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Effects of mulching on the surface energy balance and soil thermal regimes Hares, Mohammad Abu 1988

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E F F E C T S OF MULCHING ON THE SURFACE ENERGY BALANCE AND S O I L THERMAL REGIMES BY MOHAMMAD ABU HARES T H E S I S SUBMITTED IN P A R T I A L F U L F I L M E N T THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n FACULTY OF GRADUATE S T U D I E S D e p a r t m e n t o f S o i l S c i e n c e We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA S e p t e m b e r 1988 © Mohammad A . H a r e s , 1988 In 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 an 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 and 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 be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by 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 be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f 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 V a n c o u v e r , C a n a d a D a t e : \ l \ Q c W W . D E - 6 ( 2 / 8 8 ) i i ABSTRACT The e f f e c t s o f a s t r a w m u l c h , a p p l i e d e i t h e r u n i f o r m l y o r i n s t r i p s , on t h e s u r f a c e e n e r g y b a l a n c e and s o i l t h e r m a l r e g i m e s a r e i n v e s t i g a t e d u s i n g a n u m e r i c a l s o l u t i o n t o t h e t w o - d i m e n s i o n a l s o i l h e a t t r a n s f e r e q u a t i o n . The n u m e r i c a l t e c h n i q u e u s e d i s an e x t e n d e d v e r s i o n o f B a r a k a t a n d C l a r k ' s ( 1 9 6 6 ) a l t e r n a t i n g d i r e c t i o n e x p l i c i t m e t h o d . The m o d i f i e d t e c h n i q u e c a n s o l v e p r o b l e m s t h a t a r e p e r i o d i c i n t i m e a n d / o r s p a c e , on e i t h e r r e c t a n g u l a r d o m a i n s o r d o m a i n s w i t h a c u r v i l i n e a r u p p e r b o u n d a r y , and 1n w h i c h t h e s o i l t h e r m a l p r o p e r t i e s may v a r y w i t h p o s i t i o n , t i m e , o r t e m p e r a t u r e . The m e t h o d a l s o i n c o r p o r a t e s n o n - u n i f o r m g r i d s p a c i n g s t o r e d u c e c o m p u t a t i o n t i m e and t o i n c r e a s e e f f i c i e n c y . The s t r a w m u l c h i s r e p r e s e n t e d as a s i n g l e l a y e r o f c o n s t a n t s h o r t w a v e and l o n g w a v e t r a n s m i s s i v i t i e s and r e f l e c t i v i t i e s . V e r t i c a l t r a n s p o r t o f h e a t and w a t e r v a p o u r t h r o u g h t h e m u l c h i s e v a l u a t e d by b u l k t r a n s f e r e q u a t i o n s . S h a d i n g o f t h e b a r e s t r i p a n d r e f l e c t i o n o f r a d i a t i o n o n t o i t by t h e s i d e s o f t h e m u l c h s t r i p , a s w e l l a s t h e r e d u c t i o n o f t h e s k y - v i e w f a c t o r s e e n by t h e b a r e s t r i p , a r e i n c l u d e d . S o l a r i r r a d i a n c e , a i r t e m p e r a t u r e and w a t e r v a p o u r d e n s i t y , w i n d s p e e d , s o i l t h e r m a l p r o p e r t i e s and m u l c h p a r a m e t e r s a r e t h e r e q u i r e d i n p u t s . E v a p o r a t i o n i s e s t i m a t e d u s i n g a s u r f a c e r e s i s t a n c e m o d e l t o v a p o u r f l o w . The m o d e l i s c o m p a r e d w i t h m i c r o m e t e o r o l o g 1 c a l m e a s u r e m e n t s made i n s o i l s c o v e r e d w i t h b a r l e y s t r a w , e i t h e r i i i u n i f o r m l y a t r a t e s o f 2 , 10 o r 20 t / h a o r 1n 0 . 3 0 m w i d e m u l c h s t r i p s (10 t / h a r a t e ) a l t e r n a t i n g w i t h 0 . 1 0 m w i d e b a r e s t r i p s , w i t h t h e s t r i p s o r i e n t e d i n n o r t h - s o u t h , n o r t h e a s t - s o u t h w e s t , o r e a s t - w e s t d i r e c t i o n s . The r e s u l t s show t h a t f o r t h e u n i f o r m l y a p p l i e d m u l c h s o i l t e m p e r a t u r e s c a n be a d e q u a t e l y p r e d i c t e d b u t e v a p o r a t i o n i s u n d e r e s t i m a t e d . B o t h s o i l t e m p e r a t u r e s and e v a p o r a t i o n a r e u n d e r e s t i m a t e d i n t h e b a r e s t r i p . The d i f f e r e n c e s a r e a t t r i b u t e d t o t h e p e n e t r a t i o n o f w i n d e d d i e s i n t o t h e s t r a w m u l c h and t h e m l c r o s c a l e a d v e c t i o n f r o m t h e warm s t r a w s t r i p s o n t o t h e r e l a t i v e l y c o l d b a r e s t r i p s . i v I A B L E _ O F _ C O N j E N T S £ i £ e ABSTRACT i i T A B L E OF CONTENTS i v L I S T OF T A B L E S v i i L I S T OF F I G U R E S i x NOTATION x i i i ACKNOWLEDGEMENT x x i D E D I C A T I O N x x i i INTRODUCTION 1 CHAPTER 1 - EXTENSIONS OF THE A L T E R N A T I N G - D I R E C T I O N E X P L I C I T METHOD IN SOLVING TWO-DIMENSIONAL HEAT TRANSFER PROBLEMS 4 1.1 I n t r o d u c t i o n 5 1 .2 F o r m u l a t i o n o f t h e N u m e r i c a l M o d e l 7 1 .3 R e s u l t s 22 A . R e c t a n g u l a r D o m a i n s 23 C a s e I: V a r i a b l e G r i d S p a c i n g i n t h e V e r t i c a l D i r e c t i o n 23 C a s e I I : B o u n d a r y C o n d i t i o n o f t h e 3 r d K i n d a n d I n h o m o g e n e o u s T h e r m a l P r o p e r t i e s 25 C a s e I I I : T w o - d i m e n s i o n a l H o r i z o n t a l l y P e r i o d i c P r o b l e m s 28 C a s e I V : V a r i a b l e G r i d S p a c i n g i n t h e H o r i z o n t a l D i r e c t i o n 31 C a s e V : T e m p e r a t u r e D e p e n d e n c e o f T h e r m a l C o n d u c t i v i t y 33 B . C u r v i l i n e a r U p p e r B o u n d a r y 33 1 .4 ' C o n c l u s i o n s 37 1 .5 R e f e r e n c e s 38 CHAPTER 2 - OBSERVED AND PREDICTED SURFACE ENERGY BALANCES AND S O I L TEMPERATURES FOR UNIFORM AND P A R T I A L L Y MULCHED S U R F A C E S 39 2 . 1 I n t r o d u c t i o n 40 2 . 2 T h e o r y 44 A s s u m p t i o n s 44 S u r f a c e B o u n d a r y C o n d i t i o n f o r M u l c h e d S u r f a c e s 45 2 . 3 T e s t i n g and C a l i b r a t i o n o f t h e M o d e l 55 A . U n i f o r m M u l c h E x p e r i m e n t s 55 B . M u l c h and B a r e S t r i p E x p e r i m e n t s 59 V P a ^ e C . S p e c i f i c a t i o n o f t h e I n p u t P a r a m e t e r s 61 S o i l P r o p e r t i e s 61 M u l c h P r o p e r t i e s 63 R e s i s t a n c e t o H e a t and V a p o u r T r a n s f e r 66 S h o r t w a v e and L o n g w a v e R a d i a t i o n 69 2 . 4 R e s u l t s and D i s c u s s i o n 75 A . U n i f o r m l y A p p l i e d M u l c h 75 B . E f f e c t s o f S t r i p s 91 2 . 5 S e n s i t i v i t y A n a l y s i s 103 2 . 6 C o n c l u s i o n s 109 2 . 7 R e f e r e n c e s 110 CONCLUDING REMARKS 114 A P P E N D I X I A n a l y t i c a l s o l u t i o n t o t h e o n e - d i m e n s i o n a l h e a t t r a n s f e r p r o b l e m w i t h u p p e r b o u n d a r y c o n d i t i o n o f t h e 3 r d k i n d and i n h o m o g e n e o u s t h e r m a l p r o p e r t i e s 116 A P P E N D I X I I A n a l y t i c a l s o l u t i o n t o t h e t w o - d i m e n s i o n a l h o r i z o n t a l l y p e r i o d i c h e a t t r a n s f e r p r o b l e m w i t h u p p e r b o u n d a r y c o n d i t i o n o f t h e 3 r d k i n d . 120 A P P E N D I X I I I I n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on May 2 7 , 1984 123 A P P E N D I X IV C I i m a t o l o g i c a l i n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on May 2 7 , 1 9 8 4 . 124 A P P E N D I X V S o i l t e m p e r a t u r e i n i t i a l i z a t i o n p r o f i l e f o r May 2 7 , 1 9 8 4 . 125 A P P E N D I X VI I n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n 126 s t u d y on J u n e 1 4 , 1 9 8 4 . A P P E N D I X V I I C I i m a t o l o g i c a l i n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on J u n e 1 4 , 1 9 8 4 . 127 A P P E N D I X V I I I S o i l t e m p e r a t u r e i n i t i a l i z a t i o n p r o f i l e f o r J u n e 1 4 , 1 9 8 4 . 128 A P P E N D I X IX I n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on J u l y 6 , 1 9 8 4 . 129 A P P E N D I X X C I i m a t o l o g i c a l i n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on J u l y 6 , 1 9 8 4 . 130 A P P E N D I X XI S o i l t e m p e r a t u r e i n i t i a l i z a t i o n p r o f i l e f o r J u l y 6 , 1 9 8 4 . 131 A P P E N D I X X I I I n p u t " p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on S e p t e m b e r 3 , 1 9 8 4 . A P P E N D I X X I I I C I i m a t o l o g i c a l i n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on S e p t e m b e r 3 , 1 9 8 4 . A P P E N D I X X I V S o i l t e m p e r a t u r e i n i t i a l i z a t i o n p r o f i l e f o r S e p t e m b e r 3 , 1 9 8 4 . A P P E N D I X XV I n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on A p r i l 7 , 1 9 8 5 . A P P E N D I X XVI C I i m a t o l o g i c a l i n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on A p r i l 7 , 1 9 8 5 . A P P E N D I X X V I I S o i l t e m p e r a t u r e i n i t i a l i z a t i o n p r o f i l e f o r A p r i l 7 , 1 9 8 5 . A P P E N D I X X V I I I I n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on A p r i l 8 , 1 9 8 5 . A P P E N D I X X I X C I i m a t o l o g i c a l i n p u t p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s t u d y on A p r i l 8 , 1 9 8 5 . A P P E N D I X XX S o i l t e m p e r a t u r e i n i t i a l i z a t i o n p r o f i l e f o r A p r i l 8 , 1 9 8 5 . L I S T OF T A B L E S v i 1 C h a p _ t e r _ l T a b l e 1.1 T a b l e 1 .2 T a b l e 1 . 3 T a b l e 1 .4 D i f f e r e n c e s b e t w e e n t h e e x a c t and t h e n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t B 12 h f o r C a s e I a t t h e i n d i c a t e d d e p t h s . D i f f e r e n c e s b e t w e e n t h e e x a c t a n d t h e n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t • 12 h f o r C a s e I I a t t h e i n d i c a t e d d e p t h s . D i f f e r e n c e s b e t w e e n t h e e x a c t a n d t h e n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t * 12 h f o r C a s e I I I a t t h e i n d i c a t e d d e p t h s and h o r i z o n t a l p o s i t i o n s . D i f f e r e n c e s b e t w e e n t h e e x a c t and t h e n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t = 12 h and i n t h e maximum e r r o r s i n t e m p e r a t u r e b e t w e e n t h e n u m e r i c a l p r o c e d u r e s w i t h Ax c o n s t a n t and Ax v a r i a b l e f o r C a s e IV a t t h e i n d i c a t e d d e p t h s and h o r i z o n t a l p o s i t i o n s . 24 27 30 32 C h a | ) t e r _ 2 T a b l e 2 . 1 T a b l e 2 . 2 T a b l e 2 . 3 T h i c k n e s s o f t h e m u l c h ( i n cm) w i t h t i m e a t v a r i o u s a p p l i c a t i o n r a t e s . 57 T r a n s m i s s i v i t i e s o f f r e s h ( u n u s e d ) and f i e l d u t i l i z e d ( f o r one s e a s o n ) b a r l e y s t r a w a t d i f f e r e n t r a t e s o f a p p l i c a t i o n . 64 S h a d e l e n g t h (1n cm) f r o m a 4 cm l o n g ( e q u a l t o t h e m u l c h t h i c k n e s s a t t h e m i d d l e o f t h e m u l c h s t r i p ) p i n a n d f r o m t h e a c t u a l m u l c h t h i c k n e s s a t t h e edge o f t h e m u l c h s t r i p m e a s u r e d p e r p e n d i c u l a r t o t h e row and a c r o s s t h e b a r e s t r i p a t d i f f e r e n t o r i e n t a t i o n s on A p r i l 8 , 1 9 8 5 . 74 v i i i P a ^ e T a b l e 2 . 4 M e a s u r e d a n d - p r e d i c t e d e v a p o r a t i o n r a t e s (mm h ) a t v a r i o u s m u l c h a p p l i c a t i o n r a t e s on J u n e 14 a n d J u l y 6 , 1 9 8 4 . 80 T a b l e 2 . 5 M e a s u r e d and p r e d i c t e d s o i l t e m p e r a t u r e and t e m p e r a t u r e a m p l i t u d e ( ° C ) a t 0 . 0 0 5 m d e p t h a t v a r i o u s m u l c h a p p l i c a t i o n r a t e s on J u n e 14 and J u l y 6 , 1 9 8 4 . 86 T a b l e 2 . 6 M e a s u r e d a n d . p r e d i c t e d e v a p o r a t i o n r a t e s (mm h ) i n t h e b a r e s t r i p a t v a r i o u s d i r e c t i o n s on A p r i l 7 and 8 , 1 9 8 5 . 93 T a b l e 2 . 7 M e a s u r e d a n d - p r e d i c t e d e v a p o r a t i o n r a t e s (mm h ) i n t h e m u l c h s t r i p a t v a r i o u s d i r e c t i o n s on A p r i l 7 and 8 , 1 9 8 5 . 95 T a b l e 2 . 8 M e a s u r e d and p r e d i c t e d s o i l t e m p e r a t u r e and t e m p e r a t u r e a m p l i t u d e ( ° C ) a t 0 . 0 0 5 m d e p t h i n t h e b a r e s t r i p a t v a r i o u s d i r e c t i o n s on S e p t e m b e r 3 , 1984 and A p r i l 7 and 8 , 1 9 8 5 . 99 T a b l e . 2 . 9 E f f e c t o f c h a n g e s i n p a r a m e t e r s on t h e e v a p o r a t i v e f l u x e s and s o i l t e m p e r a t u r e s . 104 L I S T OF F I G U R E S P a ^ e C h a j ) t e r _ l F i g u r e 1.1 P a r t o f t h e f i n i t e d i f f e r e n c e g r i d , w i t h v a r i a b l e s p a c i n g i n b o t h d i r e c t i o n s , u s e d on t h e r e c t a n g u l a r d o m a i n . T r a n s i t i o n l i n e s a r e 1n b o l d . 10 F i g u r e 1 .2 The f i n i t e d i f f e r e n c e g r i d u s e d w i t h t h e s i n u s o i d a l u p p e r b o u n d a r y s p e c i f i e d by e q u a t i o n ( 1 4 ) . 16 F i g u r e 1 .3 The v a r i o u s n o d a l s i t u a t i o n s o c c u r r i n g a t t h e b e g i n n i n g o f rows i n t e r s e c t i n g t h e s i n u s o i d a l u p p e r b o u n d a r y s p e c i f i e d by e q u a t i o n ( 1 4 ) . The s i t u a t i o n s ( i ) t o ( 1 v ) d e s c r i b e d i n t h e t e x t c o r r e s p o n d t o F i g s . 1 . 3 a t o 1 . 3 d , r e s p e c t i v e l y . 18 F i g u r e 1.4 N u m e r i c a l l y c a l c u l a t e d s u r f a c e t e m p e r a t u r e s a t t h e p e a k , a t t h e m i d p o i n t o f t h e r i d g e on t h e l e f t s i d e , i n t h e f u r r o w on t h e l e f t s i d e , and f r o m C a s e 1 1 ( 1 ) f o r a 24 h p e r i o d . 36 C h a £ t e r _ 2 F i g u r e 2 . 1 F i g u r e 2 . 2 A p a r t i a l v i e w o f an i d e a l i z e d c o n f i g u r a t i o n o f 0 . 3 0 m w i d e m u l c h s t r i p a l t e r n a t i n g w i t h 0 . 1 0 m b a r e s o i l s t r i p ; ( a ) t h e b a r e s t r i p i s s h a d e d , and (b ) t h e b a r e s t r i p i s p a r t l y s u n l i t . The s o l i d s q u a r e s i n d i c a t e t h e t h e r m o c o u p l e s p o s i t i o n s i n t h e z - x ( s o i l ) doma i n . T y p i c a l m e a s u r e d a l b e d o s f o r t h e b a r e and u n i f o r m l y m u l c h e d p l o t s . The m e a s u r e m e n t s w e r e made on J u n e 1 0 , May 2 4 , J u n e 1 and J u n e 1 4 , 1984 f o r t h e 0 , 2 , 1 0 , and 20 t / h a r a t e s , r e s p e c t i v e l y . 43 62 Pa£e F i g u r e 2 . 3 The m e a s u r e d ( i s o l a t e d s y m b o l s ) R i n t h e b a r e and 20 t / h a , t h e c a l c u l a t e d ( c u r v e s ) R n i n t h e b a r e , 2 , 10 and 20 t / h a p l o t s and t h e m e a s u r e d ( s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) LE 1n t h e b a r e , 2 , 10 a n d 20 t / h a p l o t s on J u n e 1 4 , 1 9 8 4 . 76 F i g u r e 2 . 4 a S o i l - w a t e r r e t e n t i o n c u r v e s f o r s o i l s s a m p l e d a t 0 - 0 . 0 5 , 0 . 0 5 - 0 . 1 0 and 0 . 1 0 - 0 . 2 0 m d e p t h s i n t h e b a r e and 2 t / h a p l o t s . 78 F i g u r e 2 . 4 b S o i l - w a t e r r e t e n t i o n c u r v e s f o r s o i l s s a m p l e d a t 0 - 0 . 0 5 , 0 . 0 5 - 0 . 1 0 and 0 . 1 0 - 0 . 2 0 m d e p t h s i n t h e 10 and 20 t / h a p l o t s . 79 F i g u r e 2 . 5 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s i n t h e b a r e p l o t a t d e p t h s o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 . 2 5 and 0 . 5 0 m on J u n e 1 4 , 1 9 8 4 . S o i l s u r f a c e t e m p e r a t u r e s w e r e m e a s u r e d by t h e b o l o m e t e r . 82 F i g u r e 2 . 6 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s i n t h e 10 t / h a p l o t a t d e p t h s o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 . 2 5 and 0 . 5 0 m on J u n e 1 4 , 1 9 8 4 . S o i l s u r f a c e t e m p e r a t u r e s w e r e m e a s u r e d by t h e b o l o m e t e r . A l s o shown a r e t h e m u l c h s u r f a c e t e m p e r a t u r e s m e a s u r e d by t h e b o l o m e t e r . 83 F i g u r e 2 . 7 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s i n t h e 20 t / h a p l o t a t d e p t h s o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 . 2 5 a n d 0 . 5 0 m on J u n e 1 4 , 1 9 8 4 . S o i l s u r f a c e t e m p e r a t u r e s w e r e m e a s u r e d by t h e b o l o m e t e r . A l s o shown a r e t h e m u l c h s u r f a c e t e m p e r a t u r e s m e a s u r e d by t h e b o l o m e t e r . 84 F i g u r e 2 . 8 The c a l c u l a t e d s o i l t e m p e r a t u r e s i n t h e 2 t / h a p l o t a t d e p t h s o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 . 2 5 and 0 . 5 0 m on J u n e 1 4 , 1 9 8 4 . A l s o i n c l u d e d a r e t h e s o i l a n d m u l c h s u r f a c e t e m p e r a t u r e s m e a s u r e d by t h e b o l o m e t e r . F i g u r e 2 . 9 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s a t d e p t h s o f 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 . 2 5 and 0 . 5 0 m i n t h e 2 t / h a p l o t on May 2 7 , 1 9 8 4 . A l s o i n c l u d e d a r e t h e c a l c u l a t e d s o i l s u r f a c e tempe r a t u r e s . F i g u r e 2 . 1 0 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) t e m p e r a t u r e s a t d e p t h s o f 0 . 0 , 0 . 0 0 8 , 0 . 0 1 6 , 0 . 0 2 4 , 0 . 0 3 6 , 0 . 0 4 8 and 0 . 0 6 m i n t h e m u l c h l a y e r f o r t h e a p p l i c a t i o n r a t e o f 20 t / h a on J u n e 1 4 , 1 9 8 4 . M u l c h s u r f a c e t e m p e r a t u r e s w e r e m e a s u r e d by t h e b o l o m e t e r . F i g u r e 2 . 1 1 The m e a s u r e d ( i s o l a t e d s y m b o l s ) . R i n t h e b a r e and t h e N E - S W , and t h e c a l c u l a t e d R n ( c u r v e s ) i n t h e b a r e , N - S , NE-SW and E-W s t r i p p l o t s and t h e m e a s u r e d ( s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) LE i n t h e b a r e , N - S , NE-SW and E-W s t r i p p l o t s on A p r i l 8 , 1 9 8 5 . F i g u r e 2 . 1 2 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s a t 0 . 0 0 5 m d e p t h i n t h e b a r e , and i n t h e b a r e s t r i p o f t h e N - S , NE-SW and E-W s t r i p p l o t s on A p r i l 8 , 1 9 8 5 . F i g u r e 2 . 1 3 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l s u r f a c e t e m p e r a t u r e s i n t h e b a r e , and i n t h e b a r e s t r i p o f t h e N - S , NE-SW and E-W s t r i p p l o t s on A p r i l 8 , 1 9 8 5 . The s u r f a c e t e m p e r a t u r e s w e r e m e a s u r e d by t h e b o l o m e t e r . x i i Pa£e F i g u r e 2 . 1 4 The m e a s u r e d ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s a t 0 . 0 0 5 m d e p t h 1n t h e b a r e , and i n t h e b a r e s t r i p o f t h e N - S , NE-SW and E-W s t r i p p l o t s on S e p t e m b e r 3 , 1 9 8 4 . 