAR TIME (HOURS) 16.0 18.0 . 20.0 22.0 . 24.0 5.36 - Diurnal course of modelled and observed sensi b l e heat flux e s Model M R U G . 3 . 1 9 1 8 IM) O O B S E R V E D O E O M O D E L L E D 0 E o—a -~g_=gK - 8 - ^ 8 = 3 — a — a s-0.0 2.0 4.0 6.0 B.O 10.0 12.0 14.0 SOLAR TIME (HOURS) i i 16.0 1 6.0 20.0 22.0 2 4.0 5.37 - Diurnal course of modelled and observed l a t e n t heat fluxes: Model M 1 -9-flUG. 3. 1978 (A) a OBSERVED OH O MODELLED OH 0.0 2.0 4.0 6.0 10.0 12.0 I'.O SOLAR TIME (HOURS) IG.O lfi.0 20.0 . 22.3 24.0 ^ 5.38 - D i u r n a l course of modelled and observed s e n s i b l e heat f l u x e s Model A RUG. 3. 1978 IB) B ffl • OBSERVED OE Q MODELLED OE 0.0 2.0 4.0 6.0 10.0 12.0 14.0 SOLAR TIME (HOURS) IG.O 1B.0 20.0 22.0 24.0 5 . 3 9 - Di u r n a l - c o u r s e of modelled and observed l a t e n t heat f l u x e s : Model A 115 5H •si HUG. 3. 1918 ICI —m-• OBSERVED OH CD MODELLED QH - a— f f i—g—a -1 1 1 1 — 0.0 2.0 4.0 6.0 a.o — I 1 1 10.0 12.0 14.0 SOLAR TIME (HOURS) 16.0 18.0 20.0 22.0 24.0 F i g . 5.40 - D i u r n a l course of modelled and observed s e n s i b l e heat f l u x e s : Model C 5H AUG. 3. 1918 IC) . a—a a OBSERVED 0E O MODELLED 0E ~T [~ 0.0 2.0 4.0 6.0 a.o 10.0 12.0 14.0 SOLAR TIME IHOURS) 16.0 18.0 20.0 22.0 24.0 F i g . 5.41 - D i u r n a l course of modelled and observed l a t e n t heat f l u x e s : Model C 116 JULY 3) . 1978 tfi) • OBSERVED OH O MODELLED OH -T-ID.O 12.0 14.0 SOLAR TIME (HOURS) F i g . 5.42 - Diurnal course (July 31, 1978) of modelled and observed sensible heat f l u x e s : Model A F i g . 5.43 - Diurnal course (July 31, 1978) of modelled and observed l a t e n t heat f l u x e s : Model A 117 the p a t t e r n s e x h i b i t e d by the modelled f l u x e s because the o b s e r v a t i o n s do not suggest any c o n s i s t e n t d i u r n a l p a t t e r n s upon which to base t h i s assess-ment. . In f a c t , the o b s e r v a t i o n s o f t e n e x h i b i t i r r e g u l a r hour-to-hour v a r i a b i l i t y which the models could not be expected to d u p l i c a t e . This seems to be an a r t i f a c t of the urban environment. O v e r a l l , the r e s u l t s show the m o d e l l i n g of the t u r b u l e n t f l u x e s to be r a t h e r poor. Since the modelled net r a d i a t i o n agrees reasonably w e l l w i t h the observed, i n a l l cases, the problem i s c l e a r l y r e l a t e d to the models' a b i l i t i e s to p a r t i t i o n t h i s a v a i l a b l e energy between the three s u r f a c e heat f l u x e s . I n S e c t i o n 1.1 the f a c t o r s c o n t r o l l i n g the p a r t i t i o n i n g were l i s t e d . I t i s p o s s i b l e t h a t by c o n s i d e r i n g these i n d i v i d u a l l y the causes of the poor, agreement between the modelled and observed t u r b u l e n t f l u x e s could be i d -e n t i f i e d . U n f o r t u n a t e l y , they are s t r o n g l y i n t e r r e l a t e d so t h a t i t i s d i f -f i c u l t to i s o l a t e t h e i r i n d i v i d u a l e f f e c t s . N e v e r t h e l e s s , i t appears t h a t the s u r f a c e moisture i s the most s i g n i f i c a n t f a c t o r (Sec. 5.3). 5.3 Surface M o i s t u r e Three f i n d i n g s i n d i c a t e t h a t i t i s probably the s u r f a c e moisture r e p r e s e n t a t i o n which p l a y s the l a r g e s t r o l e i n determining the p a r t i t i o n i n g of the a v a i l a b l e net r a d i a t i o n . F i r s t , the o b s e r v a t i o n s f o r the 1978 and 1980 p e r i o d s show t h a t the s u r f a c e moisture c o n d i t i o n s are very important i n determining the energy balance i n the r e a l world (Sec. 3.3). Second, i n the s e n s i t i v i t y analyses of Chapter 4, i t was d i s c o v e r e d that the mod-e l l e d t u r b u l e n t f l u x e s are very s e n s i t i v e to the v a l u e chosen f o r the i n p u t parameter d e s c r i b i n g the s u r f a c e moisture. T h i r d , Model A proved to be best of the three models at reproducing the t u r b u l e n t f l u x e s observed i n the 1977 and 1978 p e r i o d s . The tendency toward b i a s , i n t h i s case,-was not as 118 pronounced as i t was for e i t h e r of Models M or C. As noted, t h i s may be a t t r i b u t e d to the f a c t that the Bowen r a t i o , input to t h i s model, happens to be reasonably representative of the observations f o r these two years. In 1980, when the average observed Bowen r a t i o was much lower, the r e s u l t s were not nearly as good. In other words, i f the moisture status of the surface can be adequately represented the model can be expected to produce reasonable values f o r the turbulent f l u x e s . Considering the importance of the surface moisture i t i s unfortunate that the two methods used to represent i t , the moisture a v a i l a b i l i t y f a c t o r and the Bowen r a t i o , are inadequate. The confusion involved i n assigning values to these parameters was emphasized i n Chapter 4. Moreover, holding both values constant, f o r the given s i t e , i s a gross o v e r s i m p l i f i c a t i o n