102 NOTATION Chap_te r _ l a f ( x , t ) f N H c i . j F r a c t i o n o f Az a s s o c i a t e d w i t h t h e v e r t i c a l i n t e r s e c t i o n i n t h e f i n i t e d i f f e r e n c e n o d a l n e t w o r k F r a c t i o n o f Ax a s s o c i a t e d w i t h t h e h o r i z o n t a l i n t e r s e c t i o n i n t h e f i n i t e d i f f e r e n c e n o d a l n e t w o r k V o l u m e t r i c h e a t c a p a c i t y o f t h e med i urn L e n g t h b e t w e e n t h e p o i n t o f n o r m a l a t t h e c u r v e d b o u n d a r y and ( i , j ) node L e n g t h b e t w e e n t h e p o i n t o f n o r m a l a t t h e c u r v e d b o u n d a r y and ( 1 - 1 , j ) node L e n g t h b e t w e e n ( 1 - 1 , j - 1 ) and ( i - l , j ) n o d e s on t h e c u r v e d b o u n d a r y L e n g t h b e t w e e n ( 1 - 1 , j - 1 ) and ( i - 2 , j ) n o d e s on t h e c u r v e d b o u n d a r y A r b i t r a r y f u n c t i o n o f t i m e i n t h e h o r i z o n t a l d i r e c t i o n A r b i t r a r y f u n c t i o n o f t i m e a t t h e p o i n t o f n o r m a l on t h e c u r v e d b o u n d a r y S o i l h e a t f l u x d e n s i t y n o r m a l t o t h e s u r f a c e S o i l h e a t f l u x d e n s i t y i n t h e v e r t i c a l d i r e c t i o n L a t t i c e p a r a m e t e r s u s e d i n t h e t w o - d i m e n s i o n a l g r i d d i m e n s i o n ! e s s d i m e n s i o n l e s s , - 3 o r - l J m C m m m m W m -2 W m W m W m -2 -2 -2 Maximum d e p t h o f t h e c u r v e d s u r f a c e m T h e r m a l c o n d u c t i v i t y o f t h e med ium W m d i m e n s i o n l e s s -1 V 1 k N A v e r a g e o f t h e k ' s b e t w e e n N on t h e c u r v e d b o u n d a r y and mode m i s t h e number o f f i n i t e d i f f e r e n c e n o d e s 1n t h e h o r i z o n t a l d i r e c t i on N . N 1 . N 2 P o i n t o f n o r m a l on t h e c u r v e d b o u n d a r y n L a t t i c e p a r a m e t e r a s s o c i a t e d w i t h t i m e P P e r i o d o f i n t e r e s t T T e m p e r a t u r e t T i m e U T e m p e r a t u r e U N T e m p e r a t u r e a t t h e p o i n t o f n o r m a l on t h e c u r v e d b o u n d a r y V T e m p e r a t u r e X Maximum h o r i z o n t a l d i s t a n c e x H o r i z o n t a l d i s t a n c e Z Maximum v e r t i c a l d i s t a n c e z V e r t i c a l d i s t a n c e Z D e p t h o f t h e c u r v e d s u r f a c e a t a n y x b e l o w t h e h o r i z o n t a l p l a n e a A c o n s t a n t (= 0 o r 1) a d T h e r m a l d i f f u s i v i t y o f t h e med ium 8 H e a t t r a n s f e r c o e f f i c i e n t B N H e a t t r a n s f e r c o e f f i c i e n t a t t h e p o i n t o f n o r m a l on t h e c u r v e d b o u n d a r y Y L a t e r a l f r e q u e n c y (= 2TT/X] A C h a n g e i n A t T i m e i n c r e m e n t x i v W m C d i m e n s i o n l e s s d i m e n s i o n l e s s d i m e n s i o n l e s s d o r h °C d , h o r s °C °C °C m m m m m d i m e n s i o n l e s s m 2 -1 m s . W m C I I m ~ 2 o ~ - l W m C -1 m d i m e n s i o n l e s s h o r s XV Ax Az A n IT W S p a c e i n c r e m e n t i n t h e x - d i r e c t 1 o n m S p a c e i n c r e m e n t i n t h e z - d i r e c t 1 o n m R e a s o n a b l e n e s s ( s t a b i l i t y ) c r i t e r i o n d i m e n s i o n l e s s -1 L a t e r a l f r e q u e n c y ( = 2 i r / X ) on t h e c u r v e d b o u n d a r y Sum o f t h e p r o d u c t s A c o n s t a n t A c o n s t a n t ( = X / 4 ) m d i m e n s i o n l e s s d i m e n s i o n l e s s m A n g u l a r f r e q u e n c y f o r a d a i l y c y c l e h " 1 o r s ~ * = 2TT/P Chap_te r_2 m w V o l u m e t r i c h e a t c a p a c i t y o f t h e a t m o s p h e r e J m " 3 V 1 V o l u m e t r i c h e a t c a p a c i t y o f t h e s o i l J m 3 °C 1 The h e i g h t a t w h i c h a i r t e m p e r a t u r e i s g i ven M o l e c u l a r d i f f u s i v i t y f o r t h e s e n s i b l e h e a t M o l e c u l a r d i f f u s i v i t y o f w a t e r v a p o u r E n h a n c e m e n t f a c t o r F r a c t i o n o f t h e m u l c h a t t h e e f f e c t i v e m u l c h t e m p e r a t u r e t h a t d i r e c t l y e x c h a n g e s l o n g w a v e e n e r g y w i t h t h e s o i l s u r f a c e b e l o w T o r t u o s i t y f a c t o r V i e w f a c t o r o f one m u l c h w a l l a s s e e n f r o m t h e o p p o s i t e w a l l m m 2 -1 m s m 2 -1 m s d i m e n s i o n l e s s d i m e n s i o n l e s s d i m e n s i o n l e s s d i m e n s i o n l e s s m V i e w f a c t o r o f t h e s u n l i t p a r t o f t h e v e r t i c a l m u l c h w a l l as s e e n f r o m a p o i n t on t h e s u r f a c e o f t h e b a r e s t r i p d i m e n s i o n l e s s V i e w f a c t o r o f t h e v e r t i c a l m u l c h w a l l t o t h e l e f t o r r i g h t a s s e e n f r o m a p o i n t on t h e s u r f a c e o f t h e b a r e s t r i p d i m e n s i o n l e s s V i e w f a c t o r o f t h e s h a d e d p a r t o f t h e v e r t i c a l m u l c h w a l l S o i l s u r f a c e h e a t f l u x d e n s i t y Sun ' s h o u r a n g l e S e n s i b l e h e a t f l u x d e n s i t y a t t h e s o i l s u r f a c e T h i c k n e s s o f t h e m u l c h l a y e r d i m e n s i o n l e s s W m " 2 d e g r e e s W m " 2 m T h i c k n e s s o f t h e s h a d e d p a r t o f m t h e v e r t i c a l m u l c h w a l l S e n s i b l e h e a t f l u x d e n s i t y a t t h e W m t o p o f t h e m u l c h s u r f a c e von Ka rman c o n s t a n t (=0.4) S o i l t h e r m a l c o n d u c t i v i t y L a t e n t h e a t f l u x d e n s i t y L a t e n t h e a t f l u x d e n s i t y f r o m t h e m u l c h L a t e n t h e a t f l u x d e n s i t y f r o m t h e s o i l T o t a l l a t e n t h e a t f l u x d e n s i t y ( = L E + L E J x m o ' -2 d i m e n s i o n l e s s W m " 1 V 1 L a t e n t h e a t o f v a p o r i z a t i o n o f J Kg w a t e r -1 W m W m -2 W m W m -2 -2 I n c o m i n g l o n g w a v e r a d i a t i o n W m" f l u x d e n s i t y a t a n y p o i n t i n t h e b a r e s t r i p I n c o m i n g l o n g w a v e r a d i a t i o n W m f l u x d e n s i t y a t t h e s o i l s u r f a c e -2 O u t g o i n g l o n g w a v e r a d i a t i o n f l u x d e n s i t y f r o m t h e s o i l s u r f a c e W m x v i i Incoming longwave r a d i a t i o n f l u x d e n s i t y a t the mulch s u r f a c e O u t g o i n g longwave r a d i a t i o n f l u x d e n s i t y from the mulch s u r f a c e P o r o s i t y o f the mulch Net r a d i a t i o n f l u x d e n s i t y a t the mulch s u r f a c e W m -2 -2 W m d i m e n s i o n l e s s W m~ 2 n .h m .v m s s .b 'd .b ) _ m Net r a d i a t i o n f l u x d e n s i t y a t the W m s o i l s u r f a c e A e r o d y n a m i c r e s i s t a n c e t o heat s m" t r a n s f e r A e r o d y n a m i c r e s i s t a n c e to vapour s m" t r a n s f e r R e s i s t a n c e of the mulch to heat s m" t r a n s f e r R e s i s t a n c e of the mulch to vapour s m" t r a n s f e r R e s i s t a n c e of the mulch to s m" e v a p o r a t i on S o i l s u r f a c e r e s i s t a n c e to s m" e v a p o r a t i on Beam s o l a r r a d i a t i o n f l u x d e n s i t y W m i n c i d e n t on any p o i n t 1n the bare s t r i p Incoming d i f f u s e s o l a r r a d i a t i o n W m f l u x d e n s i t y a t any p o i n t 1n the bare s t r i p Incoming d i f f u s e s o l a r r a d i a t i o n W m f l u x d e n s i t y Beam s o l a r r a d i a t i o n f l u x d e n s i t y W m i n c i d e n t on the s u n l i t v e r t i c a l mulch wal1 -2 -2 -2 -2 -2 x v i i i S" Incoming shor twave r a d i a t i o n f l u x W m d e n s i t y a t the s o i l s u r f a c e O u t g o i n g s h o r t w a v e r a d i a t i o n f l u x W m " 2 d e n s i t y from the s o i l s u r f a c e S L e n g t h of the shadow of the mulch m p w a l l i n a d i r e c t i o n p e r p e n d i c u l a r t o the bare s t r i p S? Incoming shor twave r a d i a t i o n f l u x W m d e n s i t y a t the mulch s u r f a c e u -2 S£ O u t g o i n g shor twave r a d i a t i o n f l u x W m d e n s i t y from the mulch s u r f a c e "5 S l o p e o f the s a t u r a t e d vapour Kg m ~ 3 ° C _ 1 d e n s i t y c u r v e a t T T Tempera ture °C t Time d , h , o r s T Average t e m p e r a t u r e °C T A i r t e m p e r a t u r e °C T K Average t e m p e r a t u r e K T K a A i r t e m p e r a t u r e K T K m M u l c h t e m p e r a t u r e K T K s S o i l t e m p e r a t u r e K T m M u l c h t e m p e r a t u r e °C T Q S o i l s u r f a c e t e m p e r a t u r e °C T„ m a „ Maximum t e m p e r a t u r e a t the s o i l °C 0 m a x s u r f a c e T s S o i l t e m p e r a t u r e °C x H o r i z o n t a l d i s t a n c e m XIX x ' x " u z_ a m m .L 'm C 6. vm H o r i z o n t a l d i s t a n c e from a p o i n t on the bare s t r i p to e i t h e r the l e f t o r r i g h t mulch w a l l W i d t h o f the bare s t r i p Windspeed S u r f a c e roughness l e n g t h H e i g h t a t w h i c h windspeed i s g i v e n S o l a r a l t i t u d e R e f l e c t i v i t y of the mulch to longwave r a d i a t i o n R e f l e c t i v i t y o f the mulch to s h o r t w a v e r a d i a t i o n R e f l e c t i v i t y o f the s o i l to longwave r a d i a t i o n R e f l e c t i v i t y o f the s o i l to s h o r t w a v e r a d i a t i o n S o l a r d e c l i n a t i o n a n g l e A b s o r p t i v i t y o f the mulch A b s o r p t i v i t y o f the s o i l A z i m u t h a n g l e of the s t r i p S o l a r z e n i t h a n g l e V o l u m e t r i c w a t e r c o n t e n t i n the mulch l a y e r E f f e c t i v e a t m o s p h e r i c e m i s s i v i t y A c o n s t a n t Water vapour d e n s i t y i n the a i r Water vapour d e n s i t y i n the mulch 1 a y e r S a t u r a t e d w a t e r vapour d e n s i t y m m -1 m s m m degree s d i m e n s i o n l e s s d i m e n s i o n l e s s d i m e n s i o n l e s s d i m e n s i o n l e s s deg rees d i m e n s i o n l e s s d i m e n s i o n l e s s deg rees degree s 3 , 3 m /m d i m e n s i o n l e s s d i m e n s i o n l e s s - 3 kg m kg m - 3 kg m - 3 XX * — _ p y (T) S a t u r a t e d w a t e r vapour d e n s i t y a t T kg m S t e f a n - B o l t z m a n n C o n s t a n t (=5.67 x 1 0 " 8 ) - 3 W m " 2 K " 4 m • T r a n s m i s s i v i t y of the mulch to longwave r a d i a t i o n T r a n s m i s s i v i t y o f the mulch to s h o r t w a v e r a d i a t i o n L a t i tude S o l a r a z i m u t h a n g l e d i m e n s i o n l e s s d i m e n s i o n l e s s degree s degree s ACKNOWLEDGEMENTS The funds f o r the r e s e a r c h p r o j e c t were p r o v i d e d by g r a n t s f rom the M i n i s t r y of A g r i c u l t u r e and F i s h e r i e s , and the N a t u r a l S c i e n c e 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 . My s t i p e n d was p r o v i d e d by c o n t r a c t s from the M i n i s t r y o f A g r i c u l t u r e and F i s h e r i e s , the N a t u r a l S c i e n c e 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 , and a U . B . C . Gradua te Summer F e l l o w s h i p . The b u r s a r y r e c e i v e d from the U . B . C . awards o f f i c e t o d e f r a y the house r e n t s i s a p p r e c i a t e d . I w i s h t o e x p r e s s my deepes t sense of g r a t i t u d e to D r . M . D . Novak , who s e r v e d as R e s e a r c h S u p e r v i s o r t h r o u g h o u t the p e r i o d o f the r e s e a r c h work and 1n the p r e p a r a t i o n of the m a n u s c r i p t . A p p r e c i a t i o n i s a l s o due to o t h e r members o f the c o m m i t t e e : D r . T . A . B l a c k , D r . J . de V r i e s , D r . L . M . S t a l e y and D r . J . C . Keng f o r t h e i r c o n t r i b u t i o n s . D r . M . D . Novak p r o v i d e d a s s i s t a n c e i n d e r i v i n g the a n a l y t i c a l s o l u t i o n s appended as I and I I . I w i s h to thank the f a c u l t y and s t a f f o f the Department o f S o i l S c i e n c e and the B i o m e t e o r o l o g y group f o r t h e i r h e l p d u r i n g my y e a r s o f s t u d y a t U . B . C . In p a r t i c u l a r , D r . L . M . L a v k u l i c h , Head o f the Department and D r . M . D . Novak f o r s h a r i n g i n p e r s o n a l a r e a s . H e l p f rom Tom N i c o l and R . D . P a r a c h o n i a k , N u m e r i c a l A n a l y s i s , U . B . C . Comput ing C e n t r e i n g e t t i n g the computer programs f o r C h a p t e r 2 to work i n the a t t a c h e d p r o c e s s o r i s a p p r e c i a t e d . Thanks t o A l N e i g h b o u r , F i e l d Manager , U . B . C . P l a n t S c i e n c e R e s e a r c h S t a t i o n and h i s s t a f f f o r p r e p a r i n g the e x p e r i m e n t a l s i t e s e v e r a l t i m e s . S p e c i a l t h a n k s to J eeva Jonahs f o r her e x c e l l e n t word p r o c e s s i n g o f the m a n u s c r i p t . F i n a l l y , I e x p r e s s my g r a t i t u d e to my w i f e Ferdous A r a Hares and c h i l d r e n Shoaev and Sharmin f o r t h e i r p a t i e n c e and encouragement to c o m p l e t e t h i s w o r k . My w i f e ' s s a c r i f i c e to t a k e out t ime from her c a r e e r i s h i g h l y commendable . x x i i DEDICAT ION To my f a t h e r who has p a s s e d away i n t h e m i d d l e o f t h i s s t u d y and t o my a i l i n g m o t h e r who has l o n g been a w a i t i n g t h o u s a n d s o f m i l e s away i n a r e m o t e v i l l a g e o f B a n g l a d e s h t o s e e h e r s o n . 1 INTRODUCTION C o n s e r v a t i o n t i l l a g e p r a c t i c e s , w h i c h a r e on the s t e a d y i n c r e a s e i n N o r t h A m e r i c a , c o n t r o l s o i l e r o s i o n and d e g r a d a t i o n . A l t h o u g h c rop y i e l d s have o f t e n been found to be comparab le t o , or b e t t e r t h a n , t h o s e o b t a i n e d w i t h c o n v e n t i o n a l t i l l a g e , few f a r m e r s have a c c e p t e d 1t In Canada (Gauer e t a l , 1 9 8 2 ) . C o n s e r v a t i o n t i l l a g e sys tems u s u a l l y l e a v e a l l o r p a r t o f the r e s i d u e s from the p r e v i o u s c rop on the s o i l s u r f a c e . Crop r e s i d u e s l e f t a f t e r a t i l l a g e o p e r a t i o n a f f e c t the w a t e r and heat b a l a n c e s of a s o i l o v e r the g r o w i n g s e a s o n . The d e l a y i n the s p r i n g t i m e s o i l warming i s a ma jor c o n c e r n 1n the e x p a n s i o n and a d o p t i o n o f c o n s e r v a t i o n t i l l a g e . Lower s o i l t e m p e r a t u r e s e a r l y i n the season can d e l a y seed g e r m i n a t i o n , r e t a r d p l a n t g rowth and t h e r e f o r e s h o r t e n the g r o w i n g p e r i o d and reduce the u l t i m a t e c rop g rowth a n d / o r y i e l d (Tanner et a l , 1 9 8 7 ) . S o i l t h e r m a l and m o i s t u r e reg imes may be m o d i f i e d by m u l c h i n g and s u r f a c e r i d g i n g . However , l i t t l e r e s e a r c h has been c o n d u c t e d on r e s i d u e management and s u r f a c e c o n f i g u r a t i o n s d e s p i t e the f a c t t h a t one o f the ma jor o b s t a c l e s to c r o p p r o d u c t i o n i n c o n s e r v a t i o n t i l l a g e f a r m i n g i s r educed p l a n t g rowth e a r l y i n the s e a s o n . U n t i l r e c e n t l y , the m a j o r i t y of r e s e a r c h on the p r a c t i c e s o f m u l c h i n g and s u r f a c e r i d g i n g c o n s i s t e d m a i n l y o f o b s e r v a t i o n s o f s o i l t e m p e r a t u r e s and m o i s t u r e c o n t e n t s f o r v a r i o u s t r e a t m e n t s and l o c a t i o n s . Most of t h i s work has been c o n d u c t e d o u t s i d e the 2 C a n a d i a n p r a i r i e s . R e l a t i v e l y few c o m p r e h e n s i v e models w h i c h p r e d i c t the e f f e c t s o f the se p r a c t i c e s have been d e v e l o p e d . T h i s prompted the a u t h o r to d e v e l o p , as p a r t o f t h i s t h e s i s , a p h y s i c a l l y - b a s e d model to p r e d i c t the e f f e c t s o f m u l c h i n g , e i t h e r u n i f o r m l y or i n s t r i p s , on the s u r f a c e energy b a l a n c e and s o i l t h e r m a l r e g i m e s . A s i m i l a r model has a l s o been d e v e l o p e d to s t u d y the s o i l t h e r m a l and m o i s t u r e reg imes i n a s i n u s o i d a l r i d g e , but has y e t to be t e s t e d a g a i n s t f i e l d measurement s . F i e l d e x p e r i m e n t s took p l a c e a t 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 ' s P l a n t S c i e n c e R e s e a r c h S t a t i o n from Sep tember , 1983 to A p r i l , 1985 ; f i r s t , to e v a l u a t e the e f f e c t s o f s t r a w mulch a p p l i e d u n i f o r m l y a t r a t e s o f 2 , 10 or 20 t / h a , and s e c o n d , to d e t e r m i n e the e f f e c t s o f 0 .30 m wide mulch s t r i p s (10 t / h a r a t e ) a l t e r n a t e d w i t h 0 .10 m bare s t r i p s . The f o r m u l a t i o n of the model and c o m p a r i s o n w i t h the e x p e r i m e n t a l r e s u l t s form the b a s i s o f two c h a p t e r s . In C h a p t e r 1 f o r m u l a t i o n o f a n u m e r i c a l method to p r e d i c t s o i l t h e r m a l reg imes i s p r e s e n t e d . B a r a k a t and C l a r k ' s (1966) a l t e r n a t i n g d i r e c t i o n e x p l i c i t (ADE) method i s e x t e n d e d to s o l v e one- and two - d i m e n s i o n a l heat t r a n s f e r prob lems t h a t a re p e r i o d i c i n t ime a n d / o r s p a c e , on e i t h e r r e c t a n g u l a r domains or domains w i t h a c u r v i l i n e a r upper b o u n d a r y , and i n w h i c h the s o i l t h e r m a l p r o p e r t i e s may v a r y w i t h p o s i t i o n , t ime or t e m p e r a t u r e . These e x t e n s i o n s were n e c e s s a r y to meet the r e q u i r e m e n t s t h a t a r i s e i n a g r i c u l t u r a l s o i l s c o v e r e d w i t h s t r i p s a n d / o r formed i n t o r i d g e s . 3 In C h a p t e r 2 a model d e s c r i b i n g the e f f e c t s o f m u l c h e s , a p p l i e d e i t h e r u n i f o r m l y or i n s t r i p s , on the s u r f a c e energy b a l a n c e 1s p r e s e n t e d . T h i s model 1s combined w i t h i n the ADE method so t h a t both the s u r f a c e energy b a l a n c e and s o i l t h e r m a l reg imes are s i m u l t a n e o u s l y d e t e r m i n e d . The model i s then t e s t e d and c a l i b r a t e d u s i n g the e x p e r i m e n t a l d a t a from the 1983-85 s t u d i e s a t U . B . C . The e f f e c t s o f changes i n p e r t i n e n t s o i l and mulch p a r a m e t e r s on e v a p o r a t i o n and s o i l t e m p e r a t u r e s a re d i s c u s s e d . 4 CHAPTER 1 EXTENSIONS OF THE ALTERNATING-DIRECTION E X P L I C I T METHOD IN SOLVING TWO-DIMENSIONAL HEAT TRANSFER PROBLEMS 5 EXTENSIONS OF THE ALTERNATING-DIRECTION E X P L I C I T METHOD IN SOLVING TWO-DIMENSIONAL HEAT TRANSFER PROBLEMS B a r a k a t and C l a r k (1966) p r e s e n t e d an a l t e r n a t i n g -d i r e c t i o n e x p l i c i t (ADE) method t h a t p r o v i d e s an e f f i c i e n t a p p r o a c h to s o l v i n g t i m e - d e p e n d e n t m u l t i - d i m e n s i o n a l heat t r a n s f e r p r o b l e m s . L a r k i n (1964) and B a r a k a t and C l a r k showed t h a t f o r t w o - d i m e n s i o n a l prob lems the ADE method was s u p e r i o r i n c o m p u t a t i o n t ime and comparab le i n a c c u r a c y to the a l t e r n a t i n g - d i r e c t i o n i m p l i c i t and o t h e r i m p l i c i t methods , i n c l u d i n g the w e l l - k n o w n C r a n k - N i c o l son m e t h o d . The ADE method i s s i m p l e to program and i s u n c o n d i t i o n a l l y s t a b l e , a l t h o u g h B a r a k a t and C l a r k commented t h a t the t ime Increment ( A t ) s h o u l d not be s e t , a t l e a s t i n i t i a l l y , to more than f o u r t i m e s t h a t g i v e n by the c r i t e r i o n where a d i s the t h e r m a l d i f f u s i v i t y and Az and Ax a re the g r i d i n c r e m e n t s i n the v e r t i c a l and h o r i z o n t a l d i r e c t i o n s , r e s p e c t i v e l y . O t h e r w i s e the s o l u t i o n may not d e s c r i b e the p r o b l e m c o n s i d e r e d . B a r a k a t and C l a r k a p p l i e d the method to t r a n s i e n t heat t r a n s f e r p rob lems w h i c h a s y m p t o t i c a l l y a p p r o a c h s t e a d y - s t a t e t e m p e r a t u r e d i s t r i b u t i o n s as t i n c r e a s e s . T h e i r two-1.1 I n t r o d u c t i on ] < 0 . 5 , (1) 6 d i m e n s i o n a l i l l u s t r a t i v e examples were on r e c t a n g u l a r domains w i t h the boundary c o n d i t i o n s e x p l i c i t l y s p e c i f i e d , u n i f o r m g r i d s p a c i n g s , and homogeneous t h e r m a l p r o p e r t i e s . T h i s paper e x t e n d s the ADE method to t w o - d i m e n s i o n a l p r o b l e m s t h a t a re p e r i o d i c i n t ime and the h o r i z o n t a l d i r e c t i o n , i n w h i c h the upper boundary may be c u r v i l i n e a r , and 1n w h i c h the t h e r m a l p r o p e r t i e s may v a r y w i t h p o s i t i o n , t i m e , and t e m p e r a t u r e . To improve the c o m p u t a t i o n a l e f f i c i e n c y n o n - u n i f o r m g r i d s p a c i n g s a re i n c o r p o r a t e d so t h a t f i n e g r i d s are r e q u i r e d o n l y i n the r e g i o n s o f l a r g e t e m p e r a t u r e g r a d i e n t s . A boundary c o n d i t i o n o f the 3 r d k i n d i s assumed on the upper b o u n d a r y . The boundary t e m p e r a t u r e s and f l u x e s a re not s p e c i f i e d e x p l i c i t l y on the v e r t i c a l b o u n d a r i e s but a re d e t e r m i n e d i m p l i c i t l y from the p e r i o d i c i t y i n the h o r i z o n t a l d i r e c t i o n . The n u m e r i c a l c o m p u t a t i o n s a re compared w i t h the a n a l y t i c a l s o l u t i o n s of some r e l e v a n t i l l u s t r a t i v e p r o b l e m s . The e x t e n s i o n s o f the ADE method d i s c u s s e d i n t h i s c h a p t e r a r o s e out of the i n v e s t i g a t i o n of the e f f e c t s o f s u r f a c e " m u l c h i n g " and " r i d g i n g " on the t h e r m a l reg imes of a g r i c u l t u r a l s o i l s r e p o r t e d i n p a r t i n c h a p t e r 2 . A l t h o u g h some o f the examples p r e s e n t e d a re i n t h i s c o n t e x t the r e s u l t s s h o u l d be a p p l i c a b l e to s i m i l a r prob lems i n o t h e r f i e l d s . 7 1.2 F o r m u l a t i o n _ o f _ t h e _ N u m e r i c a I _ M o d e i The u n s t e a d y heat t r a n s f e r e q u a t i o n i n two d i m e n s i o n s f o r an inhomogeneous i s o t r o p i c medium can be w r i t t e n a s : C 8 t = fz t k l z ] + fx t k l x 3 , * 2 ) where T ( z , x , t ) i s the t e m p e r a t u r e , z i s the d e p t h ( p o s i t i v e downward) , x i s the h o r i z o n t a l d i s t a n c e , t 1s the t i m e , and C ( z , x , t , T ) and k ( z , x , t , T ) a re the v o l u m e t r i c heat c a p a c i t y and the t h e r m a l c o n d u c t i v i t y , r e s p e c t i v e l y , o f the medium. In o r d e r to s o l v e e q u a t i o n (2) f o u r boundary c o n d i t i o n s and an i n i t i a l c o n d i t i o n a re r e q u i r e d . The boundary c o n d i t i o n s may t a k e a v a r i e t y o f f o r m s ; the most g e n e r a l l i n e a r c o n d i t i o n f o r the upper boundary ( a p p r o p r i a t e to s o i l s ) i s t h e R o b i n c o n d i t i o n or the boundary c o n d i t i o n of the 3 rd k i n d : a G N ( 0 , x , t ) + B ( x , t ) T ( 0 , x , t ) = f ( x , t ) , (3) where a i s e i t h e r 1 or 0 , 8 ( x , t ) i s a heat t r a n s f e r c o e f f i c i e n t (W m " 2 ° C - 1 ) , G N ( 0 , x , t ) i s the s u r f a c e heat f l u x d e n s i t y normal t o the s u r f a c e (W m ) and f ( x , t ) i s an a r b i t r a r y f u n c t i o n (W m ) . W i t h a = 0 and B = 1 e q u a t i o n (3) becomes a boundary c o n d i t i o n of the 1s t k i n d ( D i r i c h l e t c o n d i t i o n ) , w h i l e w i t h a = 1 and 6 = 0 i t becomes a boundary 8 c o n d i t i o n o f the 2nd k i n d (Neumann c o n d i t i o n ) . For the lower boundary (z = Z) a c o n s t a n t t e m p e r a t u r e i s a s sumed, i . e . T ( Z , x , t ) = c o n s t a n t . (4) T h i s i s a p p r o p r i a t e f o r a s e m i - i n f i n i t e medium i f the l o w e r boundary dep th i s chosen l a r g e enough so t h a t t e m p e r a t u r e v a r i a t i o n s due t o f o r c i n g a t z * 0 a re i n s i g n i f i c a n t . The l a t e r a l boundary c o n d i t i o n s are not known a priori but a re d e t e r m i n e d i m p l i c i t l y from the a s s u m p t i o n t h a t the p r o b l e m i s p e r i o d i c i n the h o r i z o n t a l d i r e c t i o n , i . e . T ( z , x , t ) = T ( z , x + X , t ) . (5) T h e r e f o r e , the t e m p e r a t u r e s (and the heat f l u x d e n s i t i e s , s i n c e k i s s i m i l a r l y p e r i o d i c 1n x) a t the two l a t e r a l b o u n d a r i e s (x = 0 and x = X) are equa l a t any dep th and t i m e . There a re d i f f e r e n t schemes a v a i l a b l e i n the l i t e r a t u r e to t a k e i n t o a c c o u n t the i n h o m o g e n e i t y o f the t h e r m a l p r o p e r t i e s . I n c o r p o r a t i n g i n h o m o g e n e i t y i n B a r a k a t and C l a r k ' s scheme f o l l o w i n g O z i s i k ( 1 9 8 0 ) , the f i n i t e d i f f e r e n c e e q u i v a l e n t o f e q u a t i o n (2) f o r t h e i r IK . f u n c t i o n i s w r i t t e n • »J a s : 9 U n + 1 - U n  c n + l l i a l _1A _ L 1 , J At . n r u n _ u n . k n + l (i\n+1 - u n + 1 \ ( A z ) * . n ( u n _ ur\ . k n + l r u n + 1 - u n + 1 ^ i u ± i z 2 _ i ! ! i . i ± I iiAL liAzUiJiijjL r u b i * ( 6 ) ( A x ) 2 where the i , j s u b s c r i p t s and n s u p e r s c r i p t s a re a s s o c i a t e d w i t h the z , x , and t g r i d s , r e s p e c t i v e l y ( F i g 1 . 1 ) . By i n t e r c h a n g i n g the s u p e r s c r i p t s n and (n + 1) on the r i g h t - h a n d s i d e o f e q u a t i o n (6) a s i m i l a r e x p r e s s i o n f o r t h e i r V . . • > J f u n c t i o n i s o b t a i n e d . The c a l c u l a t i o n s f o r the U . . f u n c t i o n • » J p r o c e e d f rom l e f t t o r i g h t a c r o s s each row and from the top to the bot tom of the z - x d o m a i n , w h i l e f o r the V , . f u n c t i o n they p r o c e e d from r i g h t to l e f t and from the bot tom to the t o p . The t e m p e r a t u r e i n f i n i t e - d i f f e r e n c e f o r m , T. . , i s the average o f the U . . and V. . f u n c t i o n s , i . e . ' »J • i J A c c o r d i n g to B a r a k a t and C l a r k l o w e r o r d e r e r r o r terms c a n c e l 2 i n e q u a t i o n ( 7 ) , y i e l d i n g t r u n c a t i o n s e r r o r s o f 0 (Az ) , 2 2 0(Ax ) , and 0 ( A t ) , w h i c h makes the ADE method comparab le i n a c c u r a c y to v a r i o u s w e l l - k n o w n i m p l i c i t m e t h o d s . S o l v i n g 10 0.00 2 3 m-1 m ^ 0.10 N 0.14 0.90 1.00 0.00 0.10 0.30 x (m) 0.40 F i g u r e 1.1 P a r t o f the f i n i t e d i f f e r e n c e g r i d , w i t h v a r i a b l e s p a c i n g i n bo th d i r e c t i o n s , used on the r e c t a n g u l a r d o m a i n . T r a n s i t i o n l i n e s a re i n b o l d . 11 e x p l i c i t l y f o r u " + j i n e q u a t i o n (6) y i e l d s r n + l A t _ . n _ A t _ . n where A l , t = — . i . J « , n + l A ? n + 1 _ r n + l A t _ .n+1 . _ A t _ . n + 1 A ' i , j " L 1 , J ( A z ) 2 k 1 - l / 2 , j + ( A x ) 2 k i , j - 1 / 2 B i J + J = _ A t k n + l > / A 2 n + l I . J " ( A z ) 2 " 1 - 1 / 2 , j , n t 1 , J B 2 i . J ~ ( A z ) 2 k i + l / 2 , j / A 2 i , j (9) D 1 n+1 = k n + l / A 2 n + 1 i . J ( A x ) 2 k i , j - l / 2 / A Z 1 , j D 2n + 1 - k n / A 2 n + 1 1 i J ( A x ) 2 1 . J + l / 2 / A Z i . J * For the V , . f u n c t i o n the s u b s c r i p t s ( i - 1 / 2 ) and ( j - 1 / 2 ) a re i n t e r c h a n g e d w i t h ( i + 1 / 2 ) and ( j + 1 / 2 ) , r e s p e c t i v e l y , on the r i g h t - h a n d s i d e o f e q u a t i o n s ( 9 ) . 12 E q u a t i o n (8) r e q u i r e s t h a t the t h e r m a l p r o p e r t i e s must be known a t the ( n + l ) t h t ime s t ep t h r o u g h o u t z - x d o m a i n , w h i c h p r e s e n t s d i f f i c u l t i e s when k and C are f u n c t i o n s o f t e m p e r a t u r e . In t h i s case 1t 1s f i r s t assumed t h a t the t h e r m a l p r o p e r t i e s a t the ( n + l ) t h t ime s t e p a re g i v e n by t h o s e a t the nth t ime s t e p . For each 1 the U ^ + i ' s so c a l c u l a t e d a re then used to r e c a l c u l a t e the t h e r m a l p r o p e r t i e s and new v a l u e s of u!? + , ; t h i s p r o c e d u r e i » J c o n t i n u e s u n t i l a p r e d e t e r m i n e d c o n v e r g e n c e c r i t e r i o n i s met . T h i s i t e r a t i v e t e c h n i q u e works w e l l when the dependence o f k and C on T i s not s t r o n g ( w h i c h 1s the case f o r most s o i l s ( H i l l e l , 1 9 8 0 ) ) . A s i m i l a r p r o c e d u r e i s used f o r v n + l In f i n i t e d i f f e r e n c e form the p e r i o d i c i t y c o n d i t i o n f o r each row i s g i v e n by ud - » " : : • n o ) where (m-1) i s the number o f f i n i t e d i f f e r e n c e g r i d i n t e r s e c t i o n s (nodes) In the h o r i z o n t a l d i r e c t i o n f o r 0 < x < X (see F i g 1 . 1 ) . The j = m+1 nodes a re r e q u i r e d by e q u a t i o n (8) to c a l c u l a t e m w h i l e the j = 1 nodes a re s i m i l a r l y r e q u i r e d t o c a l c u l a t e V . 9 (no te t h a t the j = 1 and the j = m+1 nodes are o u t s i d e the d o m a i n ) ; the v a l u e s o f 13 U 1 m+1 a n d V1 1 a r e d e t e r m 1 n e d f rom the p e r i o d i c i t y c o n d i t i o n 1n the h o r i z o n t a l d i r e c t i o n . A p p l y i n g e q u a t i o n (8) to a l l nodes i n the row between j = 2 and j = m i n c l u s i v e , and c o m b i n i n g w i t h e q u a t i o n (10) f o r 1 > 3 ( i = 1 and 2 f o r r e c t a n g u l a r domains are d i s c u s s e d be low) y i e l d s Uj + l 1 { p n + l + D 1 n + 1 p n + l n + 1 D 1 n+1 1,2 m . 1 1 ,m 1 ,m 1 , m - l i , m i , m - l 1 - n D i y + J j=3 1 i J where Once u j + i i s known u " + J a t a l l the o t h e r nodes i n the row f o l l o w e x p l i c i t l y f rom e q u a t i o n ( 8 ) . For i = 2 , u j + } . 1n e q u a t i o n (8) i s not known f o r a l l i - x , j j , as i t i s f o r 1 > 3 . W r i t i n g e q u a t i o n (3) i n f i n i t e d i f f e r e n c e form y i e l d s a k n + l Ul) . B n + 1 u n + 1 - f n + 1 m i a k 2 - l / 2 , j 1 Az ' + p j u l , j " f j • ( 1 3 ) 14 S o l v i n g f o r u j + j i n terms of u!} +* w i t h e q u a t i o n ( 1 3 ) , s u b s t i t u t i n g i n t o e q u a t i o n (8) w i t h i = 2 , and r e a r r a n g i n g y i e l d s an e q u a t i o n f o r U ^ j o f e s s e n t i a l l y the same form as e q u a t i o n ( 8 ) . The a n a l y s i s f o r 1 = 2 then p r o c e e d s as f o r 1 > 3 and the u " + j a re c a l c u l a t e d c o n c u r r e n t l y u s i n g e q u a t i o n ( 1 3 ) . For v " + j the a n a l y s i s i s s i m p l e r because the boundary c o n d i t i o n a t z = Z 1s of the 1 s t k i n d ; the v " + j a re d e t e r m i n e d from e q u a t i o n (13) a f t e r r e p l a c i n g the U ' s by V s . To use a v a r i a b l e g r i d s p a c i n g c a u t i o n i s needed a t the l i n e o f t r a n s i t i o n ( a t w h i c h Az a n d / o r Ax changes m a g n i t u d e ) . The numera tor o f B l . . ( e q u a t i o n (9 ) ) i n v o l v e s the nodes ( i , j ) and ( 1 - 1 , j ) w h i l e B 2 i ^ i n v o l v e s the nodes ( 1 , j ) and ( 1 + 1 , j ) . S i m i l a r l y , the numera tor o f D1^ ^ uses the nodes ( i , j ) and ( i . j - l ) w h i l e t h a t of D2, . uses the nodes ( i , j ) and ( 1 , j + l ) . ' i J However , the numera tor of A l . , i n c l u d e s the n u m e r a t o r s of • > J both B2 . . and D2, , w h i l e the d e n o m i n a t o r f o r a l l o f the 1 » J 1 « J c o e f f i c i e n t s uses bo th [ ( i , j ) , ( 1 - 1 , j ) ] and [ ( i , j ) , ( i , j - l ) ] c o m b i n a t i o n s o f n o d e s . When ( i , j ) i s a t a t r a n s i t i o n p o i n t , the g r i d s p a c i n g s over w h i c h the c o e f f i c i e n t s a re c a l c u l a t e d may not be the same i n e i t h e r the v e r t i c a l a n d / o r the h o r i z o n t a l d i r e c t i o n (see F i g 1 . 1 ) . There a re s e v e r a l ways i n w h i c h the c o e f f i c i e n t s may be c a l c u l a t e d a t t r a n s i t i o n p o i n t s . The f o l l o w i n g t h r e e were chosen f o r c o m p a r i s o n w i t h a known 15 a n a l y t i c a l s o l u t i o n 1n o r d e r to s e l e c t the b e s t one f o r subsequent a p p l i c a t i o n s : ( i ) t o use a d i f f e r e n t Az ( a n d / o r Ax) i n the numerator and the d e n o m i n a t o r c o r r e s p o n d i n g to the g r i d s p a c i n g o v e r w h i c h the t h e r m a l c o n d u c t i v i t y i s a v e r a g e d ; ( H ) to use an average o f Az ( a n d / o r Ax) i n bo th the numera tor and the d e n o m i n a t o r ; (111) to use an average o f both Az ( a n d / o r Ax) i n the numera tor o f j and an average o f Az ( a n d / o r Ax) i n the numera tor o f B2 , . and D2, . , r e s p e c t i v e l y as t h e y i n v o l v e the t h e r m a l c o n d u c t i v i t i e s t h a t have been averaged f o r the nodes ( 1 , j ) and ( 1 + 1 , j ) , a n d / o r ( 1 , j ) and ( 1 , j + l ) . The t h e o r y and s o l u t i o n t e c h n i q u e s d e s c r i b e d thus f a r a p p l y to r e c t a n g u l a r d o m a i n s . To e x t e n d the s o l u t i o n t e c h n i q u e to a c u r v e d upper boundary ( such as a s o i l r i d g e ) f u r t h e r m o d i f i c a t i o n s a re r e q u i r e d . We c o n s i d e r a case i n w h i c h the upper boundary i s s i n u s o i d a l , i . e . , H r H r Z c = 2 + 2 s i nC*c ( x + * c ) : i * ( 1 4 > where Z c i s the dep th of the c u r v e d s u r f a c e a t any x below the h o r i z o n t a l p l a n e a t z = 0 , H i s the maximum d e p t h o f the c u r v e d s u r f a c e , A i s the l a t e r a l f r e q u e n c y (= 2 n / X ) and <t>r i s a c o n s t a n t ( F i g 1 . 2 ) . 16 1.00 0.00 0.16 x (m) 0.32 F i g u r e 1.2 The f i n i t e d i f f e r e n c e g r i d used w i t h the s i n u s o i d a l upper boundary s p e c i f i e d by e q u a t i o n ( 1 4 ) . 17 When a s u r f a c e i s bounded by a c u r v e d b o u n d a r y , i n g e n e r a l the i n t e r s e c t i o n s o f the f i n i t e d i f f e r e n c e network and the c u r v e d boundary a re not a t the f i n i t e d i f f e r e n c e nodes . C o n s i d e r i n g such a node and the l i n e segments j o i n i n g i t to i t s f o u r n e a r e s t n e i g h b o u r s , i f the boundary i n t e r s e c t s one of t h e s e segments then the node i s c a l l e d an "edge" n o d e , w h i l e i f i t i n t e r s e c t s two of the se segments then i t i s c a l l e d a " c o r n e r " node . Nodes w i t h o u t such i n t e r s e c t i o n s a re c a l l e d " i n t e r i o r " n o d e s . F u r t h e r m o r e 1n some rows t h e r e c o u l d be c o m b i n a t i o n s o f s u c c e s s i v e c o r n e r and edge nodes (see F i g 1 . 2 ) . The noda l s i t u a t i o n s a t the b e g i n n i n g o f a row r e q u i r i n g m o d i f i e d a n a l y s i s , t o be d e s c r i b e d , a re the f o l l o w i n g ( F i g 1 . 3 ) : ( i ) c o r n e r node w i t h an edge node ( i n t e r s e c t i o n 1n the upper v e r t i c a l segment) to i t s r i g h t ; ( i i ) c o r n e r node o n l y ; ( i i i ) edge node ( I n t e r s e c t i o n i n the l e f t h o r i z o n t a l segment) o n l y ; ( i v ) edge node ( i n t e r s e c t i o n i n the upper v e r t i c a l segment) w i t h a s i m i l a r edge node to i t s r i g h t . Note t h a t due to symmetry the noda l a r rangement a t the o t h e r end o f a row i s the r e v e r s e o f the a b o v e . In terms o f B a r a k a t and C l a r k ' s scheme, f o l l o w i n g O z i s i k (1980) the f i n i t e d i f f e r e n c e a p p r o x i m a t i o n o f e q u a t i o n (2) f o r a c o r n e r node , w i t h i n t e r s e c t i o n s i n the l e f t h o r i z o n t a l and upper v e r t i c a l s egments , i s ( t r e a t i n g the i n t e r s e c t i o n p o i n t s e x a c t l y as g r i d n o d e s ) : F i g u r e 1.3 The v a r i o u s noda l s i t u a t i o n s o c c u r r i n g a t the b e g i n n i n g o f rows i n t e r s e c t i n g t h e s i n u s o i d a l upper boundary s p e c i f i e d by e q u a t i o n ( 1 4 ) . The s i t u a t i o n s (1) to (1v) d e s c r i b e d 1n the t e x t c o r r e s p o n d to F 1 g s . 1.3a t o 1 . 3 d , r e s p e c t i v e l y . 19 r n + l . .n+1 t n + l nn + l f_L,i + __2__ 2 1 - 1 / 2 U _iAizI/2, . U 1 . J 1 A t + ( A z ) 2 a ( a + 1) + ( A x ) 2 b(b + .1) ' ' r n + l un . n r _ L l 2__ _ i ± l / 2 _ J . _ __2__ _ l U ± l / 2 , y n 1 At ( A z ) 2 a + 1 ( A x ) 2 b + 1 > U i , j tn+l . n . Vl / l^ i nn + l __2__ : i ± l / 2 ^ i u n ( A z ) 2 a (a + 1) u i - l , j + { A z ) 2 a + 1 u 1 + l , j n+1 + __2__ llAzilZ nn+l . __2__ : i ^ i ± l / 2 n , . + ( A x ) 2 b(b + 1) u i , j - l + , A v , 2 b + 1 U i , j + l * ( 1 5 ) (Ax) where a (<1) and b (<1) a re the f r a c t i o n s o f Az and Ax a s s o c i a t e d w i t h the v e r t i c a l and h o r i z o n t a l i n t e r s e c t i o n s , r e s p e c t i v e l y . To d e s c r i b e a c o r n e r node w i t h the i n t e r s e c t i o n i n the r i g h t h o r i z o n t a l segment , b ( b + l ) and (b+1) a re i n t e r c h a n g e d i n e q u a t i o n ( 1 5 ) . By s i m p l y c h a n g i n g the p r o p o r t i o n a l i t y f a c t o r s a a n d / o r b e q u a t i o n (15) can be r e a d i l y m o d i f i e d t o f i t any of the above c a s e s . S i m i l a r e x p r e s s i o n s e x i s t f o r the V , , f u n c t i o n . •»J The m o d i f i c a t i o n s r e q u i r e d f o r s i t u a t i o n s ( i ) to ( i v ) depend on the a s s u m p t i o n t h a t the t e m p e r a t u r e s and t h e r m a l c o n d u c t i v i t i e s a t any p o i n t l o c a t e d on the c u r v e d boundary between a d j a c e n t i n t e r s e c t i o n p o i n t s can be l i n e a r l y i n t e r p o l a t e d between the v a l u e s a t the se two p o i n t s . To i l l u s t r a t e , c o n s i d e r the c o r n e r node f o r the U . , f u n c t i o n •»J ( the A f u n c t i o n i s d i s c u s s e d l a t e r ) i n F i g 1 . 3 b . u " + J . a t 20 the ( i - l ) t h row 1s known and e q u a t i o n (15) r e p r e s e n t s the f i n i t e d i f f e r e n c e a p p r o x i m a t i o n f o r t h a t node . A p p l y i n g the s u r f a c e boundary c o n d i t i o n g i v e n by e q u a t i o n (3) y i e l d s : .jn + 1 . / ! ^ N „ n + l » „n + l _ f n + l M f i . d 2 U i , j + ( d j + P N } U N " f N ' ( 1 6 ) where d j i s the l e n g t h between N and ( 1 , j ) and k N i s the ^ ^  < n o n II a t N average o f the k ' s a t N and ( 1 , j ) . u j + l 1 n e q u a t i o n (16) i s i n t e r p o l a t e d as f o l l o w s : n n + l = * 2 ..n + 1 U--Z) u n + 1 r 17^ U N d 3 u 1 - 1 , j - 1 + 1 1 d 3 ; u 1 - l , j ' ( 1 7 ) where d 2 i s the l e n g t h between N and ( 1 - 1 , j ) and d 3 i s the l e n g t h between ( 1 - 1 , J - l ) and ( 1 - 1 , J ) . E q u a t i o n s ( 1 5 ) , (16) and (17) c o n t a i n the 4 unknowns Ul? + $ , U? + } . , u j + 1 , and U i - 1 j - l a n d a s o 1 u t 1 0 n i s n o t p o s s i b l e o n l y w i t h 3 e q u a t i o n s . T h e r e f o r e u j + } . i s i n t e r p o l a t e d between u j + } . , and u J + J . as u " - i , j • HTCIJ • ( i -b "V.i.j.,. («> where d 4 i s the l e n g t h between the nodes ( i - l , j - l ) and ( i - 2 , j ) a l o n g the c u r v e d b o u n d a r y . S u b s t i t u t i n g i n t o e q u a t i o n s (15) and (16) y i e l d s 2 l i n e a r e q u a t i o n s w i t h 2 21 unknowns . The i n t e r i o r nodes then f o l l o w d i r e c t l y from e q u a t i o n (8) as o u t l i n e d p r e v i o u s l y . The p r o c e d u r e i s s i m i l a r f o r the c o r n e r node a t the end o f the row. For s i t u a t i o n ( i ) ( F i g . 