because surface moisture content i s h i g h l y v a r i a b l e . There are fu r t h e r problems. The moisture a v a i l a b i l i t y f a c t o r (M) i s e s s e n t i a l l y used as a measure of s o i l moisture to a i d i n modifying the p o t e n t i a l evaporation rate to approximate the a c t u a l evaporation rate (eqn. 4.4). This i s p h y s i c a l l y reasonable (eg. P r i e s t l e y and Taylor, 1972) but by d e f i n i n g M i n terms of the f r a c t i o n of greenspace alone i t cannot p o s s i b l y adequately represent the a c t u a l moisture status of the surface because se v e r a l other f a c t o r s are also important. These include the time elapsed since the l a s t p r e c i p i t a t i o n event, as w e l l as i t s s i z e and duration, and the rate of l o s s of moisture since the event due to f a c t o r s such as evaporation and subsurface drainage. To account f o r these features i t might be p o s s i b l e to parameterize M based on s o i l moisture content. This was done f o r a f i e l d of rye grass (Nappo, 1975). The evaporation from an i r r i g a t e d p l o t was considered to represent 119 the p o t e n t i a l e v a p o r a t i o n r a t e . This was compared to the e v a p o r a t i o n from a n o n - i r r i g a t e d p l o t , i n which s o i l moisture was a l s o measured, to determine M as a f u n c t i o n of s o i l moisture content. Another a l t e r n a t i v e might be to use the accumulated net r a d i a t i o n s i n c e the l a s t p r e c i p i t a t i o n event as an i n d i r e c t i n d i c a t o r of s o i l moisture (Berkowicz and Prahm, 1982) because s u r f a c e d r y i n g depends to a l a r g e extent on the a v a i l a b l e energy. Un-f o r t u n a t e l y , both approaches are d i f f i c u l t to implement i n a suburban area. F i r s t , very l a r g e amounts of moisture are a p p l i e d to the s u r f a c e through the i r r i g a t i o n of greenspace so t h a t the s o i l moisture content i s extremely v a r i a b l e i n both time and space. Second, the j u x t a p o s i t i o n of pervious and impervious s u r f a c e s i n suburban areas i s thought to a f f e c t e v a p o r a t i o n r a t e s (Sec. 3.3) i n a way t h a t would be d i f f i c u l t to account f o r by a simple p a r a m e t e r i z a t i o n . Thus the moisture a v a i l a b i l i t y f a c t o r i s c l e a r l y not as simple as i t s name (and the d e f i n i t i o n used i n t h i s study) i m p l i e s . Surface moisture i s i n f l u e n c e d by much more than j u s t the f r a c t i o n of greenspace. A l l of the i n f l u e n c e s , as d i s c u s s e d above, would be very d i f f i c u l t to i n c o r p o r a t e i n t o a s i n g l e n umerical v a l u e , such as M, to c h a r a c t e r i z e s urface moisture f o r a given day and a given s i t e . This o v e r s i m p l i f i c a t i o n of a complex term i s probably the major reason why the e v a p o r a t i o n r a t e s were so p o o r l y es-timated by Models M and C. In a d d i t i o n , the f a c t that the moisture a v a i l -a b i l i t y f a c t o r d i d not account f o r the day-to-day v a r i a b i l i t y i n s u r f a c e moisture helps to e x p l a i n why the modelled t u r b u l e n t f l u x e s showed much l e s s v a r i a b i l i t y than those observed. The Bowen r a t i o , as used i n Model A, i s p o t e n t i a l l y a b e t t e r approach as i t makes s p e c i f i c a t i o n of upper l e v e l humidity unnecessary. However, 120 there are a l s o problems w i t h t h i s approach. F i r s t , as w i t h the moisture a v a i l a b i l i t y f a c t o r , there i s no simple way of e s t i m a t i n g a valu e f o r the Bowen r a t i o f o r a given s i t u a t i o n . In t h i s study i t was estimated based on energy balance o b s e r v a t i o n s . This i s h a r d l y r e a l i s t i c , however, because the need to model normally a r i s e s o n ly when o b s e r v a t i o n s are u n a v a i l a b l e . Without o b s e r v a t i o n s i t would be q u i t e d i f f i c u l t to estimate a r e p r e s e n t a t i v e v a l u e f o r the Bowen r a t i o . Second, the use of a constant Bowen r a t i o has a very a r t i f i c i a l e f f e c t on the model output. I t f o r c e s the s e n s i b l e and l a t e n t heat f l u x e s to be constant mul-t i p l e s of each other. This i s not r e a l i s t i c because the Bowen r a t i o v a r i e s , not only from day-to-day, but, e s p e c i a l l y , from hour-to-hour (Kalanda et a l . , 1980). This v a r i a b i l i t y makes i t even more d i f f i c u l t to estimate the Bowen r a t i o f o r a given s i t u a t i o n . Both the methods used to represent s u r f a c e moisture have a p h y s i c a l l y r e a l i s t i c b a s i s and are c o m p u t a t i o n a l l y simple to use i n the models. The major problem i s t h a t n e i t h e r can be e a s i l y c a l c u l a t e d , or even estimated, f o r a given s i t e on a given day. A r e l i a b l e p a r a m e t e r i z a t i o n f o r e i t h e r would be extremely