1 . 3 a ) , i n t e r p o l a t i o n p r o c e e d s as f o l 1 o w s : (1) U N 1 i s i n t e r p o l a t e d between u!J + j and U n+j y (2) U ^ } j i s i n t e r p o l a t e d between U n + | and U ^ J J + A s u i - i j + i 1 s unknown, u " + j + i i s d e t e r m i n e d s i m u l t a n e o u s l y w i t h UN +J . (3) U N 2 i s i n t e r p o l a t e d between U n + J j _ 1 a n d U i - 1 j + 1* T h e s o 1 u t 1 o n ^ s t n e n c o m p l e t e d by e l i m i n a t i o n as u s u a l . The i n t e r i o r nodes then f o l l o w as p r e v i o u s l y o u t l i n e d and the nodes a t the end o f the row a r e t r e a t e d s i m i l a r l y . The s o l u t i o n f o r s i t u a t i o n ( i i i ) ( F i g 1 .3c) 1s r a t h e r s i m p l e . As bo th the ( 1 - 1 , j ) and ( 1 - 2 , j ) nodes a re known i n t h i s c a s e , i t r e q u i r e s o n l y one I n t e r p o l a t i o n a t N to comple te the s o l u t i o n . A l s o the node ( i - 2 , j - l ) , w h i c h i s known, can be used i n the c a l c u l a t i o n I n s t e a d o f node ( 1 - 2 , j ) . The l e f t -and r i g h t - e n d s are not s i g n i f i c a n t l y d i f f e r e n t . The s o l u t i o n f o r s i t u a t i o n ( i v ) ( F i g 1.3d) i s v e r y s i m i l a r t o the s o l u t i o n f o r s i t u a t i o n (1) and w i l l not be d e s c r i b e d . For the V. , f u n c t i o n , s i m i l a r t e c h n i q u e s a re u s e d . The 1 > J m a j o r d i f f e r e n c e i s t h a t t h i s t ime the unknowns on the boundary are i n t e r p o l a t e d u s i n g the knowns from the ( i + l ) t h l i n e . I t i s i n t e r e s t i n g to note t h a t a s i m p l e s i t u a t i o n f o r 22 the U . , f u n c t i o n , such as s i t u a t i o n ( i i i ) 1s r a t h e r c o m p l i c a t e d f o r the j f u n c t i o n and v i c e v e r s a . For the r e c t a n g u l a r p a r t be low the r i d g e , where the h o r i z o n t a l b o u n d a r i e s a re p e r i o d i c i n s p a c e , the s o l u t i o n t e c h n i q u e d e s c r i b e d f o r the r e c t a n g u l a r domain remains v a l i d . 1.3 R e s u l t s In o r d e r to c o n f i r m the v a l i d i t y o f the n u m e r i c a l m e t h o d , the n u m e r i c a l c a l c u l a t i o n s a re compared w i t h a n a l y t i c a l s o l u t i o n s of one- and t w o - d i m e n s i o n a l p e r i o d i c ( w i t h t e m p o r a l p e r i o d P = 24 h) heat c o n d u c t i o n p r o b l e m s i n homogeneous and inhomogeneous s e m i - i n f i n i t e m e d i a . For the n u m e r i c a l c a l c u l a t i o n s u n i f o r m i n i t i a l c o n d i t i o n s a re assumed, e x c e p t f o r Case I i n w h i c h the t e m p e r a t u r e p r o f i l e was i n i t i a l i z e d w i t h t = 0 h v a l u e s from the e x a c t s o l u t i o n . T h e r e f o r e the r e s u l t s a re compared w i t h the a n a l y t i c a l s o l u t i o n s o n l y a f t e r the t r a n s i e n t s have s u f f i c i e n t l y d e c a y e d , i . e . , the t e m p e r a t u r e s everywhere d i f f e r e d by a t most 0 . 0 2 ° C between the b e g i n n i n g and end o f a 24 h p e r i o d . The r e s u l t s a re p r e s e n t e d f o r r e c t a n g u l a r domains and a c u r v i l i n e a r upper boundary s e p a r a t e l y . 23 A . Re c t a nfl ul_a r_D oma 1_n s Case I: Variable Grid Spacing In the Vertical Direction To d e t e r m i n e the b e s t c r i t e r i o n w i t h v a r i a b l e g r i d s p a c i n g i n the v e r t i c a l d i r e c t i o n , the n u m e r i c a l s o l u t i o n w i t h the 3 schemes d e s c r i b e d p r e v i o u s l y i s a p p l i e d t o the one-d i m e n s i o n a l t e m p e r a t u r e d i s t r i b u t i o n p r o b l e m w i t h : ( i ) T ( 0 , t ) = 10°C + 12°C s i n ( w t + TT) , ( i i ) T ( Z , t ) = 1 0 ' C , where w (= 2 i r /P) 1s the a n g u l a r f r e q u e n c y f o r a d a i l y c y c l e and Z = 1 m (used i n a l l o t h e r cases b e l o w ) . The t h e r m a l 1 1 6 p r o p e r t i e s used were k = 0 .79 W m °C and C = 1.51 x 10 -3 -1 J m °C . The e x a c t s o l u t i o n 1s g i v e n i n C a r s l a w and J aeger ( 1 9 5 9 , p . 6 4 ) . For each scheme Az v a r i e d as f o l l o w s : 0 .01 m f o r the 1 s t 0 .10 m, 0 .02 m f o r the nex t 0 .10 m, 0 .05 m f o r a n o t h e r 0 .30 m, and 0 .10 m f o r r e s t o f the c o l u m n . The d i f f e r e n c e s between the t e m p e r a t u r e s a t t * 12 h c a l c u l a t e d w i t h the e x a c t s o l u t i o n and t h o s e from the t h r e e schemes f o r z = 0 . 0 1 , 0 . 0 2 , 0 . 0 3 , 0 . 0 4 , 0 . 0 5 , 0 . 1 0 , 0 . 2 0 , 0 .40 and 0 .50 m are p r e s e n t e d 1n T a b l e 1 . 1 . The t e m p e r a t u r e d i f f e r e n c e s a t t = 12 h between the e x a c t and the n u m e r i c a l method u s i n g a u n i f o r m g r i d s p a c i n g (Az = 0 .02 m) a t z = 0 . 0 2 , 0 . 0 4 , 0 . 1 0 , 0 . 2 0 , 0 .40 and 0 .50 m are a l s o p r e s e n t e d . The maximum e r r o r s a t the se depths d u r i n g the 24 h p e r i o d a re a l s o r e p o r t e d . I t i s e v i d e n t t h a t scheme 3 i s f a r s u p e r i o r to schemes 1 and 2 ; i t p e r f o r m e d a l m o s t as w e l l as the scheme w i t h u n i f o r m g r i d s p a c i n g . The maximum 24 h e r r o r w i t h scheme 3 i s - 0 . 0 8 ° C Table 1.1 D i f f e r e n c e s between the exact and the n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t = 12 h f o r Case I a t the i n d i c a t e d d e p t h s . C r i t e r i a i n e v a l u a t i n g Depth z(m) the c o e f f i c i e n t s at t r a n s i t i o n p o i n t s 0 .01 0 .02 0 .03 0.04 0 .05 0 .10 0 .20 0 .40 0 . 50 Az between ( i , j ) 0 .17 0 .33 0.49 0.64 0 .79 1 .38 2 .25 0 .05 - 0 . 13 and ( 1 - 1 , j ) nodes <0.20>*<0.39> <0.58> <0.77> <0.95> <1 .92> <2.25> <0.46> <0. 35> d i f f e r s from Az between ( i . j ) and (1+1 , j ) nodes Az 1s an average of - 0 . 1 6 - 0 . 3 3 - 0 . 4 9 - 0 . 6 6 - 0 . 8 1 -1 .47 - 1 . 4 6 - 0 . 0 1 0 . 06 A z ' s between ( i , j ) <-0.18><-0.36> <-0.55> < -0.73> <-0.91> <-l .81> <-1.46> <-0.33> <-0. 19> and ( 1 - 1 , j ) , and ( 1 , j ) and (1+1 , j ) nodes Average of A z ' s 0 .005 0 .004 0 .002 - 0 . 0 0 3 - 0 . 0 1 -0 .04 - 0 . 0 4 0 . 0 3 0 . 02 between ( 1 , j ) and <-0.01 <-0.02> <-0.02> < -0.02> <0.02> <-0 .04> <-0.08> <-0.04> <-0. 03> ( 1 - 1 , j ) , and or 0.01> ( 1 , j ) a n d ( 1 + l , j ) nodes was used In the numerator w h i l e Az between ( 1 , j ) and ( 1 - 1 , j ) nodes or t h a t between ( i , j ) and (1+1 , j ) nodes was used 1n the denomina tor f o r the U and V f u n c t i o n s , r e s p e c t i v e l y . U n i f o r m Az - <0.0005 _ -0 .01 _ -0 .01 0 .03 0 .01 0 . 01 (= 0 .02 m <-0.01 < -0 .01 <0 .04> <0.05> <0.01> <0. 01> t h r o u g h o u t ) or 0.01> or 0.01> *The number 1n b r a c k e t < > i s the maximum e r r o r a t t h a t dep th d u r i n g a 24 h p e r i o d . 25 a t z = 0 .20 m and t = 16 h , w h i c h i s 0.36% o f the s u r f a c e t e m p e r a t u r e range ( 2 2 ° G ) c o n s i d e r e d . The maximum e r r o r w i t h u n i f o r m g r i d s p a c i n g i s 0 . 0 5 ° C a t z = 0 .16 m and t = 8 h , w h i c h i s 0.23% of t h i s r a n g e . T h e r e f o r e , scheme 3 i s adopted i n the one- and t w o -d i m e n s i o n a l t e s t cases t h a t f o l l o w . V e r i f i c a t i o n o f t h i s scheme f o r v a r i a b l e h o r i z o n t a l g r i d s p a c i n g 1s c o n s i d e r e d a f t e r the t w o - d i m e n s i o n a l ca ses have been p r e s e n t e d , as i t uses some o f the r e s u l t s of those c a s e s . Case II. Boundary Condition of the 3rd Kind and Inhomogeneous Thermal Properties The n u m e r i c a l scheme i s a p p l i e d to s o l v i n g the p r o b l e m : ( i ) G z ( 0 , t ) + 10 T ( 0 , t ) = {300 + 250 s i n ( w t - i r / 2 ) } W m ~ 2 , ( i i ) T ( Z , t ) = 1 0 ° C , ( H i ) T ( z , 0 ) = 1 0 6 C , w i t h t h e r m a l p r o p e r t i e s g i v e n a s : ( i ) k = 0 .2 W m " 1 ° C _ 1 , C = 1.1 x 1 0 6 J m " 3 ° C ~ 1 , ( i i ) k = 1.4 W m " 1 • C ' 1 , C = 3 .2 x 1 0 6 J m " 3 ° C _ 1 , ( i i i ) k = 1 .6770(z + 0 . 0 0 0 2 8 6 2 7 ) 0 , 2 6 0 6 4 C = 3 .5332 x 1 0 ° (z + 0 .00028627) k = 1.4 W r r f 1 ° C _ 1 0 .14303 } , z < 0 . 5 m, fi z > 0 . 5 m . C = 3 .2 x 1 0 ° J m " J ° C _ 1 These t h e r m a l p r o p e r t i e s a re r e f e r r e d to as " l o w " , " h i g h " , and " v a r i a b l e " i n the d i s c u s s i o n . Az v a r i e d as f o l l o w s : 0 .005 m f o r 0 .10 m, 0 .01 m f o r 0 .04 m, 0 .02 m f o r 0 .06 m, 0 .05 m f o r 0 .30 m, and 0 .10 m f o r r e s t o f the c o l u m n . 26 The d i f f e r e n c e s between the t e m p e r a t u r e s a t t = 12 h c a l c u l a t e d w i t h the a n a l y t i c a l s o l u t i o n and t h o s e f rom the n u m e r i c a l method f o r z = 0 . 0 , 0 . 0 1 , 0 . 0 5 , 0 . 1 0 , 0 .20 and 0 .40 m a r e p r e s e n t e d i n T a b l e 1 . 2 . The maximum e r r o r s d u r i n g the 24 h p e r i o d a t the se dep ths a re a l s o p r e s e n t e d . The e x a c t s o l u t i o n w i t h the t h e r m a l p r o p e r t i e s o f the form g i v e n by ( i i i ) ( w h i c h i n c l u d e s (1) and ( i i ) ) Is p r e s e n t e d i n A p p e n d i x I . The maximum t e m p e r a t u r e d i f f e r e n c e and the maximum p e r c e n t e r r o r e x p r e s s e d r e l a t i v e to the v a r i a t i o n a t the s u r f a c e a r e found to be 0 . 2 9 ° C and 0.77%, 0 . 2 8 ° C and 1.44%, and 0 . 2 8 ° C and 1.02%, r e s p e c t i v e l y , f o r the l o w , h i g h , and v a r i a b l e t h e r m a l p r o p e r t y c a s e s . The magni tude o f the maximum e r r o r a t t h e s u r f a c e f o r the h i g h t h e r m a l p r o p e r t y case i s s i m i l a r to the o t h e r two cases but because the s u r f a c e t e m p e r a t u r e range ( f o r the same f o r c i n g f u n c t i o n ) i s l o w e r ( 1 9 . 1 ° C a g a i n s t 3 8 . 0 ° C f o r the low t h e r m a l p r o p e r t y ca se ) the p e r c e n t e r r o r i s h i g h e r . The magn i tudes o f the maximum e r r o r and the maximum p e r c e n t e r r o r f o r the v a r i a b l e t h e r m a l p r o p e r t y case l i e between the h i g h and low t h e r m a l p r o p e r t y c a s e s . T h e r e f o r e the e x t e n d e d ADE method as d e s c r i b e d f o r v a r i a b l e t h e r m a l p r o p e r t i e s p e r f o r m s as e f f i c i e n t l y as w i t h c o n s t a n t t h e r m a l p r o p e r t i e s . T h i s i s h i g h l y i m p o r t a n t f o r s o i l s because s o i l t h e r m a l p r o p e r t i e s a re s t r o n g l y dependent on s o i l w a t e r c o n t e n t and b u l k d e n s i t y , bo th o f w h i c h u s u a l l y v a r y w i t h p o s i t i o n , e s p e c i a l l y w i t h depth near the s u r f a c e . T a b l e 1.2 D i f f e r e n c e s between the e x a c t and the n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t = 12 h f o r Case II a t the i n d i c a t e d d e p t h s . Depth z(m) C r i t e r i a 0 . 00 0 . 01 0 .05 0 .10 0 .20 0 .40 Low thermal p r o p e r t i e s - 0 . 15 - 0 . 15 - 0 . 1 2 - 0 . 0 7 - 0 . 0 6 0 .001 <0. 29>* <0. 26> <0.16> <0.12> <-0.08> < -0.03> High thermal p r o p e r t i e s - 0 . 14 - 0 . 13 - 0 . 0 8 - 0 . 0 3 - 0 . 0 4 0 .006 <0. 28> <0. 26> <0.21> <0.20> <0.15> < -0.04> V a r i a b l e thermal p r o p e r t i e s - 0 . 16 0 . 004 0 .02 0 .006 - 0 . 0 5 - 0 . 0 0 2 <0. 28> <0. 18> <0.15> <0.17> <0.12> < -0.04> *The number 1n b r a c k e t < > 1s the maximum e r r o r a t t h a t dep th d u r i n g a 24 h p e r i o d . 28 In the prob lems c o n s i d e r e d the heat f l u x d e n s i t i e s a re g r e a t e s t a t the s u r f a c e . The t e m p e r a t u r e g r a d i e n t used to c a l c u l a t e the s u r f a c e heat f l u x d e n s i t y 1n the n u m e r i c a l p r o c e d u r e i s e s t i m a t e d w i t h the i = 1 and 1 = 2 n o d e s . T h e r e f o r e , i t i s the g r i d s i z e near the s u r f a c e t h a t m o s t l y d e t e r m i n e s the e r r o r s , i n p a r t i c u l a r when the f l u x d i v e r g e n c e i s l a r g e . For e x a m p l e , i f the g r i d s i z e i s 0 .01 m i n s t e a d of 0 .005 m near the s u r f a c e the maximum e r r o r a t the s u r f a c e and a t z = 0 .01 m i n c r e a s e s f rom 0 . 2 9 ° C and 0 . 2 6 ° C t o 0 . 5 8 ° C and 0 . 5 1 ° C , r e s p e c t i v e l y , o r a l m o s t t w o f o l d . The maximum e r r o r s r e p o r t e d h e r e i n m o s t l y o c c u r r e d between 20 h to 2 h when the s u r f a c e f l u x d i v e r g e n c e was h i g h and the e r r o r s s u b s e q u e n t l y d e c r e a s e d w i t h a d e c r e a s e i n f l u x d i v e r g e n c e . T h i s i n d i c a t e s t h a t the e r r o r s can be reduced f u r t h e r by d e c r e a s i n g the g r i d s i z e a l t h o u g h t h i s w i l l r e q u i r e a p r o p o r t i o n a l r e d u c t i o n 1n the t ime s t e p to keep the r e s u l t s p h y s i c a l l y r e a l i s t i c . The c o m p u t a t i o n c o s t w i l l then i n c r e a s e and a compromise i s r e q u i r e d between c o s t and a c c u r a c y . Note t h a t t h e low t h e r m a l p r o p e r t y case r e q u i r e d 36 days to meet the s p e c i f i e d p e r i o d i c i t y c r i t e r i o n w h i l e the v a r i a b l e and h i g h t h e r m a l p r o p e r t y cases r e q u i r e d 25 d a y s . Case III. Two-dimensional Horizontally Periodic Problems The method i s t e s t e d w i t h the f o l l o w i n g t w o - d i m e n s i o n a l p r o b l e m s : (1) G z ( 0 , x , t ) + 10 T ( 0 , x , t ) = { [ 1 8 7 . 5 + 112 .5 s i n ( v x - i r ) ] + [150 + 100 s i n ( v x - n ) ] s i n ( w t - TT / 2 ) } W m " 2 , 29 where Y = 2 i r / X and X = 0 .4 m (11) T ( Z , x , t ) = 1 0 ' C , (111) T ( z , x , 0 ) = 1 0 ° C , w i t h t h e r m a l p r o p e r t i e s as i n Case I I ( i ) and ( i i ) . Az v a r i e d as f o r Case I I and Ax was 0 .02 m t h r o u g h o u t . The d i f f e r e n c e s between the t e m p e r a t u r e s a t t = 12 h c a l c u l a t e d w i t h the e x a c t s o l u t i o n (see A p p e n d i x I I ) and those f rom the n u m e r i c a l p r o c e d u r e f o r z = 0 . 0 0 , 0 . 0 1 , 0 . 0 5 , 0 . 1 0 , 0 .20 and 0 .40 m and f o r x = 0 . 0 0 , 0 . 1 0 , 0 .20 and 0 .30 m are p r e s e n t e d i n T a b l e 1 . 3 . Note t h a t due to the h o r i z o n t a l p e r i o d i c i t y the e r r o r s a t x=0 and x=0.40 m are equa l a t a l l dep ths and t h e r e f o r e o n l y one i s l i s t e d . The maximum e r r o r s a t t h e s e d e p t h s and h o r i z o n t a l p o s i t i o n s d u r i n g the 24 h p e r i o d are a l s o r e p o r t e d . The r e s u l t s o b t a i n e d w i t h the ADE method a re In good agreement w i t h the e x a c t s o l u t i o n . The maximum e r r o r f o r the low t h e r m a l p r o p e r t y case i s h i g h e r than the h i g h t h e r m a l p r o p e r t y case a t d e p t h . The low t h e r m a l p r o p e r t y case s t i l l i n c l u d e s the r e s i d u a l t r a n s i e n t a t g r e a t e r d e p t h s . T h i s e r r o r can be reduced f u r t h e r but because of the v e r y s low change ( ~ 0 . 0 1 ° C f o r a 24 h p e r i o d ) d u r i n g the f i n a l s t age and i t s low magn i tude compared to the e r r o r s towards the s u r f a c e i t was not c o n s i d e r e d e s s e n t i a l . When i n i t i a l i z e d to 10 °C the h i g h t h e r m a l p r o p e r t y case r e q u i r e d 24 days w h i l e the low t h e r m a l p r o p e r t y case r e q u i r e d 6-10 days more to meet the c r i t e r i o n f o r p e r i o d i c i t y i n t i m e . In a d d i t i o n , the comments made f o r the o n e - d i m e n s i o n a l cases r e g a r d i n g the g r i d s i z e towards the T a b l e 1.3 D i f f e r e n c e s between the exac t and the n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s at t = 12 h f o r Case I I I a t the i n d i c a t e d dep ths and h o r i z o n t a l p o s i t i o n s . Low thermal p r o p e r t i e s High thermal p r o p e r t i e s Depth H o r i z o n t a l p o s i t i o n x(m) H o r i z o n t a l p o s i t i o n x(m) z(m) 0.00 0 .10 0 .20 0 .30 0 .00 0 .10 0 .20 0 .30 0 .00 - 0 . 0 8 <0.17>* 0.14 <0.19> - 0 . 0 8 <0.17> - 0 . 2 9 <-0.38> - 0 . 1 0 <0.15> 0.11 <0.19> - 0 . 1 0 <0.15> - 0 . 3 1 <-0.32> 0.01 - 0 . 0 7 <0.15> 0.12 <0.17> - 0 . 0 8 <0.15> - 0 . 2 7 <-0.31> - 0 . 0 9 <0.14> 0.09 <0.17> - 0 . 0 9 <0.14> - 0 . 2 7 <-0.28> 0.05 -0 .01 <0.10> 0.08 <0.13> - 0 . 0 2 <0.08> - 0 . 1 1 <-0.12> - 0 . 0 6 <0.11> 0.04 <0.13> - 0 . 0 6 <0.11> - 0 . 1 6 <-0.16> 0.10 - 0 . 0 2 <0.16> 0 .03 <0.14> - 0 . 0 0 2 <0.13> - 0 . 0 5 <0.15> - 0 . 0 4 <0.10> 0 .005 <0.11> - 0 . 0 4 <0.10> - 0 . 0 8 <0.10> 0.20 - 0 . 0 2 <0.16> 0.01 <0.14> -0 .007 <0.14> - 0 . 0 5 <0.16> - 0 . 0 4 <0.08> - 0 . 0 2 <0.08> - 0 . 0 4 <0.08> - 0 . 0 6 <0.08> 0.40 0.07 <0.07> 0.07 <0.07> 0.07 <0.07> 0.07 <0.07> 0 .004 <-0.02> 0 .005 <-0.02> 0.004 <-0.02> 0.004 <-0.02> *The number i n b r a c k e t < > i s the maximum e r r o r a t t h a t l o c a t i o n d u r i n g a 24 h p e r i o d . 31 s u r f a c e and the t ime o f maximum e r r o r a p p l y to the two-d i m e n s i o n a l cases as w e l l . Case IV. Variable Grid Spacing in the Horizontal Direction I t has been d e m o n s t r a t e d i n Case I t h a t w i t h a v a r i a b l e g r i d s p a c i n g 1n the v e r t i c a l d i r e c t i o n scheme 3 p e r f o r m e d as w e l l as the scheme w i t h u n i f o r m g r i d s p a c i n g and i t was adopted f o r subsequent a p p l i c a t i o n s . In o r d e r to examine the e f f e c t s o f v a r i a b l e g r i d s p a c i n g i n the h o r i z o n t a l d i r e c t i o n scheme 3 i s a p p l i e d to the p r o b l e m s t a t e d i n Case I I I w i t h t h e r m a l p r o p e r t i e s as 1n Case 1 1 ( 1 ) . The g r i d s p a c i n g i n the v e r t i c a l d i r e c t i o n remained as d e s c r i b e d under Case I I and the g r i d s p a c i n g used 1n the h o r i z o n t a l d i r e c t i o n was 0 .02 m f o r the 1s t 0 .04 m, 0 .01 m f o r the nex t 0 .05 m, 0 .005 m f o r the nex t 0 .02 m, 0 .01 m f o r the nex t 0 .18 m, 0 .005 m f o r the next 0 .02 m, 0 .01 m f o r the nex t 0 .05 m, and 0 .02 m f o r the l a s t 0 .04 m. The d i f f e r e n c e s between the t e m p e r a t u r e s a t t = 12 h c a l c u l a t e d w i t h the a n a l y t i c a l s o l u t i o n and t h o s e from the n u m e r i c a l p r o c e d u r e f o r z = 0 . 0 , 0 . 0 1 , 0 . 0 5 , 0 . 1 0 , 0 . 2 0 , and 0 .40 m and f o r x = 0 . 0 0 , 0 . 1 0 , 0 . 2 0 , and 0 .30 m are p r e s e n t e d i n T a b l e 1 .4 . The maximum e r r o r s a t t h e s e d e p t h s and h o r i z o n t a l p o s i t i o n s d u r i n g the 24 h p e r i o d a re a l s o r e p o r t e d . The d i f f e r e n c e s i n maximum e r r o r u s i n g Ax u n i f o r m and Ax v a r i a b l e f o r the depths and h o r i z o n t a l p o s i t i o n s m e n t i o n e d above a re a l s o p r e s e n t e d . The e r r o r s i n g e n e r a l i n c r e a s e d near the s u r f a c e (0 - 0 .05 m) and d e c r e a s e d a t dep th ( 0 . 1 0 - 0 .20 m) . The T a b l e 1.4 D i f f e r e n c e s b e t w e e n t h e e x a c t and t h e n u m e r i c a l l y c a l c u l a t e d t e m p e r a t u r e s a t t = 12 h and i n t h e maximum e r r o r s 1n t e m p e r a t u r e b e t w e e n t h e n u m e r i c a l p r o c e d u r e s w i t h Ax c o n s t a n t and Ax v a r i a b l e f o r C a s e IV a t t h e i n d i c a t e d d e p t h s and h o r i z o n t a l p o s i t i o n s . D i f f e r e n c e s b e t w e e n t h e e x a c t D i f f e r e n c e s i n t h e maximum s o l u t i o n and t h e n u m e r i c a r me thod e r r o r s b e t w e e n t h e n u m e r i c a l w i t h v a r i a b l e Ax p r o c e d u r e s w i t h Ax c o n s t a n t and Ax v a r i a b l e D e p t h H o r i z o n t a l p o s i t i o n x(m) H o r i z o n t a l p o s i t i o n xXm) z (m) 0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 0 0 - 0 . 0 8 <0.18> 0 . 1 4 * <0.20> - 0 . 0 8 <0.18> - 0 . 3 0 < -0 .40> - 0 . 0 1 - 0 . 0 1 - 0 . 0 1 0 . 0 2 0 . 0 1 - 0 . 0 8 <0.16> 0 . 1 2 <0.19> - 0 . 0 8 <0.16> - 0 . 2 9 < -0 .34> - 0 . 0 1 - 0 . 0 2 - 0 . 0 1 0 . 0 3 0 . 0 5 - 0 . 0 5 <0.11> 0 . 0 8 <0.15> - 0 . 0 5 <0.11> - 0 . 1 7 < -0 .18> - 0 . 0 1 - 0 . 0 2 - 0 . 0 3 0 . 0 6 0 . 1 0 - 0 . 0 2 <0.11> 0 . 0 5 <0.13> 0 . 0 0 3 <0.10> - 0 . 0 6 <0.08> 0 . 0 5 0 . 0 1 0 . 0 3 0 . 0 7 0 . 2 0 0 . 0 3 <0.10> 0 . 0 5 <0.11> 0 . 0 4 <0.09> 0 . 0 1 <0.09> 0 . 0 6 0 . 0 3 0 . 0 5 0 . 0 7 0 . 4 0 0 . 0 8 <0.08> 0 . 0 8 <0.08> 0 . 0 8 <0.08> 0 . 0 7 <0.07> - 0 . 0 1 - 0 . 0 1 - 0 . 0 1 0 . 0 0 The number i n b r a c k e t < > i s t h e maximum e r r o r a t t h a t 1 o c a t i on d u r i n g a 24 h p e r i o d 33 maximum v a r i a t i o n i n e r r o r due to the change i n Ax from u n i f o r m t o v a r i a b l e was ± 0.18% ( o v e r the s u r f a c e t e m p e r a t u r e range o f 3 7 . 0 ° C ) and we, t h e r e f o r e , i n f e r t h a t the scheme 3 p e r f o r m e d as w e l l as the scheme u s i n g Ax u n i f o r m t h r o u g h o u t . Case V. Temperature Dependence of Thermal Conductivity An a n a l y t i c a l s o l u t i o n f o r T when the t h e r m a l c o n d u c t i v i t y Is a f u n c t i o n of t e m p e r a t u r e i s not a v a i l a b l e . However , when e v a l u a t i n g k as k (T) f o r s o i l s a f t e r de V r i e s ( 1 9 6 3 ) , f o r w h i c h the dependence on T i s not s t r o n g , no d i f f i c u l t y was e x p e r i e n c e d i n m e e t i n g the c o n v e r g e n c e c r i t e r i o n o f U n +] - u j ^ ( o r V ^ J - V n j ) < 0 . 5 ° C , f o l l o w i n g the i t e r a t i v e p r o c e d u r e as o u t l i n e d p r e v i o u s l y , w i t h Case I . B . C u r y i J M n e a r_Up_p_e r_B o u n da ry_ An a n a l y t i c a l s o l u t i o n f o r the c u r v e d upper boundary p r o b l e m w i t h w h i c h the n u m e r i c a l r e s u l t s can be compared i s not a v a i l a b l e . U n t i l the n u m e r i c a l method i s t e s t e d a g a i n s t f i e l d measurements we can d e m o n s t r a t e o n l y r e a s o n a b l e n e s s to some d e g r e e . In so d o i n g , the scheme i s a p p l i e d to s o l v e the p r o b l e m : (i) G N ( x , t ) + 10 T(Z , x , t ) = 300 W m + 250 W m " 2 s i n ( w t - n / 2 ) , -2 ( i i ) T ( Z , x , t ) = 1 0 ° C , ( i i i ) T ( z , x , 0 ) = 1 0 6 C , 34 w i t h t h e r m a l p r o p e r t i e s as 1n Case I I ( i ) . T h i s p r o b l e m i s I d e n t i c a l to Case I I ( i ) e x c e p t f o r the c u r v e d upper b o u n d a r y . I t e n a b l e s us to compare the t e m p e r a t u r e d i s t r i b u t i o n i n the c u r v i l i n e a r domain w i t h the o n e - d i m e n s i o n a l case a t comparable d e p t h s , as w e l l as the t e m p e r a t u r e d i s t r i b u t i o n a t the p o i n t s of symmetry w i t h i n the c u r v i l i n e a r domain I t s e l f . The p a r a m e t e r s used 1n e q u a t i o n (14) a re H c = 0 . 1 3 m, X = 0 .32 m and $ c = X / 4 . Az v a r i e d as f o l l o w s : 0 .01 m f o r the 1 s t 0 .17 m, 0 .02 m f o r the next 0 .08 m, 0 .05 m f o r the nex t 0 .25 m, and 0 .10 m f o r r e s t o f the c o l u m n , w h i l e Ax was 0 .01 m t h r o u g h o u t . I t i s r e c o g n i z e d t h a t the ADE method i s u n c o n d i t i o n a l l y s t a b l e f o r r e c t a n g u l a r domains w i t h c o n s t a n t t h e r m a l p r o p e r t i e s but c a l c u l a t i o n f o r the c u r v e d upper boundary shows t h a t the method i s s t a b l e o n l y f o r A i n e q u a t i o n (1) l e s s than a p p r o x i m a t e l y 10 . The p r o b l e m t r e a t e d here has c o r n e r nodes i n some rows w i t h d i m e n s i o n s a l m o s t n e g l i g i b l e compared to the minimum Az or A x . For e x a m p l e , a t 1 = 10 ( F i g 1.2) w i t h Az = Ax = 0 .01 m, aAz and bAx a re r e s p e c t i v e l y 0 .0001065 m and 0 .0001255 m, and A becomes e x t r e m e l y l a r g e w i t h a modest -7 2 -1 t h e r m a l d i f f u s i v i t y ( s a y , 5 x 10 m s , a t y p i c a l v a l u e f o r a loam s o i l ) f o r a At as s m a l l as 1 m1n. To a p p l y the ADE method i n such a case At must be v e r y s m a l l t o o b t a i n p h y s i c a l l y r e a l i s t i c r e s u l t s and t h e r e f o r e the c o s t s become p r o h i b i t i v e . There a re two ways to overcome t h i s d i f f i c u l t y : (1) to n e g l e c t those boundary nodes and c o n s i d e r the n e a r e s t i n t e r i o r node as the boundary node and (2) to i n t e r p o l a t e the n e a r e s t i n t e r i o r node ( 1 , j ) l i n e a r l y between the next i n t e r i o r node ( i , j + l ) and the s u r f a c e boundary node ( i - l , j - l ) (see 35 F i g 1 . 3 b ) . In o t h e r w o r d s , u"+j i s r e p l a c e d w i t h . . . b 1 n+1 1 . . . u ?!3 - F ~ + " T " i . j + i • F 7 - - 1 "T-i.j-i- <19> T h i s a d d i t i o n a l i n t e r p o l a t i o n 1s r e q u i r e d f o r 1 = 3 , 4 , 9 , 10 and 13 f o r the U i j f u n c t i o n w h i l e f o r the V i j f u n c t i o n 1t 1s a l s o r e q u i r e d f o r 1 = 8 a t l a r g e A t . The I n t e r p o l a t i o n f o r i = 8 f o r the j f u n c t i o n does not i n f l u e n c e the r e s u l t s . W i t h A l a r g e r than what I t i s a t 1 = B o r 9 t h i s k i n d o f i n t e r p o l a t i o n f o r the case o f two c o n s e c u t i v e edge nodes w i t h i n t e r s e c t i o n s i n the v e r t i c a l segments (1 = 14) d e s t a b i l i z e s the m e t h o d , i n s t e a d o f i m p r o v i n g i t . Such b e h a v i o u r i s d i f f i c u l t to e x p l a i n w i t h o u t a s t a b i l i t y a n a l y s i s . I t 1s found t h a t a f t e r a l l t h e s e i n t e r p o l a t i o n s the V , . f u n c t i o n 1s ' »J s t a b l e w i t h At two t i m e s l a r g e r than t h a t f o r the U . . ' »J f u n c t i o n . The c a l c u l a t e d s u r f a c e t e m p e r a t u r e s f o r a 24 h p e r i o d a t the peak , i n the f u r r o w on the l e f t s i d e , a t the m i d p o i n t of the r i d g e on the l e f t s i d e , and from Case 11(1) a re shown i n F i g 1 . 4 . The maximum t e m p e r a t u r e o c c u r s a t t = 13 h i n a l l c a se s and i s h i g h e s t a t the peak , f o l l o w e d by the m i d p o i n t , then the o n e - d i m e n s 1 o n a l c a s e , and f i n a l l y the f u r r o w . The minimum t e m p e r a t u r e o c c u r s a t t = 1 h and 1s h i g h e s t f o r the f u r r o w , f o l l o w e d by o n e - d i m e n s i o n a l c a s e , then m i d p o i n t of the r i d g e and f i n a l l y the peak . Because o f the shape o f the 36 0 4 8 12 16 20 24 TIME (h) F i g u r e 1.4 N u m e r i c a l l y c a l c u l a t e d s u r f a c e t e m p e r a t u r e s a t the peak , a t the m i d p o i n t o f the r i d g e on the l e f t s i d e , 1'n the f u r r o w on the l e f t s i d e , and from Case 11(1) f o r a 24 h p e r i o d . 37 r i d g e , d u r i n g the p e r i o d of maximum energy i n p u t the heat f l u x c o n v e r g e s be low the peak of the r i d g e w h i l e i t d i v e r g e s below the f u r r o w , thus r e s u l t i n g 1n a h i g h e r t e m p e r a t u r e a t the peak than the f u r r o w ; the r e v e r s e 1s t r u e when the s u r f a c e heat f l u x d e n s i t y 1s n e g a t i v e . The s u r f a c e t e m p e r a t u r e a t the m i d p o i n t o f the r i d g e l i e s between those o f the peak and the f u r r o w and 1s v e r y c l o s e to the o n e - d i m e n s i o n a l c a s e ; e v i d e n t l y c o n v e r g e n c e and d i v e r g e n c e a t t h a t p o i n t i s m i n i m a l . The d i f f e r e n c e i n maximum and minimum t e m p e r a t u r e s between the peak of the r i d g e and the o n e - d i m e n s i o n a l case i s about 3 ° and 2 . 5 ° C , r e s p e c t i v e l y . There 1s a n e g l i g i b l y s m a l l d i f f e r e n c e (< 0 . 0 5 ° C ) i n t e m p e r a t u r e between the l e f t and r i g h t s i d e s of the r i d g e , as e x p e c t e d from the symmetry o f the boundary c o n d i t i o n s . A c c o r d i n g to the se r e s u l t s the program and i n t e r p o l a t i o n t e c h n i q u e s used to a v o i d i n s t a b i l i t y seem to be v e r y r e a s o n a b l e . 1.4 C o n c l u s i o n s The a l t e r n a t i n g d i r e c t i o n e x p l i c i t method has been e x t e n d e d to s o l v e t w o - d i m e n s i o n a l h e a t - c o n d u c t i o n problems t h a t a r e p e r i o d i c i n space and t i m e , i n w h i c h the upper boundary i s c u r v i l i n e a r , and i n w h i c h the t h e r m a l p r o p e r t i e s a r e n o n u n i f o r m w i t h p o s i t i o n , t i m e , and t e m p e r a t u r e . The 38 m e t h o d was a l s o e x t e n d e d t o i n c o r p o r a t e n o n u n i f o r m g r i d s p a c i n g . T e m p e r a t u r e s c a l c u l a t e d w i t h t h e m e t h o d w e r e f o u n d t o be i n e x c e l l e n t a g r e e m e n t w i t h a n a l y t i c a l s o l u t i o n s f o r v a r i o u s o n e - ( i n c l u d i n g a n o n h o m o g e n e o u s med ium) and t w o -d i m e n s i o n a l p e r i o d i c s o l u t i o n s on r e c t a n g u l a r d o m a i n s . When t h e u p p e r b o u n d a r y was c u r v i l i n e a r ( s i n u s o i d a l ) and when k and C d e p e n d e d on T , t h e c a l c u l a t e d t e m p e r a t u r e s w e r e m a t h e m a t i c a l l y and p h y s i c a l l y r e a s o n a b l e . 1 .5 R e f e r e n c e s 1. B a r a k a t , H . Z . and J . A . C l a r k . On t h e s o l u t i o n o f t h e d i f f u s i o n e q u a t i o n s by n u m e r i c a l m e t h o d s . J . H e a t T r a n s f e r 8 8 : 4 2 1 - 4 2 7 , 1 9 6 6 . 2 . C a r s l a w , H . S . , and J . C . J a e g e r . C o n d u c t i o n o f H e a t i n S o l i d s , C l a r e n d o n P r e s s , p . 6 4 , 1 9 5 9 . 3 . de V r i e s , D . A . T h e r m a l p r o p e r t i e s o f s o i l s , i n W . R . Van W i j k ( e d . ) , P h y s i c s o f P l a n t E n v i r o n m e n t . N o r t h - H o l l a n d P u b l . C o . , C h a p . 7 , p p . 2 1 0 - 2 3 5 , 1 9 6 3 . 4 . H i l l e l , D. F u n d a m e n t a l s o f S o i l P h y s i c s , A c a d e m i c P r e s s , C h . 1 2 , p p . 2 8 7 - 3 1 7 , 1 9 8 0 . 5 . L a r k i n , B . K . Some s t a b l e e x p l i c i t d i f f e r e n c e a p p r o x i m a t i o n s t o t h e d i f f u s i o n e q u a t i o n . Ma th o f C o m p u t a t i o n 1 8 : 1 9 6 - 2 0 2 , 1 9 6 4 . 6 . O z i s i k , M . N . H e a t C o n d u c t i o n , J o h n W i l e y & S o n s , C h . 1 2 , p p . 4 7 1 - 5 2 1 , 1 9 8 0 . 39 CHAPTER 2 OBSERVED AND PREDICTED SURFACE ENERGY BALANCES AND SOIL TEMPERATURES FOR UNIFORM AND PARTIALLY MULCHED SURFACES 40 OBSERVED AND PREDICTED SURFACE ENERGY BALANCES AND SOIL TEMPERATURES FOR UNIFORM AND PARTIALLY MULCHED SURFACES 2 . 1 I n t r o d u c t i o n In a g r i c u l t u r a l p r a c t i c e , m u l c h i n g i s a consequence of r e d u c e d or c o n s e r v a t i o n t i l l a g e ( a l s o synonymous ly c a l l e d minimum or z e r o t i l l a g e ) t h a t l e a v e s v a r y i n g amounts of c rop r e s i d u e on the s o i l s u r f a c e a f t e r the s e e d i n g / p l a n t i n g o p e r a t i o n has been c o m p l e t e d . The c o n s e r v a t i o n t i l l a g e p r a c t i c e s t h a t began 1n the 1950 ' s have i n c r e a s e d a t an a c c e l e r a t e d r a t e t h r o u g h the 1980 ' s ( G r i f f i t h , 1 9 8 5 ) , m a i n l y because o f t h e i r e f f e c t i v e n e s s i n p r o t e c t i n g s o i l f rom e r o s i o n i n humid a r e a s and i n c o n s e r v i n g s o i l m o i s t u r e i n a r i d a r e a s . More r e c e n t l y , h i g h e r c o s t s of e n e r g y , equipment and l a b o u r have p r o v i d e d e x t r a I n c e n t i v e 1n the s w i t c h to c o n s e r v a t i o n t i l l a g e . I t i s e s t i m a t e d t h a t c o n s e r v a t i o n t i l l a g e p r a c t i c e s w i l l be used on up to 95% of U . S . c r o p l a n d s by 2010 ( M e y e r s , 1 9 8 3 ) . The e x p a n s i o n o f minimum t i l l a g e sys tems can o n l y o c c u r i f the a s s o c i a t e d a d v e r s e e f f e c t s can be e i t h e r e l i m i n a t e d , a v o i d e d , or a t l e a s t reduced to an a c c e p t a b l e l e v e l . In C a n a d a , s o i l t e m p e r a t u r e s and m o i s t u r e c o n t e n t s o f t e n l i m i t p l a n t g rowth ( B a i e r and Mack, 1 9 7 5 ) . In the n o r t h e r n i n t e r i o r r e g i o n s o f B r i t i s h C o l u m b i a both low s o i l t e m p e r a t u r e s and m o i s t u r e o f t e n a f f e c t y i e l d l e v e l s ( W i l l i a m s et a l , 1 9 8 0 ) . 41 The I n c r e a s e d amounts o f c r o p r e s i d u e s on the s o n a s s o c i a t e d w i t h z e r o and minimum t i l l a g e p r a c t i c e s r e t a r d the warming of the s o i l i n the s p r i n g because of the i n s u l a t i n g and r e f l e c t i n g e f f e c t s o f the r e s i d u e . T h i s s h o r t e n s the l e n g t h o f the g r o w i n g season and reduces c rop y i e l d s i n more n o r t h e r l y l a t i t u d e s (Lemon, 1956; Gauer e t a l , 1 9 8 2 ) , thus l i m i t i n g the a d o p t i o n o f the se t i l l a g e p r a c t i c e s i n the p r a i r i e s . U n t i l r e c e n t l y t h e r e appear s to have been l i m i t e d r e s e a r c h on the f l o w o f mass and energy under c r o p r e s i d u e m u l c h e s . B r i s t o w et a l . ( 1 9 8 3 ) , S h a f f e r e t a l . ( 1 9 8 3 ) , Ross e t a l . (1983 a , b ) and Chung and H o r t o n (1987) have p r e s e n t e d models t o s i m u l a t e the vapour and heat f l o w i n the s o i l -r e s i d u e - a t m o s p h e r e s y s t e m . B r i s t o w et a l . d e a l t w i t h a wheat -s t r a w r e s i d u e , S h a f f e r e t a l . w i t h c o r n r e s i d u e and Ross et a l . w i t h c h e m i c a l l y k i l l e d s t a n d i n g p a s t u r e . Chung and H o r t o n were c o n c e r n e d w i t h an opaque mulch c o v e r , i . e . , o n l y the r e f l e c t i v i t y o f the mulch was used i n the m o d e l . T h e i r model e x t e n d s H o r t o n et a l . ' s (1984 b) model by i n c l u d i n g w a t e r f l o w and by r e p l a c i n g the canopy s h a d i n g p o r t i o n w i t h a p a r t i a l l y mulched s u r f a c e c o n d i t i o n . B r i s t o w e t a l . and Ross et a l . s e p a r a t e d the sys tem i n t o l a y e r s , n a m e l y , a t m o s p h e r i c , r e s i d u e and s o i l l a y e r s , and e v a l u a t e d the t r a n s p o r t o f mass and energy 1n the a t m o s p h e r i c l a y e r by b u l k t r a n s f e r e q u a t i o n s . Both groups d i v i d e d the r e s i d u e l a y e r I n t o s u b l a y e r s and e v a l u a t e d the t r a n s p o r t of vapour and heat w i t h "eddy d i f f u s i v i t y " e q u a t i o n s ; however , 42 B r i s t o w e t a l . i g n o r e d the t r a n s p o r t o f heat by f r e e c o n v e c t i o n . S h a f f e r et a l . a v o i d e d c o n v e c t i v e t r a n s p o r t 1n the a t m o s p h e r i c and r e s i d u e l a y e r s ; they used a s t a t i s t i c a l r e g r e s s i o n f o r m u l a to c a l c u l a t e the upper boundary s o i l t e m p e r a t u r e s f rom a i r t e m p e r a t u r e s measured a t 2 m h e i g h t , and then c a l c u l a t e d the s o i l t e m p e r a t u r e p r o f i l e s n u m e r i c a l l y a f t e r Hanks e t a l . (1971) as p a r t o f a management package f o r n i t r o g e n , t i l l a g e and c rop r e s i d u e s . The s o i l t e m p e r a t u r e submodel i n c o r p o r a t e d i n S h a f f e r et a l . ' s model has been p u b l i s h e d p r e v i o u s l y e l s e w h e r e (Gupta et a l . , 1981 and 1 9 8 2 ) . Chung and H o r t o n d i d not c o n s i d e r the e f f e c t s of e i t h e r a t m o s p h e r i c s t a b i l i t y , s h a d i n g o f the bare s t r i p by the m u l c h , or c o n v e c t i v e t r a n s p o r t w i t h i n the mulch l a y e r . M o d i f i c a t i o n o f s o i l t e m p e r a t u r e reg imes i s s i g n i f i c a n t i n many e n v i r o n m e n t s , e s p e c i a l l y where the c l i m a t e 1s m a r g i n a l f o r a g r i c u l t u r a l c r o p s or d e v e l o p i n g p l a n t c o v e r s o f any t y p e . T o d a y ' s n o r t h e r n f a r m e r s f a c e a di lemma In s e l e c t i n g a t i l l a g e p r a c t i c e t h a t c o n t r o l s s o i l e r o s i o n and c o n s e r v e s s o i l m o i s t u r e , y e t t h a t warms up the s o i l o p t i m a l l y e a r l y i n the s e a s o n . To a c h i e v e t h a t d e l i c a t e b a l a n c e i t may be p o s s i b l e to use narrow bare s t r i p s ( i n w h i c h the annua l c rop i s sown) a l t e r n a t i n g w i t h w i d e r mulch s t r i p s (see F i g . 2 . 1 ) ; a l s o i t may be t h a t a p a r t i c u l a r o r i e n t a t i o n o f the se s t r i p s i s p r e f e r a b l e . The o b j e c t i v e s o f t h i s c h a p t e r are t w o - f o l d : ( i ) To d e v e l o p a p h y s i c a l l y - b a s e d n u m e r i c a l model t h a t d e t e r m i n e s the energy b a l a n c e o f the a t m o s p h e r e - m u l c h -43 SUN H IT) M U L C H S U R F A C E B A ^ E 0.30 (m) •*-0-10 (n ) Z (•) T(z,x,t) . T(z,x+X,t) T(Z fx,t) = Constant 0.25 B A R E - 0.50 -X (m) (a) (b) F i g u r e 2 .1 A p a r t i a l v i ew of an i d e a l i z e d c o n f i g u r a t i o n o f 0 .30 m w ide mulch s t r i p a l t e r n a t i n g w i t h 0 .10 m bare s o i l s t r i p ; (a) the bare s t r i p 1s s h a d e d , and (b) the bare s t r i p 1s p a r t l y s u n l i t . The s o l i d square s I n d i c a t e the t h e r m o c o u p l e s p o s i t i o n s i n the z - x ( s o i l ) d o m a i n . 44 s o i l sy s tem and the s o i l t h e r m a l reg imes u s i n g s t a n d a r d m i c r o m e t e o r o l o g 1 c a l d a t a as i n p u t s , f o r s o i l s c o v e r e d e i t h e r u n i f o r m l y or In s t r i p s w i t h a m u l c h , ( i i ) To t e s t and c a l i b r a t e t h i s model w i t h m l c r o m e t e o r o l o g i c a l measurements made on p l o t s e s t a b l i s h e d a t the U . B . C . P l a n t S c i e n c e R e s e a r c h S t a t i o n , V a n c o u v e r ( 4 9 . 1 8 ° N ) , i n 1 9 8 3 - 8 5 . 2 . 