v a l u a b l e f o r urban energy balance m o d e l l i n g because su r f a c e moisture i s c l e a r l y a major determinant of the p a r t i t i o n i n g of the net r a d i a t i o n . U n f o r t u n a t e l y , t h i s would not be a simple task given the complexity of the f a c t o r s a f f e c t i n g e v a p o r a t i o n i n a suburban area. A l t e r n a t i v e methods f o r c a l c u l a t i n g e v a p o r a t i o n which could e a s i l y be i n c o r p o r a t e d i n t o the energy b a l a n c e / s u r f a c e temperature framework of these models are not easy to come by. Two approaches f o r e s t i m a t i n g the components of the energy balance based on r e a d i l y a v a i l a b l e m e t e o r o l o g i c a l data have 121 r e c e n t l y been devised by DeBruin and H o l s t l a g (1982) and Berkowicz and Prahm (1982). They are based on a modified form of the P r i e s t l e y - T a y l o r equation and the Penman-Monteith formula, r e s p e c t i v e l y , w h i l s t both contain new and po s s i b l y promising methods f o r accounting f o r surface moisture, they are not immediately compatible with the form of the models tested here nor were they developed f o r urban areas. 5.4 Subsurface Heat Flux The modelled subsurface heat f l u x e s were compared to those c a l c u l a t e d using the parameterizations based on the observed net r a d i a t i o n (eqns. 3.2 and 3.3). The RMS e r r o r s f o r t h i s f l u x are lowest f o r Model A i n a l l cases, except the daytime t o t a l s , i n which case the erro r was lowest f o r C (Table 5.4). Models A and C c o n s i s t e n t l y overestimate the subsurface heat f l u x i n comparison with the parameterized values, while Model M c o n s i s t e n t l y underestimates t h i s term. In f a c t , at noon on most days the subsurface heat f l u x modelled by M represents only about 3% of the net r a d i a t i o n . Very small values f o r the subsurface heat f l u x were also obtained by Morgan et a l . (1977) i n t h e i r study of Sacramento, C a l i f o r n i a using Model M. The overestimation by Models A and C and the underestimation by Model M i s i n t e r e s t i n g because the three models were supplied with equal values for the subsurface thermal p r o p e r t i e s and s i m i l a r methods were used i n each to compute the subsurface temperature p r o f i l e s . The d i f f e r e n c e s between them, therefore, may be due to the f a c t that the temperature gradients used i n the c a l c u l a t i o n s of the subsurface heat f l u x are much smaller f o r Model M than they are f o r the other two models. The input mean d a i l y a i r temp-erature (used as the temperature at the base of the subsurface l a y e r i n Model M) i s never very d i f f e r e n t from the modelled surface temperatures 122 Table 5.4 - Summary of s t a t i s t i c s comparing modelled and observed W m for hourly values and MJ m d for t o t a l values; r 2 i s dimensionless) Observed Model M Model A Model C 1977 X 49.7 5.7 72.6 53.1 a 53.1 15.8 76.8 76.0 RMSE 60.6 37.3 38.7 r 2 0.63 0.93 0.79 1978 X 72.3 6.8 109.0 86.1 a 55.2 18.1 83.1 87.8 RMSE 78.1 55.3 63.9 r 2 0.61 0.81 0.50 1980 X 64.8 6.9 88.5 83.7 a 52.4 17.0 73.5 90.2 RMSE 71.9 38.7 --65.1 r 2 0.47 0.88 0.55 A l l X 67.2 6.7 97.4 82.5 years a 54.3 17.5 79.8 88.1 RMSE 74.2 47.7 62.4 r 2 0.55 0.84 0.54 Daytime X 3.4 0.3 4.7 4.0 t o t a l s a . 1.0 0.06 0.9 0.8 RMSE 3.3 1.5 1.0 r 2 - 0.07 0.43 0.33 123 (Table 5.5). U n f o r t u n a t e l y , due to the above s i t u a t i o n the comparisons p r o v i d e no c l u e as to whether or not an a p p r o p r i a t e choice was made f o r the v a l u e s of the subsurface thermal p r o p e r t i e s . F i g u r e s 5.44 to 5.46 are time s e r i e s p l o t s of the "observed" (para-meterized) and modelled subsurface heat f l u x e s f o r August 3, 1978. That f o r Model M shows t h a t the modelled f l u x i s , indeed, very s m a l l . The mod-e l l e d v a l u e s peak very e a r l y and become neg a t i v e about f o u r hours e a r l i e r than those "observed". A s i m i l a r trend i s evident f o r Model C but w i t h a more pronounced peak and much l a r g e r v a l u e s . The o v e r e s t i m a t i o n of the morning v a l u e s , and u n d e r e s t i m a t i o n of the afternoon v a l u e s , i n t h i s case, r e s u l t i n the comparatively low RMS e r r o r f o r t o t a l daytime v a l u e s . Model A reproduces the "observed" t r e n d b e t t e r than do e i t h e r of Models M or C. The curves produced by the l a t t e r two models would be more r e p r e s e n t a t i v e of those observed over s i m p l e r s u r f a c e s . S e l l e r s (1965; p.114) s t a t e s that f o r bare s o i l on c l e a r days the subsurface heat f l u x peaks one or two hours before noon, w h i l e Nunez and Oke (1977) note t h a t urban values tend to peak c l o s e r to noon. 5.5 Surface Temperatures The f o l l o w i n g assessment of modelled s u r f a c e temperatures i s based on the d i s c u s s i o n i n S e c t i o n 3.4 and on Table 5.5. The t a b l e c o n t a i n s maximum and minimum, modelled s u r f a c e temperatures and, observed a i r temperatures. The l a t t e r were obtained from Atmospheric Environment S e r v i c e o b s e r v a t i o n s made at the Vancouver I n t e r n a t i o n a l A i r p o r t . The minimum temperature values l i s t e d f o r Model C may not be the t r u e modelled minima because the f i r s t time step f o r t h i s model i s set at approximately 0600 LST. 124 F i g . 