2 T h e o r y A s s u m p t i o n s (1) The mulch i s r e p r e s e n t e d as a b u l k e d s i n g l e l a y e r w i t h c o n s t a n t s h o r t w a v e and longwave r a d i a t i v e p r o p e r t i e s . (2) The t e m p e r a t u r e ( T m ) and vapour d e n s i t y (p m ) o f t h i s l a y e r a re r e p r e s e n t a t i v e o f those o c c u r r i n g near the top o f the a c t u a l m u l c h , i . e . , a t the energy exchange s u r f a c e . (3) S o i l s u r f a c e heat and vapour f l u x d e n s i t i e s ( i . e . the f l u x e s i n the mulch) a re p r o p o r t i o n a l to the t e m p e r a t u r e and vapour d e n s i t y d i f f e r e n c e s , r e s p e c t i v e l y , between the s o i l s u r f a c e and the o v e r l y i n g mulch l a y e r , and i n v e r s e l y p r o p o r t i o n a l to e f f e c t i v e mulch t r a n s f e r r e s i s t a n c e s ( d e s c r i b e d l a t e r ) . 45 ( 4 ) The t e m p e r a t u r e s and vapour d e n s i t i e s a t v a r i o u s depths w i t h i n the mulch a re assumed t o v a r y l i n e a r l y between the c a l c u l a t e d mulch and s o i l s u r f a c e v a l u e s . Energy and vapour s t o r a g e e f f e c t s a re n e g l i g i b l e i n the mulch l a y e r . ( 5 ) Energy and vapour t r a n s p o r t i n the h o r i z o n t a l d i r e c t i o n i n the r e s i d u e and a t m o s p h e r i c l a y e r s , but not i n the s o i l , a re n e g l e c t e d . i u i l i £ i _ § O M n d . a i ^ _ C o n d i t i o n _ f o r _ M u l c h e d _ S u r f a c e s The energy b a l a n c e e q u a t i o n a t the s o i l - a i r i n t e r f a c e under a mulch i s R ° ( x , t ) - H Q ( x , t ) - L E Q ( x , t ) - G Q ( x , t ) = 0, (1) where R° Is the net r a d i a t i o n f l u x d e n s i t y (W m~ Z) ( p o s i t i v e - ? downward) , H Q i s the s e n s i b l e heat f l u x d e n s i t y (W m ) 2 ( p o s i t i v e u p w a r d ) , L E Q i s the l a t e n t heat f l u x d e n s i t y (W m ) ( p o s i t i v e u p w a r d ) , G Q i s the s o i l heat f l u x d e n s i t y (W m ) ( p o s i t i v e downward) , x i s the h o r i z o n t a l d i s t a n c e (m) and t i s the t i m e ( h ) . H Q i s g i v e n by C a [ T s ( x , 0 , t ) - T m ( x , t ) ] H 0 ( x , t ) = r , ( 2 ) m 46 w h e r e C . i s t h e v o l u m e t r i c h e a t c a p a c i t y ( a t c o n s t a n t a p r e s s u r e ) o f t h e a t m o s p h e r e ( 1 . 2 k J m C~ ) . T $ ( x , o , t ) i s t h e s o i l s u r f a c e t e m p e r a t u r e ( ° C ) , T m i s t h e e f f e c t i v e m u l c h t e m p e r a t u r e ( ° C ) and r^ i s t h e r e s i s t a n c e o f t h e m u l c h t o h e a t t r a n s f e r ( s m " 1 ) . A n a l o g o u s l y , L { P v [ T s ( x , 0 , t ) ] - p m ( x , t ) } L E Q ( x , t ) ? , (3) r m + r s * -3 w h e r e p v [ T $ ( x , o , t ) ] i s t h e s a t u r a t e d v a p o u r d e n s i t y ( k g m ) a t t h e s o i l s u r f a c e t e m p e r a t u r e , p m i s t h e e f f e c t i v e m u l c h v a p o u r d i f f u s i v i t y ( k g m ~ 3 ) , r^ i s t h e r e s i s t a n c e o f t h e -1 s m u l c h t o v a p o u r t r a n s f e r ( s m ) and r* i s t h e s o i l s u r f a c e r e s i s t a n c e t o e v a p o r a t i o n (s m - 1 ) . The n e t r a d i a t i o n f l u x d e n s i t y r e c e i v e d by t h e m u l c h ( f r o m b o t h a b o v e and b e l o w ) i s g i v e n by O x . t ) . S< - S « + S » - S j • I ? - L j + - , ( 4 ) w h e r e S a n d L d e n o t e s h o r t w a v e and l o n g w a v e r a d i a t i o n f l u x d e n s i t i e s , r e s p e c t i v e l y , and s u b s c r i p t t ( t o p ) r e f e r s t o f l u x d e n s i t i e s a b o v e t h e m u l c h , and t h e s u p e r s c r i p t s d a n d u r e f e r t o t h e i n c o m i n g and o u t g o i n g f l u x d e n s i t i e s , r e s p e c t i v e l y . The e n e r g y b a l a n c e e q u a t i o n a t t h e e f f e c t i v e m u l c h e n e r g y e x c h a n g e s u r f a c e i s 47 R j | ( x , t ) = H t ( x , t ) - H Q ( x , t ) + L E m ( x , t ) . (5) _ 2 Here L E m i s the l a t e n t heat f l u x d e n s i t y (W m ) ( p o s i t i v e upward) o f the mulch and i s g i v e n by L { p v [ T m ( x , t ) ] - p m ( x , t ) } L E m ( x . t ) = , (6) where r™ i s the r e s i s t a n c e of the mulch to e v a p o r a t i o n (s m " 1 ) ; t h i s a p p l i e s i f the mulch i s wet or i s a t r a n s p i r i n g canopy ( i n w h i c h case i s the canopy r e s i s t a n c e , McNaughton and J a r v i s , 1 9 8 3 ) . The t o t a l l a t e n t heat f l u x d e n s i t y a t any x i s the sum of the s o i l and mulch f l u x d e n s i t i e s , i . e . , L E . ( x . t ) = LE ( x , t ) + L E m ( x , t ) . (7) A n a l o g o u s t o e q u a t i o n ( 2 ) , H t ( x , t ) and L E t ( x , t ) a re g i v e n by H t ( x , t ) = r (8) and 48 L [ p ™ ( x . t ) - p v ( t ) ] L E . ( x , t ) = , (9) r a where T a ( £ ) and p w ( t ) a re the known v a r i a t i o n s i n a i r a v t e m p e r a t u r e and vapour d e n s i t y , r e s p e c t i v e l y , a t s c r e e n h e i g h t ( d a ) , and rj] and r^ are the a e r o d y n a m i c r e s i s t a n c e s to heat a a a and vapour t r a n s f e r , r e s p e c t i v e l y . C o m b i n i n g e q u a t i o n s ( 3 ) , ( 6 ) , (7) and ( 9 ) , and l i n e a r i z i n g the p * ( T ) f u n c t i o n as P j ( T ) - P * ( T ) + s p ( T - T) y i e l d P v ( t ) ? P ~SP P j ( x . t ) - { — + — — T s ( x , 0 , t ) + - - T m ( x , t ) + a m s s ( P * ( T ) - S T ) ( ~ ~ - ? + ~ ) } / (10) P r m + r s r s 1 1 1 r : r „ + r^ r™ a m s s where T i s the average s u r f a c e t e m p e r a t u r e o v e r a day and Ip i s the s l o p e o f the s a t u r a t e d vapour d e n s i t y c u r v e a t T . S u b s t i t u t i n g e q u a t i o n s ( 2 ) , (6) and (8) i n t o e q u a t i o n (5) and then c o m b i n i n g w i t h e q u a t i o n (10) one o b t a i n s R n = b l V * * * ) + b 2 T a ( t ) + b 3 p v ( t ) + b 4 T s ^ * 0 * * ) + b 5 ' (11) where b [ • c a <;* + :E> + ; i p c1 - l / < r s B i ' i r _ r m r _ a m s "5 • " L " r s rl Bl> b 4 r - - v! - " - 'p ' fT < r * + r s ' B i J "J • : s cpv<T> - " - ( - 7 -7 : 1 + ; i 5 ) / B i ) s m s s D _ I_ . I ' I_ 1 _v _v ^ ..s m * r , r m + r , r „ a m s s The s h o r t w a v e r a d i a t i o n f l u x d e n s i t y r e c e i v e d a t the s o i l s u r f a c e (under the mulch) and the o u t g o i n g r a d i a t i o n f l u x d e n s i t i e s f rom the s o i l and the mulch s u r f a c e a re g i v e n by = b j S d • b 2 S d (12) = b3 s d 50 where . s * m 1 = 1 n s _s 1 - a_ a_ s m b 2 = a s b j 3 m z m t ~ and &t a re the shor twave t r a n s m i s s 1 v i t y and r e f l e c t i v i t y of m m J the m u l c h , r e s p e c t i v e l y , and i s the s o i l a l b e d o . S i m i l a r l y , the longwave r a d i a t i o n f l u x d e n s i t y r e c e i v e d a t the s o i l s u r f a c e and the longwave r a d i a t i o n f l u x d e n s i t i e s o u t g o i n g from the s o i l and the mulch s u r f a c e a r e g i v e n by d L d L 4 L 4 L Q = b j L t + b 2 o T K s ( x , 0 , t ) + b 3 o T K m ( x , t ) L o - bi L t + b 5 ^ ( x . O . t ) + b^ o T 4 m ( x , t ) (13) L t - b 7 L ? + b 8 o T 4 s ( x , 0 , t ) + blQ o T 4 m ( x , t ) , where b 1 1 - a1 a L m s a L + (1 - fm) et ^ m s v m' s b  1 - a a m s 51 f m m 1 -• « i b 3 >i • « i "!i--I >i -I »l t m » and e m a re the t r a n s m i s s i v i t y , r e f l e c t i v i t y and m m m J J a b s o r p t i v i t y o f the m u l c h , r e s p e c t i v e l y , aj: and a re the r e f l e c t i v i t y and a b s o r p t i v i t y o f the s o i l , r e s p e c t i v e l y , f i s the f r a c t i o n of the mulch a t the e f f e c t i v e mulch t e m p e r a t u r e t h a t d i r e c t l y exchanges longwave energy w i t h the s o i l s u r f a c e b e l o w , and o ( = 5 . 6 7 x l 0 ~ 8 W m~ 2 K ~ 4 ) i s the S t e f a n - B o l t z m a n n c o n s t a n t , f v a r i e s between 0 and 1, 0 f o r a v i r t u a l l y opaque l a y e r ( e . g . , a deep s t r a w mulch) and 1 f o r a v i r t u a l l y t r a n s p a r e n t l a y e r ( e . g . , the a i r gap be low a p o l y e t h y l e n e f i l m ) . C o m b i n i n g the e q u a t i o n s (4) and (11) t h r o u g h ( 1 3 ) , 4 4 — 4 — 3 — and l i n e a r i z i n g the T .^ terms as T„ = T + 4T„ (T-T) y i e l d s T m ( x . t ) = b m S j + b m L? + b^ T a ( t ) + b m p v ( t ) + + b m T s ( x , 0 , t ) + b m , (14) w h e r e b l = > b®/B 2 b 2 = = bpB 2 b 3 = - - -5 / B 2 b 4 1 / B 2 b 5 = • <>5 ) / B 2 b 6 1 ) / B 2 B 2 . b l = = 1 " b j + >; b 2 = = 1 - >\ + b 3 = »l b 4 = b « . b§ 4oTj j 53 b 6 ' b 4 4 o T K = (b§ + b J ) o T 3 ( T K - 4T) For the s o i l s u r f a c e , the net r a d i a t i o n f l u x d e n s i t y i s found as R ° ( x , t ) . Sd0 - + L d 0 - L»0. (15) C o m b i n i n g the e q u a t i o n s ( 1 ) , ( 2 ) , ( 3 ) , ( 1 0 ) , (14) and ( 1 5 ) , 4 l i n e a r i z i n g the terms as p r e v i o u s l y , and r e a r r a n g i n g y i e l d s G ( x . o . t ) + c - J + - b m - - - - ^ - » ( b m + b m b m ) r m ^ + r s r m r m + r s 5 9 - [ ( b l ; - bjp + (bjj - b ^ ) b m ] 4 o T 3 } T s ( x , 0 , t ) = {(b\ - b » ) + ( b L - b L ) 4 o T 3 b m + -I b m r m + -Tk--1 bT bS>SJ + " bi> + <b3 " b 6 > 4 o T K b2 m s C a . m , L .m . m, , d , r / U L . L> . T 3 .m + " h b 2 + b 2 b 9 ^ L t + <<b3 - b 6 > 4 o T K b 3 m m s 54 C + " h b 3 + ~v s b 3 b 9 } T a ( t ) + { ( b 3 " b 6 ) 4 o T K b 4 r m r_ + r m m s C . a . m . L ,.m Km . m , , + Th b 4 + <b7 + b 4 b 9 > 3 P v ( t > r m r m + r s + { (b^ - b ^ ) b m 4 o T 3 + o T 3 ( T K - 4 T ) [ ( b ! j - bj:) + . / u L L L I - I a .m , L r . m .m , .m * / T \ + ( b 3 " b 6 ) ] + ~h b 6 + ~"~"s C b 6 b 9 + b 1 0 " ? v ( T > m m s + S p T ] } , (16) where m b S - V C ( r 2 + r!> B . i 3 b S - V < r s B!> bTo - <p^ ( T > - v , ( : v - ; ~ i + ^ ) / B i m S S For a s o i l c o v e r e d w i t h mulch s t r i p s , the t h e o r y a p p l i e s d i r e c t l y to the mulch l a y e r s . I t a l s o can a p p l y to the bare s t r i p s by s e t t i n g ri2 and r ^ , and t h i c k n e s s o f the mulch (H m ) m m v m' 55 to s u f f i c i e n t l y s m a l l but n o n - z e r o v a l u e s , s e t t i n g the s h o r t and longwave t r a n s m i s s i v i t i e s to 1 and i n c l u d i n g the s h a d i n g and v iew f a c t o r e f f e c t s due to the s u r r o u n d i n g mulch when s p e c i f y i n g S d and L d ( t o be d e s c r i b e d l a t e r ) . In C h a p t e r 1 an ADE f i n i t e d i f f e r e n c e model was d e v e l o p e d to d e t e r m i n e the s o i l t e m p e r a t u r e s i n r e c t a n g u l a r domains s u b j e c t to a c o n s t a n t t e m p e r a t u r e boundary c o n d i t i o n on the bo t tom s u r f a c e , h o r i z o n t a l l y p e r i o d i c c o n d i t i o n s on the v e r t i c a l s u r f a c e s , and a boundary c o n d i t i o n of the 3 rd k i n d on the upper s u r f a c e . S i n c e e q u a t i o n (16) i s o f t h i s form and the c o n d i t i o n s on the o t h e r s u r f a c e s a re a p p r o p r i a t e , the ADE model i s d i r e c t l y a p p l i c a b l e to the case o f a l t e r n a t i n g bare and mulch s t r i p s . The model a l s o a p p l i e s to e i t h e r bare or u n i f o r m l y mulched s o i l s by a s s i g n i n g the same ( a p p r o p r i a t e ) p a r a m e t e r s and t h e r m a l p r o p e r t i e s to both s t r i p s . 2 . 3 T e s t l n f l _ a n d_C al_j.b r a tj^o n_o f _ t h e_M o d §1 A . U n i f o rm_Mulch_Exp_eriments The s t u d y was c o n d u c t e d on the U . B . C . P l a n t S c i e n c e R e s e a r c h S t a t i o n from Sep tember , 1983 to J u l y , 1984. A 26 m x 26 m a r e a was p l o u g h e d , l e v e l e d and f i r m l y p a c k e d . The f i e l d was d i v i d e d i n t o f o u r equa l p l o t s , t h r e e o f w h i c h were u n i f o r m l y c o v e r e d w i t h b a r l e y s t r a w at r a t e s o f 2 , 10 , and 20 56 t / h a , w h i l e the f o u r t h was m a i n t a i n e d bare as a r e f e r e n c e . The s o i l 1s a Bose s e r i e s , c o n s i s t i n g o f loamy sand (83.2% s a n d , 10.0% s i l t and 6.8% c l a y ) f o r the upper 0 . 3 m, be low w h i c h i t 1s a sand (93.5% s a n d , 4.5% s11 t and 2.0% c l a y ) . The b u l k d e n s i t y was measured ( s t a n d a r d c o r e method) to be 1050 --3 -3 1200 kg m f o r the upper 0 - 0 .20 m and 1200 - 1300 kg m be low 0 . 2 m d e p t h . S o i l o r g a n i c m a t t e r c o n t e n t e s t i m a t e d by c o m b u s t i o n o f s o i l samples a t 4 0 0 ° C was 6-9% near the s u r f a c e and l e s s than 4% below 0 . 3 m depth on dry-mass b a s i s . The s u r f a c e energy b a l a n c e components and s o i l t h e r m a l and m o i s t u r e reg imes were m o n i t o r e d on a l l p l o t s o n l y u n t i l June 5 , 1984. The t e m p e r a t u r e measurement sys tem from the 2 t / h a p l o t was then I n s t a l l e d w i t h i n the 0 .06 m ( n o t i c e t h a t the t h i c k n e s s o f the mulch changed c o n s i d e r a b l y w i t h f i e l d a g i n g , T a b l e 2 . 1 ) t h i c k s t r a w mulch i n the 20 t / h a p l o t a t d e p t h s o f 0 . 0 0 8 , 0 . 0 1 6 , 0 . 0 2 4 , 0 . 0 3 6 , 0 .048 and 0 .060 m to m o n i t o r the t e m p e r a t u r e p r o f i l e i n the mulch l a y e r . However , m o n i t o r i n g o f w a t e r c o n t e n t s and o t h e r p h y s i c a l and c l i m a t i c v a r i a b l e s i n the 2 t / h a p l o t was c o n t i n u e d as u s u a l u n t i l the t e r m i n a t i o n of the f i e l d e x p e r i m e n t . I n s t a n t a n e o u s s o i l t e m p e r a t u r e s were measured h o u r l y i n q u a d r u p l i c a t e a t z = 0 .005 m f o r the bare and a t z = 0 . 0 0 5 , 0 . 0 1 , 0 .025 and 0 .05 m f o r the 2 , 10 and 20 t / h a p l o t s , and i n d u p l i c a t e a t z = 0 . 0 1 , 0 . 0 2 5 , 0 . 0 5 , 0 .10 and 0 .25 m f o r the bare and a t z = 0 .10 and 0 .25 m f o r the 2 , 10 and 20 t / h a p l o t s . Measurement a t 0 .50 m depth was not r e p l i c a t e d i n any o f the p l o t s . Two c o p p e r - c o n s t a n t a n t h e r m o c o u p l e s , c o n n e c t e d Tab le 2.1 Th ickness o f the mulch l a y e r ( In cm) w i t h t ime a t v a r i o u s a p p l i c a t i o n r a t e s f o r the u n i f o r m l y a p p l i e d mulch exper iment . Rate ( t / h a ) Date ' 2.0 10.0 20.0 1-12-83 1.0 5.3 8.9 17-2-84 0.9 4.5 7.6 15-5-84 0.8 4.2 7.0 14-6-84 0.8 4.0 6.0 6-7-84 0.8 4.0 6.0 58 i n p a r a l l e l w e r e p l a c e d a t t h e same d e p t h a b o u t 0 . 5 0 m a p a r t f o r t h e d u p l i c a t e , w h i l e a p a i r o f two t h e r m o c o u p l e s , c o n n e c t e d i n p a r a l l e l w e r e p l a c e d a t t h e same d e p t h a b o u t 0 . 5 0 m a p a r t f o r t h e q u a d r u p l i c a t e m e a s u r e m e n t s . T e m p e r a t u r e d i f f e r e n c e s b e t w e e n d e p t h s w e r e m e a s u r e d w i t h t h e r m o c o u p l e s w h i l e t h e a b s o l u t e t e m p e r a t u r e s a t 0 . 5 0 and 0 . 8 5 m d e p t h s w e r e m e a s u r e d w i t h F D - 3 0 0 s i l i c o n d i o d e s t o + 0 . 1 ° C . S u r f a c e t e m p e r a t u r e s w e r e p e r i o d i c a l l y m o n i t o r e d w i t h a B a r n e s P R T - 1 0 -L i n f r a r e d t h e r m o m e t e r ( I R T ) t o + 3 ° C . H o u r l y a v e r a g e s o f R R w e r e m e a s u r e d w i t h two S w i s s t e c o S - l n e t r a d i o m e t e r s m o u n t e d a t 1 m h e i g h t . One r a d i o m e t e r was p e r m a n e n t l y l o c a t e d on t h e b a r e p l o t w h i l e t h e s e c o n d was l o c a t e d on one o f t h e t h r e e m u l c h e d p l o t s , w i t h t h e t r a n s f e r f r o m one p l o t t o a n o t h e r o c c u r r i n g e v e r y 2 - 3 d a y s . H o u r l y a v e r a g e s o f S d and s" o r S ^ w e r e m e a s u r e d u s i n g K i p p and Zonen CM5 p y r a n o m e t e r s . The p y r a n o m e t e r s w e r e l o c a t e d on one o f t h e f o u r p l o t s , w i t h t h e t r a n s f e r o f t h e s" o r s|j p y r a n o m e t e r a l s o o c c u r r i n g e v e r y 2 - 3 d a y s . I n s t a n t a n e o u s a i r t e m p e r a t u r e s and v a p o u r p r e s s u r e s w e r e m o n i t o r e d h o u r l y u s i n g p s y c h r o m e t e r s b u i l t by t h e U . B . C . b i o m e t e o r o l o g y g r o u p . H o u r l y a v e r a g e s o f w i n d s p e e d and d i r e c t i o n a t 1 m h e i g h t w e r e m o n i t o r e d u s i n g a C a s e l l a a n e m o m e t e r and a C l i m e t 0 1 2 - 6 C w i n d v a n e , r e s p e c t i v e l y . M o i s t u r e c o n t e n t s w e r e o c c a s i o n a l l y m e a s u r e d g r a v i m e t r i c a l l y b o t h i n t h e s o i l and t h e s t r a w m u l c h . E v a p o r a t i o n r a t e s w e r e m e a s u r e d o n l y on J u n e 14 and J u l y 6 , 1984 by w e i g h i n g m i c r o - l y s i m e t e r s ( i n s t a l l e d t h e p r e v i o u s n i g h t , two r e p l i c a t e s p e r p l o t ) e v e r y two h o u r s d u r i n g t h e 59 d a y t i m e . The m i c r o - l y s i m e t e r s were c o n s t r u c t e d by t a p i n g t o g e t h e r two s o i l b u l k d e n s i t y c o r e s (each 76 mm l o n g and 74 mm 1 . d . ) . B o a s t and R o b e r t s o n (1982) r e p o r t e d t h a t f o r a m o l l i s o l and f o r " e v a p o r a t i v i t y " r a n g i n g from 2 to 9 mm/day a 70 mm l o n g m i c r o - l y s i m e t e r ( w i t h 76 mm i . d . ) p r o v i d e s an a c c u r a t e measurement to w i t h i n 0 .5 mm f o r 1 or 2 d a y s , d e p e n d i n g on i n i t i a l wetness ( the d r i e r the s o i l the l o n g e r the p e r i o d a m i c r o - l y s i m e t e r can be i n c o n t i n u o u s u s e ) . June 14 and J u l y 6 , 1984 were s e l e c t e d to compare the measured and m o d e l l e d energy f l u x d e n s i t i e s and s o i l t h e r m a l r e g i m e s . For the 2 t / h a p l o t , f o r w h i c h t e m p e r a t u r e measurements were not a v a i l a b l e on the se d a y s , one o t h e r day (May 2 7 , 1984) was randomly s e l e c t e d f o r c o m p a r i s o n . June 14 and May 27 were c h a r a c t e r i z e d by m a i n l y c l e a r and sunny s k i e s w h i l e J u l y 6 was c l o u d y i n the morn ing and became c l e a r a t n o o n . On bo th June 14 and J u l y 6 the s o i l was m o d e r a t e l y w e t ; however , the bare p l o t was w e t t e r and the mulch was d r i e r on J u l y 6 . B . M u J_ c h_a n d_B a r e_S t rie_E xge r ime n t s T h i s s t u d y was c o n d u c t e d on the s i t e used f o r the u n i f o r m mulches f rom A u g u s t , 1984 to A p r i l , 1985 . The p l o t s were r e -e s t a b l i s h e d a f t e r p l o u g h i n g , l e v e l i n g and p a c k i n g o f the s o i l . The p l o t s i z e was reduced to 10 m x 10 m due to the a n t i c i p a t e d e x t e n s i v e manual l a b o u r r e q u i r e d i n making the mulch s t r i p s and k e e p i n g them i n p l a c e a g a i n s t d i s r u p t i o n by the w i n d . M u l c h s t r i p s ( b a r l e y s t r a w ) o f 0 . 3 m w i d t h and 10 t / h a r a t e were a l t e r n a t e d w i t h bare s t r i p s o f 0 .10 m w i d t h . 60 The s t r i p s were o r i e n t e d 1n n o r t h - s o u t h ( N - S ) , n o r t h e a s t -s o u t h w e s t (NE-SW), and e a s t - w e s t (E-w) d i r e c t i o n s ; a bare p l o t was m a i n t a i n e d as a r e f e r e n c e . I n s t a n t a n e o u s s o i l t e m p e r a t u r e s were measured h o u r l y i n q u a d r u p l i c a t e a t z = 0 . 0 0 5 , 0 .02 and 0 .10 m f o r the bare p l o t , and i n d u p l i c a t e a t z = 0 .25 m f o r the bare and a t z = 0 . 