5.44 - Diurnal course of modelled and observed subsurface heat f l u x e s : Model M AUG. 3. 1978 (A) • OBSERVED QG CD MODELLEO OG 10.0 12.0 14.0 SOLAR TIME (HOURS) F i g . 5.45 - Diur n a l course of modelled and observed subsurface heat f l u x e s : Model A 125 5.46 - D i u r n a l course of modelled and observed subsurface heat f l u x e Model C 126 Tab le 5 . 5 - D a i l y maximum and minimum mode l led s u r f a c e temperatures and observed a i r temperatures Tab le 5 . 5 a - Model M Date Observed M o d e l l e d min max range min max rangi Sept, . 10, 1977 9 . 3 17.6 8 . 3 12.3 15.6 3 . 3 Sept, . 2 5 , 1977 7 . 0 15.2 8 . 2 9 .2 13.4 4 . 2 J u l y 2 0 , 1978 14 .1 2 6 . 5 12.4 12.8 17.7 4 . 9 J u l y 2 9 , 1978 13.2 24 .7 11.5 16.6 2 1 . 1 4 . 5 J u l y 3 0 , 1978 13.2 2 6 . 4 13.2 17 .0 2 1 . 5 4 .6 J u l y 3 1 , 1978 11.5 2 4 . 0 12 .5 15 .0 19.8 4 . 8 Aug. 1 , 1978 12 .1 23 .7 11.6 15 .0 19 .9 4 . 9 Aug. 2 , 1978 12.6 2 4 . 6 12 .0 16.2 20.7 4 . 5 Aug. 3 , 1978 14 .1 25 .7 11.6 17.7 2 2 . 0 4 . 3 Aug. 4 , 1978 13.5 2 4 . 4 10.7 15.5 20 .5 5 . 0 Aug. 8 , 1978 16.4 3 0 . 1 13.7 19 .9 2 4 . 3 4 . 4 J u l y 2 4 , 1980 12 .0 2 2 . 5 . 10.5 15 .1 19.7 4 . 6 J u l y 26 , 1980 13 .2 2 2 . 9 9.7 . 1 5 . 9 20 .4 4 . 5 J u l y 2 7 , 1980 14.3 . 23 .4 . 9 . 1 16.7 2 1 . 1 4 . 4 J u l y 2 8 , 1980 14.9 2 0 . 9 6 . 6 16 .0 19 .0 3 . 0 J u l y 3 0 , 1978 11.4 2 1 . 6 10.2 14.6 19 .1 4 . 5 Aug. 9 , 1980 12.9 2 4 . 1 11.2 16.2 2 0 . 6 4 . 4 Aug. 10, 1980 13.9 2 6 . 1 12.2 17.4 2 1 . 8 4 . 4 127 Tab le 5 .5b - Model A Date Observed M o d e l l e d min max range min max range Sept, . 10, 1977 9 . 3 17.6 8 . 3 - 1 . 2 2 4 . 1 2 5 . 3 Sept, . 2 5 , 1977 7 . 0 15 .2 8 . 2 - 1 . 9 19.5 2 1 . 4 J u l y 2 0 , 1978 14.1 2 6 . 5 12.4 8 . 5 5 1 . 2 4 2 . 7 J u l y 2 9 , 1978 13.2 2 4 . 7 11.5 4 . 5 3 4 . 8 3 0 . 3 J u l y 3 0 , 1978 13.2 26 .4 13.2 3 . 8 3 2 . 1 2 8 . 3 J u l y 3 1 , 1978 11.5 2 4 . 0 12.5 4 . 2 3 2 . 6 28 .4 Aug. 1 , 1978 12 .1 2 3 . 7 11.6 4 . 6 3 2 . 9 2 8 . 3 Aug. 2 , 1978 12.6 2 4 . 6 12 .0 4 . 2 ' 31 .7 2 7 . 5 Aug. 3 , 1978 1 4 . 1 2 5 . 7 11.6 5 . 2 3 1 . 9 26 .7 Aug. 4 , 1978 13 .5 2 4 . 4 10.7 4 . 6 35 .4 3 0 . 8 Aug. 8 , 1978 16.4 3 0 . 1 13.7 8 . 2 4 0 . 4 3 2 . 2 J u l y 24 , 1980 12.0 2 2 . 5 10.5 4 . 9 2 9 . 8 . 2 4 . 9 J u l y 26 , 1980 13.2 2 2 . 9 9 .7 5 .2 3 1 . 5 2 6 . 3 J u l y 2 7 , 1980 14 .3 2 3 . 4 9 . 1 6 . 3 3 2 . 1 25 .8 J u l y 2 8 , 1980 14.9 2 0 . 9 6 .6 2 .9 3 0 . 6 27.7 J u l y 3 0 , 1980 11.4 2 1 . 6 10 .2 6 .9 2 9 . 3 22 .4 Aug. 9 , 1980 12.9 2 4 . 1 11 .2 4 . 1 3 1 . 0 2 6 . 9 Aug. 10 , 1980 13.9 2 6 . 1 12.2 11.3 3 4 . 6 2 3 . 3 128 Table 5.5c - Model C Date Observed Modelled min max range min max range Sept. 10, 1977 Sept. 25, 1977 J u l y 20, 1978 J u l y 29, 1978 J u l y 30, 1978 J u l y 31, 1978 Aug. 1, 1978 Aug. 2, 1978 Aug. 3, 1978 Aug. 4, 1978 Aug. 8, 1978 J u l y 24, 1980 J u l y 26, 1980 J u l y 27, 1980 J u l y 28, 1980 J u l y 30, 1980 Aug. 9, 1980 Aug. 10, 1980 9.3 17.6 7.0 15.2 14.1 26.5 13.2 24.7 13.2 26.4 11.5 24.0 12.1 23.7 12.6 24.6 14.1 25.7 13.5 24.4 16.4 30.1 12.0 22.5 13.2 22.9 14.3 23.4 14.9 20.9 11.4 21.6 12.9 24.1 13.9 26.1 8.3 4.9 8.2 7.0 12.4 11.4 11.5 11.1 13.2 11.9 12.5 12.5 11.6 12.7 12.0 11.7 11.6 13.7 10.7 14.5 13.7 15.9 10.5 13^5 9.7 13.7 9.1 13.9 6.6 9.1 10.2 9.4 11.2 10.8 12.2 10.4 28.3 23.4 25.6 18.6 38.5 27.1 35.4 24.3 32.7 20.8 29.9 17.4 31.8 19;i 36.5 24.8 41.1 27.4 30.4 15.9 36.8 20.9 37.5 24.0 35.1 21.4 34'.9 21.0 35.5 26.4 35.3 25.9 41.4 30.6 32.9 22.5 129 Even without observed s u r f a c e temperatures i t i s probably reasonable to conclude that the temperatures modelled by M are u n r e a l i s t i c . T his con-c l u s i o n i s simply based on the f a c t t h a t the d i u r n a l ranges of modelled su r f a c e temperatures f a l l w i t h i n those of the observed a i r temperatures. Other users of Model M have found s i m i l a r s m a l l d i u r n a l temperature ranges. O u t c a l t (1972c) f i n d s a range of 5.4° C f o r an o l d r e s i d e n t i a l area, and s l i g h t l y higher ranges of 8.5° C and 9.1° C f o r a c i t y centre and new sub-d i v i s i o n , r e s p e c t i v e l y . On the other hand, O u t c a l t obtained a range of as hig h as 19.3° C f o r a c o n s t r u c t i o n s i t e . Greene (1980) r e p o r t s a range of about 5° C f o r a low d e n s i t y r e s i d e n t i a l area. The l a r g e s t range he found, about 15° C, was f o r a freeway. As would be expected, the modelled temp-e r a t u r e ranges are not as s m a l l f o r those s u r f a c e s c h a r a c t e r i z e d by a low su r f a c e moisture content and roughness l e n g t h . The s u r f a c e temperatures modelled by A and C are more r e a l i s t i c . The maximum s u r f a c e temperatures, as w e l l as t h e i r d i u r n a l ranges, are s i m i l a r to those of the o b s e r v a t i o n s c i t e d i n Chapter 3. The minimum temperatures modelled by A a l s o appear reasonable, whereas those modelled by C seem h i g h as they are u s u a l l y q u i t e s i m i l a r to the observed minima. This may, however, be due to the modelled v a l u e s not being the t r u e minima ( i . e . the f u l l d i u r n a l regime i s not a v a i l a b l e so that the minimum cannot be a s c e r t a i n e d ) . One f u r t h e r aspect of the modelled s u r f a c e temperatures should be con-s i d e r e d . Surface temperatures modelled by M peak at s o l a r noon f o r every day c o n s i d e r e d , w h i l e those modelled by A peak between 1500 and 1700 LST and those modelled by C peak at times between 1030 and 1430 LST. Based on the d i s c u s s i o n i n Chapter 3, i n which i t was s t a t e d that s u r f a c e temperatures should peak i n mid-afternoon, i t seems th a t the d i u r n a l course of the temp-130 e r a t u r e s modelled by A are most r e a l i s t i c . I t has, been shown th a t the su r f a c e temperatures modelled by M are q u i t e u n r e a l i s t i c : the val u e s cover a very s m a l l range and are almost p e r f e c t l y symmetric about s o l a r noon. The temperatures modelled by A and C appear to be more reasonable. F o r t u n a t e l y , there seems to be an exp l a n -a t i o n f o r these f i n d i n g s . Greene (1980) found that by running Model M w i t h h o u r l y , r a t h e r than mean d a i l y , input v a l u e s f o r a i r temperature, humidity, wind speed, and a i r p r e s s u r e , the d i u r n a l s u r f a c e temperature range was s i g n i f i c a n t l y i n -creased. In a d d i t i o n , the val u e s peaked i n mid-afternoon, r a t h e r than at s o l a r noon. S i m i l a r l y , C a r l s o n and Boland (1978) found that u n l e s s the uppe l e v e l temperature i n Model C was allowed to vary over the model-day, as a r e s u l t of warming from below, the d i u r n a l s u r f a c e temperature range, and l a g of the peak v a l u e beyond s o l a r noon, were underestimated. (The upper l e v e l s p e c i f i c humidity i n Model C i s a l s o v a r i e d but t h i s change was found to be so s m a l l t h a t i t had l i t t l e e f f e c t on s u r f a c e temperatures (C a r l s o n and Boland, 1978)). I n both Models A and C the upper l e v e l temperature i s allowed to vary w i t h the depth of the mixed l a y e r , whereas i n Model M t h i s temperature i s hel d constant. This would appear to be the cause of the un-r e a l i s t i c nature of the temperatures modelled by M as compared to those mod-e l l e d by A and C. The f i n d i n g s of Greene (1980) imply that a l l meteorologic c o n d i t i o n s need v a r y , whereas the present f i n d i n g s seem to i n d i c a t e t h a t i t may o n l y be necessary to vary the temperature. 5.6 Mixed Layer Heights The mixed l a y e r h e i g h t s modelled by A and C were compared to the mixed 131 l a y e r height o b s e r v a t i o n s made i n the summer of 1978. On seven of the t e s t days there were s u f f i c i e n t data f o r comparison. Because Model C does not c o n t a i n c a l c u l a t i o n s f o r the decrease i n the mixed l a y e r depth a f t e r i t s maximum v a l u e has been reached, the a n a l y s i s was not extended beyond t h a t p o i n t (normally about 1700 LST). The r e s u l t s of the a n a l y s i s are summarized i n Table 5.6 and F i g u r e s 5.47 and 5.48. The RMS e r r o r f o r Model A i s g r e a t e r than t h a t f o r Model C and, on the average, both models overestimate the mixed l a y e r h e i g h t . The s c a t t e r p l o t s show th a t the s m a l l e r v a l u e s are o f t e n underestimated by both models w h i l e the higher v a l u e s are g r o s s l y overestimated. The poor r e s u l t s might be expected because the modelled s e n s i b l e heat f l u x e s , upon which the mixed l a y e r h e i g h t s s t r o n g l y depend, were q u i t e d i f f e r e n t from those observed. The standard d e v i a t i o n s of the observed and modelled mixed l a y e r h e i g h t s show that the models g r e a t l y overestimate the observed v a r i a b i l i t y . August 3, 1978 was chosen f o r a time s e r i e s p l o t of the mixed l a y e r Table 5.6 - Summary of s t a t i s t i c s comparing modelled and observed mixed - 2 l a y e r h e i g h t s (X, a , and RMSE are i n m; r i s dimensionless) Observed Model A Model C a RMSE 2 332.6 130.8 541.7 408.6 409.6 0.31 543.1 343.3 345.4 0.45 132 F i g . 5.48 - Modelled v s . observed mixed l a y e r h e i g h t s : Model C (1978) 133 h e i g h t s ( F i g . 5.49). On-^this date the s e n s i b l e heat f l u x modelled by A was q u i t e s i m i l a r , to t h a t observed ( F i g . 