0 0 5 , 0 . 0 2 , 0 .10 and 0 .25 m f o r the s t r i p p l o t s . D u p l i c a t e and q u a d r u p l i c a t e measurements were made as d e s c r i b e d p r e v i o u s l y . For the s t r i p p l o t s the s e n s o r l o c a t i o n s a l o n g the x - a x i s were a t 0 .15 m ( m i d d l e o f the mulch s t r i p ) , 0 .28 m ( f rom E to W f o r the N - S , f rom SE to NW f o r the NE-SW and from S to N f o r the E-W o r i e n t e d p l o t s ) , and 0 .35 m ( m i d d l e o f the bare s t r i p ) f o r z = 0 . 0 0 5 , 0 .02 and 0 .10 m and o n l y a t x = 0 .28 m f o r z = 0 .25 and 0 .50 m ( F i g . 2 . 1 ) . M o i s t u r e c o n t e n t s and o t h e r p h y s i c a l and m i c r o m e t e o r o l o g i c a l v a r i a b l e s were m o n i t o r e d as i n the u n i f o r m mulch e x p e r i m e n t . E v a p o r a t i o n r a t e s were measured on A p r i l 7 and 8 , 1985 i n the m i d d l e o f the bare and mulch s t r i p s (two r e p l i c a t e s i n the bare s t r i p i n each p l o t e x c e p t NE-SW and no r e p l i c a t e i n the mulch s t r i p ) as d e s c r i b e d p r e v i o u s l y . From an a g r i c u l t u r a l s t a n d p o i n t , the s p r i n g t i m e s o i l t h e r m a l and m o i s t u r e regimes a re h i g h l y c r i t i c a l and so t h e s e days were s e l e c t e d to t e s t the m o d e l . In a d d i t i o n , m o d e l l e d and measured s o i l t e m p e r a t u r e p r o f i l e s a re compared f o r September 3 , 1984. At t h a t t i m e i t 1s presumed the mulch had not y e t s e t t l e d and the s h a d i n g d u r i n g the d a y t i m e o f the bare s t r i p by the mulch was e x p e c t e d to be o f g r e a t e r i m p o r t a n c e . September 3 was a c l e a r 61 day w i t h t h e d r i e s t s o i l c o n d i t i o n s d u r i n g t h i s s t u d y . A p r i l 7 was a c l o u d y day w h i l e A p r i l 8 was a m a i n l y sunny day both w i t h wet s o i l c o n d i t i o n s due to e x t e n s i v e r a i n f a l l i n the p r e v i o u s w e e k s . C . S £ e c i . 11 c a tj .o n_o f _ t h e_I np u t _ P a r am e t e r s Soil Properties: The p r o f i l e s o f s o i l v o l u m e t r i c heat c a p a c i t y (C ) were c a l c u l a t e d from measured v a l u e s o f s o i l m o i s t u r e , o r g a n i c m a t t e r , and b u l k d e n s i t y u s i n g the r e l a t i o n s h i p s g i v e n i n de V r i e s ( 1 9 6 3 ) . The p r o f i l e s o f s o i l t h e r m a l c o n d u c t i v i t y ( k $ ) were c a l c u l a t e d u s i n g the method of de V r i e s (1963) w i t h the r e c e n t c o r r e c t i o n s to the c a l c u l a t i o n o f the d i f f u s i o n c o e f f i c i e n t of w a t e r vapour i n a i r (de V r i e s and P h i l i p , 1 9 8 6 ) . Quar tz c o n t e n t i n the s o i l was not measured and the v a l u e f o r the t h e r m a l c o n d u c t i v i t y o f the s o i l s o l i d s was adopted from Sepaskah and Boersma (1979) based on the measured s o i l t e x t u r e , e . g . , loamy sand i n t h i s c a s e . For the bare p l o t the a | used was an average o v e r the d a y t i m e p e r i o d . I t g e n e r a l l y v a r i e d somewhat due to the d r y i n g o f the s o i l s u r f a c e d u r i n g the d a y t i m e but t h i s v a r i a t i o n was not c o n s i d e r e d s i g n i f i c a n t (see F i g . 2 . 2 ) . The v a l u e s o f a | under the m u l c h , w h i c h were not measured d i r e c t l y , were reduced by the r a t i o o f the s o i l s u r f a c e m o i s t u r e c o n t e n t of the bare p l o t t o the s o i l s u r f a c e m o i s t u r e c o n t e n t under the mulch f o r m o i s t u r e c o n t e n t s i n the range from " f i e l d c a p a c i t y " t o s a t u r a t i o n . For d r y to f a i r l y dry (< 15% by mass) s o i l c o n d i t i o n s the same v a l u e was used f o r 62 > LU UL LU CC o DC LU Q_ 50 - MULCH RATE (t/ha) 0 * 2 o 10 D 20 o 25 i i i i T r MODERATELY DRY SOIL/MULCH SURFACE -o e -A A-8 12 16 TIME (h, PST) 20 24 F i g u r e 2 . 2 T y p i c a l measured a l b e d o s f o r the bare and u n i f o r m l y mulched p l o t s . The measurements were made on June 10 , May 2 4 , June 1 and June 14, 1984 f o r the 0 , 2 , 10 , and 20 t / h a r a t e s , r e s p e c t i v e l y . 63 bo th bare and mulched p l o t s . S i n c e the a l b e d o o f the bare p l o t was not measured on a l l o f the t e s t d a y s , the measured v a l u e from the day c l o s e s t t o 1t was u s e d , v a l u e s were not measured d i r e c t l y but were e s t i m a t e d as v a r y i n g from 0 .93 f o r d r y t o 0 .96 f o r wet s o i l s (Fuchs and T a n n e r , 1968; Tanner e t a l . , 1 9 8 7 ) . Mulch Properties: Measured v a l u e s o f f o r " f r e s h " and " u s e d " (on the p l o t s f rom f a l l , 1983 to s p r i n g , 1984) b a r l e y s t r a w a t d i f f e r e n t r a t e s o f a p p l i c a t i o n a re p r e s e n t e d i n T a b l e 2 . 2 . The measurements were made by m o u n t i n g a square ( 1 . 5 m x 1.5 m) wooden frame l i n e d w i t h t r a n s p a r e n t a c r y l i c p l a s t i c f i l m a t 0 .13 m above an u p - f a c i n g K i p p and Zonen CM5 p y r a n o m e t e r . B a r l e y s t r a w a t r a t e s o f 0 . 5 , 1, 2 , 5 , 10 or 20 t / h a was p l a c e d u n i f o r m l y i n the frame and the r a t i o o f the s o l a r t r a n s m i t t e d to t h a t above the mulch was measured a t 10 min i n t e r v a l s d u r i n g the d a y t i m e on c l e a r days (one r a t e per day) i n the summer, 1984. The v a l u e s p r e s e n t e d i n T a b l e 2 . 2 a re a v e r a g e s f o r 7 hours from 8 to 15 PST, c o r r e c t e d f o r the t r a n s m i t t a n c e o f the a c r y l i c f i l m and the frame ( 0 . 8 8 ) . The l o w e r v a l u e s f o r the used s t r a w are a t t r i b u t e d m a i n l y to the s m a l l e r p a r t l y decomposed p i e c e s o f s t r a w t h a t s e t t l e d a t the bo t tom ( t h e used s t r a w was n o t i c e a b l y b r i t t l e and e a s i l y s h a t t e r e d ) and i n p a r t to s o i l d e p o s i t s w i t h low r e f l e c t i v i t y and h i g h a b s o r p t i v i t y t h a t adhered to the s t r a w , thus r e d u c i n g f o r w a r d s c a t t e r i n g t h r o u g h the m u l c h . S i g n i f i c a n t s c a t t e r ( 0 . 4 0 to 0 .99 ) i n the 10 m i n . v a l u e s (not shown) e x i s t e d f o r a p p l i c a t i o n r a t e s l o w e r than 5 t / h a , most o f w h i c h was due to Table 2.2 Transmisslvltles of fresh (unused) and f ield util ized (for one season) barley straw at different rates of application. Rate (t/ha) s* 1 in TSut1l1zed/ m T S fresh m Fresh (unused) Field Utilized 0.5 0.77 0.41 0.53 1 0.51 0.26 0.51 2 0.34 0.16 0.47 5 0.17 0.08 0.47 10 0.09 0.06 0.67 20 0.07 0.06 0.86 *Average for 7 hours measured 9 10 min Intervals from 8 to 15 PST. 65 the s p a t i a l n o n u n i f o r m 1 t y i n the r e s i d u e d i s t r i b u t i o n . The a p p r o p r i a t e v a l u e s of x * f rom T a b l e 2 . 2 were used on the t e s t d a y s . No measurements o f x^ were made; the v a l u e s used were e s t i m a t e d to be equa l to the c o r r e s p o n d i n g v a l u e s o f x ^ . The aj* v a l u e s were c a l c u l a t e d from measured v a l u e s o f m s " / S d u s i n g e q u a t i o n (12) i n the f o l l o w i n g form % ~i • i " ) s To be p h y s i c a l l y r e a l i s t i c , o n l y the n e g a t i v e r o o t o f the d i s c r i m i n a n t Is c o n s i d e r e d . However , f o r x * < 25%, m m can be d e t e r m i n e d w i t h i n an e r r o r of ~ 1% as s " / S d , and so t h i s s i m p l e p r o c e d u r e was u s e d . S i n c e v a l u e s o f s" f o r any p a r t i c u l a r mulch a p p l i c a t i o n r a t e was not measured on a l l of the t e s t d a y s , the measured v a l u e s f rom the day c l o s e s t to i t were u s e d , aj^ was not m e a s u r e d ; 1t was c o n s i d e r e d to be v e r y s m a l l and a v a l u e of 0 .02 was u s e d . At l o w e r a p p l i c a t i o n r a t e s the bare s o i l and r e s i d u e s u r f a c e s e x i s t i n p a t c h e s w h i l e i n s t r i p s t h e y a re m a i n t a i n e d i n b a n d s . For such a c o m p o s i t e s o i l / r e s i d u e s u r f a c e the measured r e f l e c t i v i t y showed t h a t i t was a w e i g h t e d average of s o i l and mulch components i n p r o p o r t i o n to the bare and c o v e r e d f r a c t i o n s . For e x a m p l e , the measured r e f l e c t i v i t y ( a v e r a g e d from 5 to 18 PST) f o r the E-W o r i e n t e d s t r i p p l o t on Augus t 2 8 , 1984 was 0 .27 w h i l e the c a l c u l a t e d r e f l e c t i v i t y was a l s o the same ( ( 0 . 3 0 x 3 + 0 . 1 8 ) / 4 ) . a * (=0.30) was t a k e n 66 from the measurements made o v e r a f r e s h mulch s u r f a c e , a p p l i e d u n i f o r m l y a t 10 t / h a r a t e and a * ( = 0 . 1 8 ) was a measured v a l u e o v e r the bare p l o t on September 1 s t , 1984. Note t h a t the a p p l i e d mulch was v e r y f r e s h and the s o i l was 1n i t s d r i e s t c o n d i t i o n i n a l l the p l o t s d u r i n g t h a t p e r i o d , so t h a t p r e s u m a b l y no v a r i a t i o n i n between the bare and s t r i p p l o t s due t o v a r i a t i o n i n s o i l wetnes s o c c u r r e d . Tanner et a l . (1987) i n d i c a t e s i m i l a r f i n d i n g s . However , the model r e q u i r e s the v a l u e s f o r i n d i v i d u a l s o i l and mulch e l e m e n t s . R e s i s t a n c e t o Heat and Vapour Transfer: For the bare p l o t , r * was d e t e r m i n e d by t r i a l and e r r o r u n t i l the measured and p r e d i c t e d d a y t i m e t o t a l LE or the maximum s o i l t e m p e r a t u r e a t 0 .005 m ( f o r those days when LE was not measured) a g r e e d to w i t h i n + 25 W m~ 2 o r + 1 ° C , r e s p e c t i v e l y . The r^ t h u s o b t a i n e d was then a p p l i e d to the u n i f o r m s t r i p p l o t s when the s o i l was o f s i m i l a r w a t e r c o n t e n t . When the s o i l m a t r i c p o t e n t i a l exceeded - 1500 k P a , w h i c h was the case u s u a l l y under a t h i c k m u l c h , was assumed to be z e r o (Novak and B l a c k , 1 9 8 5 ) . Assuming vapour d i f f u s i o n as the o n l y mechanism f o r m o i s t u r e f l o w t h r o u g h the m u l c h , r ^ i s g i v e n by (Fuchs and T a n n e r , 1967 ; Novak and B l a c k , 1985) rl = H m / [ f f ( P m " 0 „ m ) D u ] m m L t x m vm' v J (18) 67 where f t 1s the t o r t u o s i t y f a c t o r , P m and 8 v m a r e , r e s p e c t i v e l y , the p o r o s i t y and v o l u m e t r i c w a t e r c o n t e n t o f the mulch l a y e r , and D y ( 2 . 4 2 x 1 0 ' 5 m 2 s - 1 a t 2 0 ° C ) 1s the m o l e c u l a r d l f f u s l v i t y o f w a t e r v a p o u r . Assuming the p a r t i c l e d e n s i t y o f the b a r l e y s t r a w as e q u i v a l e n t to t h a t of s o i l _ 3 o r g a n i c m a t t e r (1300 kg m ) , P m was c a l c u l a t e d to be ~ 0 .98 f o r a l l a p p l i c a t i o n r a t e s . Because o f the h i g h p o r o s i t y of the r e s i d u e s f t = 0 .90 was u s e d . On the o t h e r h a n d , i n p a r t because o f the v e r y low b u l k d e n s i t y o f the r e s i d u e s , the v a l u e s o f 8 v m on the t e s t d a y s , on an average f o r the e n t i r e mulch l a y e r , were a l w a y s found to be l e s s than 1%. By a n a l o g y , r ^ i s o b t a i n e d from e q u a t i o n (18) by r e p l a c i n g D y w i t h D h , where D h ( 2 . 1 5 x 1 0 " 5 m 2 s - 1 a t 2 0 ° C ) i s the m o l e c u l a r d i f f u s i v i t y f o r s e n s i b l e h e a t . K i m b a l l and Lemon (1971) r e p o r t e d t h a t the energy f l u x d e n s i t y o f heptane t h r o u g h a 20 mm t h i c k l a y e r o f wheat s t r a w i n c r e a s e d l i n e a r l y w i t h wind speed measured a t 2 .7 a n d / o r 4 .7 m. Campbe l l et a l . (1980) a l s o r e p o r t e d t h a t the f l u x d e n s i t y a c r o s s c l o t h i n g and a n i m a l f u r i n c r e a s e d l i n e a r l y w i t h w i n d s p e e d o v e r a range from 0 .5 to 5 m s " 1 measured 1n a w i n d -t u n n e l . T h i s i n d i c a t e s t h a t f o r c e d t u r b u l e n t c o n v e c t i o n was s i g n i f i c a n t w i t h i n the r e s i d u e , r e s u l t i n g i n an enhancement of heat and vapour t r a n s p o r t i n 1 t . T h e r e f o r e i n the model the d e n o m i n a t o r o f (18) ( f o r both vapour and h e a t ) was m u l t i p l i e d by an "enhancement f a c t o r " E f t h a t was d e t e r m i n e d by m a t c h i n g the measured and p r e d i c t e d LE ( d a y t i m e t o t a l ) f o r the 20 t / h a 68 p l o t on June 14, 1984, w i t h a p r e a d j u s t e d z Q t h a t y i e l d e d d a y t i m e mulch s u r f a c e t e m p e r a t u r e s 1n agreement w i t h those measured w i t h the b o l o m e t e r . The v a l u e thus found was 4 . 5 and was a p p l i e d to a l l o t h e r a p p l i c a t i o n r a t e s and t e s t d a y s . For f o r c e d c o n v e c t i o n a e r o d y n a m i c r e s i s t a n c e s a re g i v e n by r v r h i^VfoLl^ iVfo* M(n  a a • <19> where d_ and z M a r e , r e s p e c t i v e l y , the h e i g h t s a t w h i c h T a ( t ) a u a and p u ( t ) and w i n d s p e e d (u) a re g i v e n , z i s the s u r f a c e V U roughness l e n g t h and k i s von Karman ' s c o n s t a n t ( = 0 . 4 0 ) . E q u a t i o n (19) i s s t r i c t l y v a l i d o n l y under n e u t r a l c o n d i t i o n s when the l o g a r i t h m i c wind p r o f i l e p r e v a i l s . D u r i n g the d a y t i m e f o r c e d and f r e e c o n v e c t i o n c o - e x i s t (Thorn, 1975) and a w i n d p r o f i l e c o r r e c t i o n i s r e q u i r e d to a c c o u n t f o r the e f f e c t s o f a t m o s p h e r i c i n s t a b i l i t y . T h i s i s u s u a l l y done i t e r a t i v e l y u t i l i z i n g the Monin-Obukhov l e n g t h s c a l i n g f a c t o r or R i c h a r d s o n number or both to d e t e r m i n e the d i a b a t i c i n f l u e n c e f u n c t i o n s . In the model I t i s done r o u g h l y by d i v i d i n g e q u a t i o n (19) by a " c o r r e c t i o n f a c t o r " w h i c h , i s > 1 f o r u n s t a b l e c o n d i t i o n s and < 1 f o r s t a b l e c o n d i t i o n s . The degree o f s t a b i l i t y or i n s t a b i l i t y i n the f l o w depends on the t u r b u l e n c e a t or near the s u r f a c e boundary l a y e r . For r e l a t i v e l y smooth s u r f a c e s , the f l o w near the s u r f a c e i s e x p e c t e d to be l e a s t t u r b u l e n t and a c o r r e c t i o n f a c t o r o f 1.5 69 i s c o n s e r v a t i v e l y used i n the m o d e l , a v a l u e chosen from the range o f 1 ( n e u t r a l ) to ~ 3 ( u n s t a b l e ) f o r the d i a b a t i c i n f l u e n c e f u n c t i o n s ( f o r heat and w a t e r v a p o u r ) as a f u n c t i o n o f a t m o s p h e r i c s t a b i l i t y g i v e n i n Campbel l ( 1 9 7 7 ) . A rough c o r r e c t i o n i s f e l t to be adequate s i n c e i n r e a l i t y z Q ' s f o r heat and vapour d i f f e r f rom t h a t f o r momentum ( B r u t s a e r t , 1982) i n a manner t h a t opposes the e f f e c t s o f i n s t a b i l i t y d u r i n g the d a y t i m e . The z 's were d e t e r m i n e d by t r i a l and e r r o r so t h a t the o J model agreed w i t h midday s o i l (bare p l o t ) and mulch (mulched p l o t s ) s u r f a c e t e m p e r a t u r e s measured by the b o l o m e t e r . The -4 v a l u e s thus found v a r y from 5 x 10 m f o r the bare p l o t to -3 2 x 10 m f o r the mulched p l o t s , i n agreement w i t h o t h e r s t u d i e s f o r such s u r f a c e s ( B r u t s a e r t , 1 9 8 2 ) . E v a p o r a t i o n from the mulch was assumed to be n e g l i g i b l e m 1 9 and r s = 1 x 10 was u s e d , but f o r the s t r i p p l o t s under wet c o n d i t i o n s i t was found t h a t a v a l u e of < 500 s m - 1 was r e q u i r e d to e x p l a i n the measured s o i l e v a p o r a t i o n r a t e s i n the mulch s t r i p s ( d i s c u s s e d l a t e r ) . Shortwave and Longwave Radiation: For the b a r e , u n i f o r m l y mulched p l o t s and the mulch s t r i p s the i n c o m i n g s h o r t w a v e r a d i a t i o n i s e q u a l , i . e . , S° = S° and the i n c o m i n g d 4 longwave r a d i a t i o n , l _ t = €a<>TK a , where £ a i s the e f f e c t i v e a t m o s p h e r i c e m i s s i v i t y and T ^ a i s the a b s o l u t e a i r t e m p e r a t u r e a t s c r e e n h e i g h t . However f o r the bare s t r i p and L d v a r y 70 w i t h x as t h e y depend upon the s h a d i n g and v i e w f a c t o r e f f e c t s due to the s u r r o u n d i n g v e r t i c a l mulch " w a l l s " . The d i f f u s e r a d i a t i o n a t any p o i n t i n the bare s t r i p ( S J J ) 1s g i v e n by S K • (1 " fi~ fI)s2 + aid + 0(-^*)s!! + aj.fjj S b b x m m' d m v m m / x 2 ' d m m m (20) 1 r where f m and f -are the v i ew f a c t o r s o f the v e r t i c a l mulch m m w a l l s t o the l e f t or r i g h t as seen from the p o i n t ( F i g . 2.1), f i s the v i ew f a c t o r o f one mulch w a l l as seen from the w o p p o s i t e w a l l , i s the d i f f u s e p a r t o f S d and 1s d e t e r m i n e d f rom the measured s£ f o l l o w i n g Gates (1980), f ° i s the v i ew f a c t o r o f the s u n l i t p a r t o f the v e r t i c a l mulch w a l l (may be on the l e f t o r the r i g h t s i d e d e p e n d i n g on the l o c a t i o n o f the sun) as seen from the p o i n t on the s u r f a c e o f the bare s t r i p ( i . e . , x ) , S b i s the beam s o l a r r a d i a t i o n f l u x d e n s i t y i n c i d e n t on the s u n l i t v e r t i c a l mulch w a l l ( i t i s the beam i n t e n s i t y t i m e s the c o s i n e of the a n g l e between the beam and the normal to the s u n l i t v e r t i c a l mulch w a l l ) . S i m i l a r l y , the longwave r a d i a t i o n a t any p o i n t i n the bare s t r i p (LJlJ) i s g i v e n by am(fI+ 0 < - - r - - > « a < » - <21> 71 £ a i s c a l c u l a t e d u s i n g the e q u a t i o n s g i v e n i n Buchan (1982) a and Idso ( 1 9 8 0 ) , as d e t a i l e d i n Novak and B l a c k ( 1 9 8 5 ) . 1 r C and f ' a re g i v e n by f£ « (1 - x ' / v V 2 + H 2 ) / 2 , (22) where x' i s the h o r i z o n t a l d i s t a n c e from the p o i n t to e i t h e r the l e f t o r r i g h t w a l l , r e s p e c t i v e l y , w h i l e f i s g i v e n by f w = 71 + ( x ' ' / H m ) 2 - x " / H m . (23) where x " i s the w i d t h o f the s t r i p . f b i s o b t a i n e d by s u b t r a c t i n g f m n f rom f £ ( o r as the case may be) where f * n , the v i e w f a c t o r o f the shaded p a r t of the v e r t i c a l mulch w a l l i s o b t a i n e d from e q u a t i o n (22) r e p l a c i n g H m by H m n , t h i c k n e s s o f the shaded p a r t of the v e r t i c a l mulch w a l l . C a l c u l a t i n g the beam r a d i a t i o n a t any p o i n t on the bare s t r i p r e q u i r e s t r a c k i n g of the shadow c a s t by the mulch w a l l s , The l e n g t h o f the shadow of the w a l l i n a d i r e c t i o n p e r p e n d i c u l a r to the s t r i p i s g i v e n by S p = H m t a n 6 z S l n ( * - C ) , (24) where 8 Z i s the s o l a r z e n i t h a n g l e g i v e n by 72 cos8 = s 1 n T s l n S + costy cosS c o s ( h ) = s i n a _ , (25) Z a ij; i s the s o l a r a z i m u t h a n g l e measured c l o c k w i s e from n o r t h g i v e n by cos\J> = ( s i n a f i sinij) - s 1 n S ) / c o s a a costy, (26) and C 1s the a z i m u t h a n g l e of the s t r i p . In (25) and (26) <J> i s the l a t i t u d e , a a (= TT / 2 - 8 , ) i s the s o l a r a l t i t u d e , 5 i s a z the s o l a r d e c l i n a t i o n a n g l e g i v e n by ( R a d k e , 1982) 5 = 0 .39038 + 23 .27 s 1 n [ ( 3 6 0 / 3 6 5 ) ( J D - 8 2 . 