5.38). The corresponding v a l u e s modelled by C were i n c l u d e d f o r comparison. This p l o t shows c l e a r l y t h a t -even on a day when Model A has p r e d i c t e d the s e n s i b l e heat f l u x q u i t e w e l l , the modelled mixed l a y e r h e i g h t s are s t i l l v e r y d i f f e r e n t from those observed. I t should a l s o be noted that the gradual i n c r e a s e i n depth over the day, as modelled by C, i s f a r more r e a l i s t i c than the sudden i n c r e a s e modelled by A. In both cases, the modelled v a l u e s peak i n l a t e a f t e r n o o n , w h i l e the observed peak i n e a r l y or mid-afternoon. The l a t e a f t ernoon peak, as w e l l as the l a r g e v a l u e s , are c h a r a c t e r i s t i c of the mixed l a y e r depth e v o l u t i o n observed over e x t e n s i v e homogeneous s u r f a c e s (eg. Carson, 1973). The ob-served mixed l a y e r h e i g h t s , w i t h t h e i r e a r l y peaks and s m a l l v a l u e s , are c h a r a c t e r i s t i c of l o c a t i o n s at which meso-scale a d v e c t i o n i s s i g n i f i c a n t (Steyn and Oke, 1982). As Vancouver i s a c o a s t a l c i t y , the mixed l a y e r depths used i n t h i s study have been a f f e c t e d by the sea breeze c i r c u l a t i o n . The f a c t t h a t Model A, as used h e r e i n (Sec. 2.2.3.1), and Model C do not account f o r a d v e c t i o n may be a major reason why they have produced mixed l a y e r depths q u i t e d i f f e r e n t to those observed. A model of mixed l a y e r depth, which i n c l u d e s a d v e c t i o n i n a more gen e r a l way than i n Model A, has been developed by Steyn and Oke (1982). This model has been t e s t e d at two c o a s t a l s i t e s , i n c l u d i n g Vancouver, and shown to give good r e s u l t s . As such, i t may prove worthwhile to i n c o r p o r a t e i t i n t o the energy balance models. I f the models can be r e v i s e d i n t h i s manner they may not o n l y produce mixed l a y e r depths t h a t are c l o s e r to those observed, but other aspects of the models' output may a l s o be improved. 134 5.49 - D i u r n a l course of the modelled and observed mixed l a y e r height Models A and C 135 5.7 Conclusions The preceding a n a l y s i s made i t p o s s i b l e to evalu a t e the performance of the three models. I t was shown t h a t , i n g e n e r a l , the modelled net r a d -i a t i o n agreed f a i r l y w e l l w i t h that observed. This was mainly due to the f a c t t h a t c l e a r sky s o l a r r a d i a t i o n can be a c c u r a t e l y c a l c u l a t e d . The mod-e l l e d t u r b u l e n t f l u x e s , on the other hand, were i n poor agreement w i t h those observed. I t was decided t h a t the inadequacy of the methods used to s i m u l a t e s u r f a c e moisture was the major cause of t h i s . This was thought to be p a r t -i c u l a r l y t r u e f o r Models A and C because the t u r b u l e n t f l u x e s modelled by M were a l s o v e r y s e n s i t i v e to the input mean d a i l y a i r temperature. I t i s l i k e l y that other f a c t o r s c o n t r i b u t e d to the inadequate m o d e l l i n g of the t u r b u l e n t f l u x e s , but none could be i s o l a t e d . The modelled subsurface heat f l u x e s were a l s o i n poor agreement w i t h those "observed". I t was d i f f i c u l t to assess the modelled s u r f a c e temperatures as no ob s e r v a t i o n s were a v a i l a b l e . However, by c o n s i d e r i n g the d i s c u s s i o n , i n Chapter 3, on the nature of urban su r f a c e temperatures, i t was p o s s i b l e to conclude that those modelled by M were u n r e a l i s t i c i n almost a l l a s p e c t s , w h i l e those modelled by A and C appeared to be much more reasonable. T h i s d i s c r e p a n c y was thought to have occurred because i n Models A and C the upper l e v e l temperature i s v a r i a b l e , whereas i n Model M the upper l e v e l temperature i s h e l d constant. F i n a l l y , the observed e v o l u t i o n of the mixed l a y e r depth was p o o r l y d u p l i c a t e d by both Models A and C. The major reason f o r t h i s seemed to be that the observed mixed l a y e r depths were i n f l u e n c e d by meso-scale a d v e c t i o n , whereas the models d i d not account f o r t h i s process. When the study was begun i t was thought t h a t i t might be p o s s i b l e to choose a "best" model from the th r e e . I t seems, however, t h a t i t i s e a s i e r 136 to choose the "worst" - model:observation comparisons were c o n s i s t e n t l y poorest f o r Model M. In a d d i t i o n , the r e s u l t s of the s e n s i t i v i t y analyses f o r t h i s model were u n r e a l i s t i c i n se v e r a l cases, whereas those f o r Models A and C were u s u a l l y more reasonable. The rather o v e r s i m p l i f i e d nature of Model M, exemplified, p a r t i c u l a r l y , by the use of a mean d a i l y a i r temp-erature as the constant upper boundary temperature, l i m i t s i t s performance. Models A and C are both improvements over Model M seemingly because each includes a mixed la y e r scheme and a more d e t a i l e d c o n s i d e r a t i o n of atmos-pheric d i f f u s i v i t y . In a d d i t i o n , the s o l a r r a d i a t i o n c a l c u l a t e d by Model C, and used i n Model A, was c l o s e r to that observed than was the s o l a r r a d i a t i o n c a l c u l a t e d using the comparatively simple routine of Model M. The f o l l o w i n g comparison of Models A and C i l l u s t r a t e s why i t was de-cided that a "best" model could hot be chosen. To begin with, the er r o r i n modelling the subsurface heat f l u x was smaller f o r Model A than i t was for C. Model A was also b e t t e r able to reproduce the "observed" d a i l y trend of these values. On the other hand, the e r r o r s i n modelling the net long-wave r a d i a t i o n and the mixed layer depths were smaller f o r Model C. Moreover, the gradual increase i n the mixed la y e r depth, as modelled by C, was more r e a l i s t i c than the rather abrupt increase as modelled by A. Both models produced seemingly reasonable surface temperatures, although Model A was, again, better able to reproduce the expected d a i l y trend. F i n a l l y , i t i s d i f f i c u l t to compare the turbulent heat fluxes produced by these two models because of the d i f f e r e n t methods used to represent surface moisture. For example, i t was decided th a t the good agreement between the turbulent fluxes modelled by A, and those observed i n 1977 and 1978, was simply a r e s u l t of the input Bowen r a t i o being representative for these two years. 137 This study makes i t p o s s i b l e to o f f e r some recommendations f o r urban energy balance m o d e l l i n g . F i r s t , the model should c o n t a i n the s o l a r r a d -i a t i o n c a l c u l a t i o n s of Model C. Second, the net long-wave r a d i a t i o n might be improved i f i t were c a l c u l a t e d u s i n g a s u r f a c e e m i s s i v i t y a p p r o p r i a t e to the s u r f a c e type and an e f f e c t i v e atmospheric e m i s s i v i t y determined u s i n g the equation of Idso (1981). In a d d i t i o n , the incoming component of the long-wave r a d i a t i o n should probably be computed as a f u n c t i o n of the a i r temperature. T h i r d , the model should a l s o c o n t a i n mixed l a y e r dynamics. I t i s suggested t h a t an approach such as t h a t of Steyn and Oke (1982) be used to help preserve g e n e r a l i t y . Most i m p o r t a n t l y , a p a r a m e t e r i z a t i o n f o r the moisture a v a i l a b i l i t y f a c t o r or the Bowen r a t i o must be developed o r , i f t h i s i s not p o s s i b l e , an a l t e r n a t i v e method f o r s i m u l a t i n g s u r f a c e moisture c o n d i t i o n s must be i n t r o d u c e d . No s p e c i f i c recommendation could be made i n t h i s regard. As the s u r f a c e moisture r e p r e s e n t a t i o n appears to be the s i n g l e most important f a c t o r i n determining the p a r t i t i o n i n g of the net r a d i a t i o n the above suggestions f o r improving energy balance m o d e l l i n g w i l l be of l i t t l e v a lue unless the moisture problem can be s o l v e d . U n f o r t u n a t e l y , no d e f i n i t e c o n c l u s i o n s could be made concerning the appropriateness of the v a l u e s used f o r the subsurface thermal p r o p e r t i e s ; the r e l a t i v e m e r i t s of the methods used i n the atmospheric d i f f u s i v i t y c a l c u l a t i o n s ; or whether the e x t r a com-p l e x i t y of Model C i s necessary. T h i s study has shown th a t the p r a c t i c a l a p p l i c a t i o n of urban energy balance models i s s e v e r e l y l i m i t e d . F i r s t , i t i s extremely d i f f i c u l t to c o n f i d e n t l y a s s i g n v a l u e s to the s u r f a c e c h a r a c t e r i s t i c s of complex urban t e r r a i n . Without previous measurements the s u r f a c e albedo could be l i t t l e more than a rough e s t i m a t e ; the roughness l e n g t h , though i t can be c a l c u l a t e d 138 using well-known formulae, w i l l contain a large margin of uncertainty; and the subsurface thermal pr o p e r t i e s can, at best, be reasonable general-i z a t i o n s . The problems with the inputs d e s c r i b i n g surface moisture are many and, as such, were considered i n d e t a i l . Second, the "surface" i t s e l f i s not c l e a r l y defined i n urban areas. This adds f u r t h e r uncertainty to the values of the input parameters and would create confusion i n applying the model output. T h i r d , the models were found to be rather poor at pre-d i c t i n g the surface heat f l u x e s . They would, therefore, be too u n r e l i a b l e for most a p p l i c a t i o n s (Sec. 1.3) but may have some l i m i t e d use. To begin with, the models might s t i l l be valuable f o r teaching purposes as the sen-s i t i v i t y analyses showed them to respond r e a l i s t i c a l l y , i n most cases, to changes i n the input parameters. Their l i m i t a t i o n s , however, should be stressed. Further, the models could be used as an a l t e r n a t i v e to measure-ment, where values of surface energy fluxes are necessary f o r a i r p o l l u t i o n purposes or f o r urban planning, but only i n s i t u a t i o n s where evaporation i s known to be i n s i g n i f i c a n t . In such cases, however, the e f f e c t of the subsurface thermal p r o p e r t i e s , on the output, w i l l probably" increase. This presents some problems of i t s own because the values f o r these inputs have been shown to be d i f f i c u l t to determine with any c e r t a i n t y . On the other hand, i n any s i t u a t i o n i n which evaporation would be s i g n i f i c a n t the use of these models i s questionable. CHAPTER SIX  SUMMARY OF CONCLUSIONS The purpose of t h i s research was to v e r i f y three urban energy balance models using observations c a r r i e d out i n a suburban area. I t was shown that the three models were s u f f i c i e n t l y d i f f e r e n t to warrant each being tested, and that the observations against which they were v e r i f i e d were r e l i a b l e . The f o l l o w i n g i s a l i s t of the conclusions which were reached. 1. Problems a r i s e i n applying these models to urban areas because of the great complexity of the surface, exemplified by the wide v a r i e t y of materials and geometric c o n f i g u r a t i o n s of which i t i s composed. Moreover, the nature of the "surface" i s not c l e a r l y defined. The above makes i t very d i f f i c u l t to assign values, with any degree of c e r t a i n t y , to those inputs which des-c r i b e the surface. This i s p a r t i c u l a r l y discouraging because these are the inputs to which the models are, i n general, most s e n s i t i v e . I t i s s t i l l not known whether the values chosen, i n t h i s study, for the roughness length and the substrate thermal p r o p e r t i e s were a c t u a l l y representative of the surface f o r which the observations were made. It was decided that the value chosen f o r the moisture a v a i l a b i l i t y f a c t o r was d e f i n i t e l y not representative. 2. A l l three models were able to p r e d i c t the net r a d i a t i o n quite w e l l but they were unable to p a r t i t i o n t h i s r a d i a t i o n between the three surface heat fluxes i n a way that corresponded with the observations. Several f a c t o r s could be responsible f o r t h i s but the inadequate representation of surface moisture i s probably the most important. No a l t e r n a t i v e approaches could be recommended because of the complexity of the f a c t o r s which a f f e c t surface moisture i n a suburban area. 139 140 3 . The modelled s u r f a c e temperatures could not be q u a n t i t a t i v e l y assessed because no o b s e r v a t i o n s were a v a i l a b l e . A v ery q u a l i t a t i v e assessment, however, showed those modelled by M to be unreasonable, w h i l e those of A and C appeared more reasonable. 4. Models A and C c o u l d not d u p l i c a t e the observed e v o l u t i o n of the mixed l a y e r h e i g h t . The major reason f o r t h i s seems to be t h a t the models do not a l l o w f o r meso-scale a d v e c t i o n : a process which s t r o n g l y a f f e c t e d the ob-served mixed l a y e r depths. 5 . Due to t h e i r v e r y nature a l l models are a s i m p l i f i c a t i o n of r e a l i t y . Model M, however, i s too simple to perform r e a l i s t i c a l l y . T his was shown i n the assessment of the model output, as w e l l as i n the s e n s i t i v i t y a n a l y s e s . Due to t h e i r c o m p l e x i t y , and probably t h e i r c l o s e r r e p r e s e n t a t i o n of r e a l i t y , Models A and C performed b e t t e r than d i d Model M. I t was not p o s s i b l e , however, to determine whether the added complexity of Model C was necessary. This would have been a v a l u a b l e c o n c l u s i o n as i t i s d e s i r a b l e that models be as simple as p o s s i b l e . 6. This study made i t p o s s i b l e to put f o r t h a l i s t of suggestions f o r urban energy balance m o d e l l i n g . I t must be s t r e s s e d that without improved methods f o r r e p r e s e n t i n g s u r f a c e m o i s t u r e , and f o r determining v a l u e s f o r the other s u r f a c e c h a r a c t e r i s t i c s , the models w i l l be of l i t t l e p r a c t i c a l use i n urban areas. 141 REFERENCES Ackerman, T.P., 1976: "A study of the i n f l u e n c e of aerosols on urban boundary l a y e r s with p a r t i c u l a r a p p l i c a t i o n s to the Los Angeles basin." Ph.D. t h e s i s , U n i v e r s i t y of Washington, 224 pp. Ackerman, T.P., 1977: "A model of the e f f e c t of aerosols on urban climates with p a r t i c u l a r a p p l i c a t i o n s to the Los Angeles basin." J. Atmos.  S c i . , 34, 531-547. A r n f i e l d , A.J., 1982: "An approach to the estimation of the surface rad-i a t i v e p r o p e r t i e s and r a d i a t i o n budgets of c i t i e s . " P h y s i c a l Geog- raphy, 3, 97-122. 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