2 ) ] , (27) where JD i s the J u l i a n day number, and h i s the s u n ' s hour a n g l e . For 0 < C < TT /2 , I f C < * < C + n . the shadow i s c a s t onto the s t r i p by the l e f t mulch w a l l ( r e l a t i v e t o an o b s e r v e r f a c i n g i n the C + n d i r e c t i o n ) . I f \|> < C o r > C + i t . the shadow i s c a s t by the r i g h t mulch w a l l . The beam r a d i a t i o n , i n c i d e n t on any p o i n t 1n the bare s t r i p i s 0 , 1f s h a d e d , and , i f s u n l i t . As s h a d i n g 1s the d e t e r m i n i n g f a c t o r i n c a l c u l a t i n g the beam r a d i a t i o n i m p i n g i n g onto the bare s t r i p 1t i s p r o f o u n d l y i m p o r t a n t to know the e f f e c t i v e t h i c k n e s s of the mulch t h a t a c t u a l l y c a s t s shade on the bare s t r i p . I t was o b s e r v e d t h a t due to i t s l o o s e n a t u r e the s t r a w d i d not s t a c k v e r t i c a l l y a t 73 the edges , but r a t h e r c o l l a p s e d g r a d u a l l y and s p r e a d out over the bare s t r i p . T h i s caused a r o u n d i n g e f f e c t a t the edges and the a c t u a l shade c a s t on the bare s t r i p was l e s s than what s h o u l d have o c c u r r e d as suming the i d e a l geometry i n d i c a t e d i n F i g . 2 . 1 . A c o m p a r i s o n between the shade l e n g t h c a s t by the mulch s t r i p and a p i n of h e i g h t e q u i v a l e n t to the mulch t h i c k n e s s , measured a c r o s s the bare s t r i p on A p r i l 8 , 1985 i s p r e s e n t e d i n T a b l e 2 . 3 to d e m o n s t r a t e the d i s c r e p a n c y i n v o l v e d . I t was found by t r i a l and e r r o r t h a t the a c t u a l shade t h a t was c a s t on the bare s t r i p was t h a t o f an e q u i v a l e n t mulch w i t h a t h i c k n e s s of about h a l f o f the t h i c k n e s s t h a t e x i s t e d a t the m i d d l e of the mulch s t r i p . A l t h o u g h the mulch t h i c k n e s s was reduced to h a l f w i t h r e s p e c t to the s h a d i n g e f f e c t , i t remained unchanged i n the c a l c u l a t i o n of the r!) o r r*. m m Those s o i l and mulch p r o p e r t i e s t h a t a r e not e x p l i c i t l y s p e c i f i e d i n t h i s s e c t i o n , o t h e r r e l a t e d i n p u t p a r a m e t e r s and c l i m a t o l o g i c a l i n p u t p a r a m e t e r s are g i v e n i n a p p e n d i c e s I I I t h r o u g h X X . 74 Table 2.3 Shade length (In cm) from a 4 cm long (equal to the mulch thickness at the middle of the mulch strip) pin and from the actual mulch thickness at the edge of the mulch strip measured perpendicular to the row and across the bare strip at different orientations on April 8, 1985. Time Orientations (PST) N-S NE-SW E-W 8 9 10 11 12 13 14 15 16 17 18 10.5 7.0 4.5 2.3 1.5 0.0 2.0 4.5 7.5 >10.0 >10.0 6.0 3.0 1.5 0.5 0.0 0.0 0.5 2.0 4.0 MO.O >10.0 8.5 7.0 5.0 4.5 3.5 3.5 2.0 0.2 1.5 6.0 MO.O 5.0 3.5 2.0 1.5 1.0 1.0 0.5 0.0 1.0 4.0 >10.0 0.0 2.0 3.5 4.0 4.0 4.0 4.5 5.5 6.0 6.5 7.0 0.0 1.0 0.5 1.5 1.0 1.5 2.0 2.5 4.0 4.0 6.0 75 2 . 4 Re s u i t s_and_p_lscuss 1.0n A number of the i m p o r t a n t p a r a m e t e r s r e q u i r e d by the model were not measured i n d e p e n d e n t l y , i . e . , z Q , r ^ , k s , E f , and the c o r r e c t i o n f a c t o r f o r a e r o d y n a m i c s t a b i l i t y . These were d e t e r m i n e d by f i t t i n g to some of the measured d a t a . The use o f the model i n t h i s " i n v e r s e " or " c a l i b r a t i o n " mode p r o v i d e s i n s i g h t s i n t o the p h y s i c s o f some o f the p r o c e s s e s o c c u r r i n g i n the s o i l - r e s i d u e - a t m o s p h e r e sys tem t h a t are not f u l l y u n d e r s t o o d . However , any model t e s t i n g then depends upon the agreement between measurements and p r e d i c t i o n s on o t h e r d a t a , o r of the same d a t a on o t h e r days a n d / o r p l o t s . The r e s u l t s a re p r e s e n t e d and d i s c u s s e d f o r a s e t o f i n p u t p a r a m e t e r s s u b s e q u e n t l y r e f e r r e d to as " b a s i c " . The e f f e c t s o f the v a r i a t i o n o f some o f the p a r a m e t e r s a re d i s c u s s e d r e l a t i v e to the b a s i c p a r a m e t e r s i n a s e n s i t i v i t y a n a l y s i s s e c t i o n . A . U n l f o r m l y _ A r j £ l l e d_M u l c h The measured and t h e o r e t i c a l l y c a l c u l a t e d R„ and LE f o r n June 14, 1984 a t the d i f f e r e n t mulch a p p l i c a t i o n r a t e s a re p r e s e n t e d i n F i g . 2 . 3 . The m o d e l l e d v a l u e s o f R n d i f f e r from measured v a l u e s by a t most 50 W m ( i n the bare p l o t a t 7 .5 and 12 .5 h ) ; however , t h e y agreed w e l l o v e r a l l t h r o u g h o u t the 24 h p e r i o d . The e r r o r s i n measured and computed R n f o r J u l y 6 , 1984 (no t p r e s e n t e d ) are s i m i l a r . The m o d e l l e d d a y t i m e t o t a l L E , w i t h E f = 4 . 5 ( o b t a i n e d from the 20 t / h a p l o t as i n d i c a t e d p r e v i o u s l y ) agrees to w i t h i n 3.47. o f the measured 76 6 0 0 f1 i 1 1 1 1 1 i i TIME (h, PST) F i g u r e 2 . 3 The measured ( i s o l a t e d s y m b o l s ) R i n the bare and 20 t / h a , the c a l c u l a t e d J c u r v e s ) R i n the b a r e , 2 , 10 and 20 t / h a p l o t s and trie measured ( s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) LE 1n the b a r e , 2 , 10 and 20 t / h a p l o t s on June 14, 1984. 77 v a l u e i n the 10 t / h a p l o t on June 14, 1984, but o v e r e s t i m a t e s by 47% 1n the 2 t / h a p l o t . Agreement f o r t h i s p l o t can be improved e i t h e r by a d j u s t i n g r * as done f o r the b a r e p l o t or by r e d u c i n g E f by about one h a l f . The second o p t i o n i s p h y s i c a l l y u n r e a s o n a b l e as E f s h o u l d i n c r e a s e as the mulch a p p l i c a t i o n r a t e d e c r e a s e s . S i n c e the measured s o i l m o i s t u r e c o n t e n t p r o f i l e f o r the 2 t / h a p l o t (~ 31% by volume on average i n the 0 - 0 .21 m w i t h i n s i g n i f i c a n t v a r i a t i o n w i t h d e p t h ) on June 14 were s i m i l a r to those i n the 10 and 20 t / h a p l o t s , and c o r r e s p o n d e d to a t e n s i o n of < 100 kPa (see F i g s . 2 . 4 a and b ) , r * = 0 was s e e m i n g l y j u s t i f i e d . An r^ o f 300 s m _ 1 was needed to make the m o d e l l e d d a y t i m e c u m u l a t i v e LE i n the bare p l o t agree w i t h the measurements ; i n c o r p o r a t i n g an r^ of a s i m i l a r magni tude i n the 2 t / h a p l o t would make the m o d e l l e d c u m u l a t i v e comparab le w i t h the measured v a l u e . I t 1s c o n j e c t u r e d t h a t s o i l v a r i a b i l i t y was the ma jor r ea son t h a t the h i g h measured s o i l m o i s t u r e c o n t e n t on t h a t day c o e x i s t e d w i t h an a p p a r e n t r^ > 0 . A l s o the uneven d i s t r i b u t i o n o f the mulch a t the low a p p l i c a t i o n r a t e (2 t / h a ) o v e r the m i c r o - l y s i m e t e r c o u l d have l e d to an e r r o r i n the measurement s . S i n c e p r e d i c t e d s o i l t e m p e r a t u r e s f o r the 2 t / h a p l o t cannot be compared a g a i n s t measured v a l u e s f o r t h e s e d a t e s a l t e r n a t i v e s to improve the agreement i n LE were not p u r s u e d i n d e t a i l . On J u l y 6 , w i t h E f = 4 . 5 , LE f o r the 10 and 20 t / h a p l o t s i s u n d e r e s t i m a t e d by 23 to 28% w h i l e f o r the bare and 2 t / h a p l o t s i t i s o v e r e s t i m a t e d by 12 and 24%, r e s p e c t i v e l y ( T a b l e 2 . 4 ) . 78 0.6 I I I 1 1 1 1 1 1 ( 1 1 111Mil 0.4 0.2 0.0 BARE 0.0 - 0.05 m 1-0.05 - 0.10 m 0.10 - 0.20 m 2 t/ha 0.0 - 0.05 m 0.05 - 0.10 m 0.10 - 0.20 m. • • • i,i 11 i i i 1111111 l nn -10 -100 -1000-2000 SOIL MATRIC POTENTIAL (kPa) F i g u r e 2 . 4 a S o i l - w a t e r r e t e n t i o n c u r v e s f o r s o i l s sampled a t 0 - 0 . 0 5 , 0 . 0 5 - 0 . 1 0 and 0 . 1 0 - 0 . 2 0 m dep ths i n the bare and 2 t / h a p l o t s . 79 SOIL MATRIC POTENTIAL (kPa) F i g u r e 2 .4b S o i l - w a t e r r e t e n t i o n c u r v e s f o r s o i l s sampled a t 0 - 0 . 0 5 , 0 . 0 5 - 0 . 1 0 and 0 . 1 0 - 0 . 2 0 m dep ths i n the 10 and 20 t / h a p l o t s . 80 Table 2.4 Measured and predicted evaporation rates (mm h - 1 ) at various mulch application rates on June 14 and July 6, 1984. June 14, 1984 Mulch application rate (t/ha) 0 2 10 20 Period (PST) Meas. Pred. Meas. Pred. Meas. Pred. Meas. Pred. 5-7 0.061 0.026 0.038 0.044 0.006 0.016 0.005 0.010 7-9 0.223 0.092 0.116 0.128 0.030 0.030 0.009 0.017 9-11 0.226 0.179 0.151 0.237 0.040 0.048 0.019 0.024 11-13 0.205 0.225 0.195 0.291 0.058 0.056 0.045 0.027 13-15 0.144 0.231 0.192 0.309 0.083 0.064 0.034 0.032 15-17 0.077 0.139 0.120 0.175 0.057 0.049 0.027 0.026 17-19 0.026 0.052 0.034 0.059 0.028 0.027 0.019 0.016 Total* 1.924 1.888 1.692 2.486 0.604 0.580 0.316 0.304 July 6, 1984 5-7 0.063 0.008 0.034 0.026 0.015 0.020 0.012 0.015 7-11 0.191 0.101 0.133 0.151 0.045 0.039 0.026 0.024 11-13 0.206 0.242 0.327 0.351 0.146 0.077 0.073 0.045 13-15 0.149 0.261 0.355 0.369 0.108 0.086 0.066 0.051 15-17 0.112 0.222 0.165 0.314 0.101 0.083 0.088 0.050 17-20 0.039 0.101 0.064 0.121 0.057 0.051 0.048 0.034 Total* 1.941 2.173 2.486 3.087 1.091 0.841 0.726 0.520 •Total = ^(hourly rates x time Interval); Meas. • Measured; Pred. = Predicted. Note: Predicted evaporation rates are taken at the middle of the period of consideration. 81 The measured v a l u e s o f LE f o r the bare p l o t a r e t y p i c a l of s o i l - l i m i t e d e v a p o r a t i o n , i . e . , LE peaks e a r l y i n t h e " m o r n i n g and then d e c l i n e s d u r i n g the d a y t i m e . A l t h o u g h the rf. f o r the bare p l o t was a d j u s t e d to match the measured and c a l c u l a t e d d a y t i m e t o t a l L E , F 1 g . 2 . 3 shows t h a t the s u r f a c e r e s i s t a n c e model f a i l e d to p r e d i c t the h o u r l y r a t e s a c c u r a t e l y . T h i s i s i n agreement w i t h Fuchs and Tanner ( 1 9 6 7 ) , who e v a l u a t e d the s u r f a c e r e s i s t a n c e model on a P l a i n f i e l d s a n d , and i n d i s a g r e e m e n t w i t h Novak and B l a c k (1985) who found t h a t the s u r f a c e r e s i s t a n c e model gave good agreement on a " r e l a t i v e l y w e t " Monroe s i l t l o a m . The model p r e d i c t e d the v a r i a t i o n i n e v a p o r a t i o n r a t e under the mulch r e a s o n a b l y c o r r e c t l y . E v a p o r a t i o n r a t e s d e c l i n e d c o n s i d e r a b l y under the mulch w i t h i n c r e a s i n g mulch r a t e . E v a p o r a t i o n f o r the mulched p l o t s was a p p a r e n t l y l i m i t e d by the energy a v a i l a b l e f o r the phase c h a n g e , i n c o n t r a s t w i t h the bare p l o t f o r w h i c h the h y d r a u l i c c o n d u c t i v i t y o f the s o i l was l i m i t i n g . The measured and c a l c u l a t e d s o i l t e m p e r a t u r e s a t depths of 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 .25 and 0 .50 m f o r the b a r e , 10 t / h a and 20 t / h a p l o t s f o r June 14, 1984 a re p r e s e n t e d i n F i g s 2 . 5 , 2 .6 and 2 . 7 , r e s p e c t i v e l y . A l s o shown are the s o i l and mulch s u r f a c e t e m p e r a t u r e s measured by the b o l o m e t e r . The p r e d i c t e d and IRT measured s u r f a c e t e m p e r a t u r e f o r the bare p l o t compare w e l l because zQ was chosen to match w i t h the IRT measured s u r f a c e t e m p e r a t u r e s . P r e d i c t e d v a l u e s o f s o i l t e m p e r a t u r e s show t r e n d s s i m i l a r to the measured 82 50 35 DEPTH {m) - 0.000 • -0.005 A -0.010 o -0.025 • • 0.050 o -0.100 v -0.250 • -0.500 • • 20 JUNE 14,1984 if// ° * 8 12 16 TIME (h, PST) 20 24 F i g u r e 2 . 5 The measured ( I s o l a t e d symbol s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s i n the bare p l o t a t dep ths o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 .25 and 0 . 5 0 m on June 14 , 1984. S o i l s u r f a c e t e m p e r a t u r e s were measured by the b o l o m e t e r . 83 o L U DC < DC L U Q L L U h-o CO 50 35 DEPTH (m) 0.000 • • 0.005 A • 0.010 o -0.025 • • 0.050 o • 0.100 v • 0.250 • • 0.500 X X JUNE 14,1984 x MULCH SURFACE x 20 8 12 16 TIME (h, PST) 20 24 F i g u r e 2 . 6 The measured ( i s o l a t e d symbol s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s i n the 10 t / h a p l o t a t dep ths o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 .25 and 0 .50 m on June 14 , 1984. S o i l s u r f a c e t e m p e r a t u r e s were measured by the b o l o m e t e r . A l s o shown are the mulch s u r f a c e t e m p e r a t u r e s measured by the b o l o m e t e r . 84 V LU c r L U Q. UJ I-O CO 50 i i 35 DEPTH {m 0.000 • 0.005 A 0.010 o 0.025 • 0.050 o 0.100 v 0.250 • 0.500 • X X X JUNE 14,1984 -X X X 20 MULCH SURFACE x 8 12 16 TIME (h, PST) 20 24 F i g u r e 2 .7 The measured ( I s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s 1n the 20 t / h a p l o t a t dep ths o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 .25 and 0 .50 m on June 14, 1984. S o i l s u r f a c e t e m p e r a t u r e s were measured by the b o l o m e t e r . A l s o shown a re the mulch s u r f a c e t e m p e r a t u r e s measured by the b o l o m e t e r . 85 v a l u e s but a r e o v e r e s t i m a t e d near the s u r f a c e . D i f f e r e n c e s of over 8 ° C and 2 °C were o b s e r v e d a t 0 .005 m d e p t h f o r the bare and mulched p l o t s , r e s p e c t i v e l y . For the bare p l o t i t i s s u s p e c t e d t h a t the spot where the t h e r m o c o u p l e s were p l a c e d was w e t t e r than i t s s u r r o u n d i n g s . A bare s o i l u s u a l l y d r i e s i n p a t c h e s ( J a c k s o n e t a l . , 1 9 7 6 ) , m o s t l y due to s p a t i a l v a r i a b i l i t y , and i t was noted t h a t p a t c h i n e s s had been o c c u r r i n g on June 14. U n c e r t a i n t i e s i n assumed z Q , c a l c u l a t e d rj? or r a , C c and k c may a l s o p a r t i a l l y a c c o u n t f o r a a s s the d i s c r e p a n c i e s . P r e d i c t e d and IRT measured s o i l s u r f a c e t e m p e r a t u r e s under the mulch agree w i t h i n the l i m i t o f measurement e r r o r s . T h i s c o n f i r m s t h a t the m o d e l , though i t t ends t o o v e r e s t i m a t e s l i g h t l y , p r e d i c t s the t h e r m a l reg imes a d e q u a t e l y f o r the bare and mulched s o i l c o n d i t i o n s . Agreement between the measured and p r e d i c t e d s o i l s u r f a c e t e m p e r a t u r e f o r J u l y 6 i s s i m i l a r to t h a t f o r June 14 f o r the bare p l o t (same z Q was used) and even b e t t e r f o r the 10 and 20 t / h a p l o t s w i t h E f = 4 . 5 (see T a b l e 2 . 5 ) . However , i f E f i s i n c r e a s e d to improve agreement between the measured and c a l c u l a t e d LE f o r the 10 and 20 t / h a p l o t s then the p r e d i c t e d t e m p e r a t u r e s are somewhat o v e r e s t i m a t e d ( e x p l a i n e d under s e n s i t i v i t y a n a l y s i s ) . A t the s u b - s u r f a c e p o s i t i o n s i n the 10 and 20 t / h a p l o t s on both days the model p r e d i c t e d the t e m p e r a t u r e f l u c t u a t i o n s a c c u r a t e l y and was a l w a y s i n phase w i t h the measurements . T h i s i s t r u e f o r the bare p l o t as w e l l once the b i g d i f f e r e n c e a t the s u r f a c e i s r e c o g n i z e d . The 86 Table 2.5 Measured and predicted soil temperature and temperature amplitude (°C) at 0.005 m depth at various mulch application rates on June 14 and July 6, 1984. June 14, 1984 Soil temperature at 0.005 m Temperature amplitude at 13 h (PST) at 0.005 m Mulch Application rate (t/ha) Meas. Pred. Meas. Pred. 0 32.4 39.9 10.5 13.7 2 - 29.1 - 7.9 10 21.6 23.5 4.3 5.1 20 18.3 21.0 3.0 4.0 July 6, 1984 0 31.0 38.4 9.8 13.0 2 - 26.7 - 6.9 10 21.6 21.7 4.1 4.2 20 19.5 19.8 3.0 3.2 Meas. = Measured; Pred. » Predicted. 87 model p r e d i c t e d t h a t the d i u r n a l v a r i a t i o n d e c l i n e d w i t h I n c r e a s i n g mulch r a t e . F i g . 2 . 8 shows the p r e d i c t e d s o i l t e m p e r a t u r e s a t d i f f e r e n t dep ths a l o n g w i t h the IRT measured mulch and s o i l s u r f a c e t e m p e r a t u r e s f o r the 2 t / h a p l o t on June 14, 1984. I t 1s d i f f i c u l t to measure the mulch s u r f a c e t e m p e r a t u r e f o r the 2 . 0 t / h a r a t e as the b o l o m e t e r " s e e s " the bare s o i l b e l o w . The measurements were made by h o l d i n g the i n s t r u m e n t as c l o s e to the s t r a w as p o s s i b l e . However , t h i s may have a l l o w e d the i n s t r u m e n t t o see some of i t s own shade as w e l l . The agreement between the measured and p r e d i c t e d s o i l s u r f a c e t e m p e r a t u r e s i n d i c a t e s t h a t they are r e a s o n a b l e . F i g . 2 . 9 shows the measured and p r e d i c t e d s o i l t e m p e r a t u r e s a t the same dep ths i n the 2 t / h a p l o t f o r May 2 7 , 1984. The p r e d i c t e d v a l u e s a re s i m i l a r to the measured v a l u e s and a re i n phase bo th a t the s u r f a c e and s u b s u r f a c e p o s i t i o n s . However the model t e n d s to o v e r e s t i m a t e s l i g h t l y , as seen p r e v i o u s l y w i t h o t h e r mulch r a t e s . Though the d i s c r e p a n c i e s can be e x p l a i n e d i n terms o f z Q , rjjj, C g and k $ , and measurement e r r o r , s t i l l i t i s d i f f i c u l t to a s s e s s the c o n t r i b u t i o n o f a " s i n g l e f a c t o r " , i n p a r t because e v a p o r a t i o n was not m e a s u r e d . The measured and m o d e l l e d t e m p e r a t u r e s w i t h i n the mulch l a y e r a t dep ths of 0 . 0 , 0 . 0 0 8 , 0 . 0 1 6 , 0 . 0 2 4 , 0 . 0 3 6 , 0 .048 and 0 .06 m ( f rom the 20 t / h a p l o t ) a re p r e s e n t e d 1n F 1 g . 2 . 1 0 . The model t ends to o v e r e s t i m a t e s l i g h t l y , however they are i n phase and a c c u r a t e w i t h i n the l i m i t s o f measurement e r r o r s . Agreement i s e s p e c i a l l y good i n the a f t e r n o o n , but 1s p o o r e r 88 50 ( J LU cr OC LU Q_ LU I-o CO 35 i i i DEPTH (m) 0.000 • 0.005 0.010 0.025 0.050 0.100 0.250 0.500 x MULCH SURFACE JUNE 14,1984 -20 8 12 16 TIME (h, PST) 20 24 F i g u r e 2 . 8 The c a l c u l a t e d s o i l t e m p e r a t u r e s i n the 2 t / h a p l o t a t dep ths o f 0 . 0 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 .25 and 0 .50 m on June 14, 1984. A l s o i n c l u d e d a re the s o i l and mulch s u r f a c e t e m p e r a t u r e s measured by the b o l o m e t e r . 89 i i i i • i • • 1 1 1 1 0 4 8 12 16 20 24 TIME (h, PST) F i g u r e 2 .9 The measured ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s a t dep ths of 0 . 0 0 5 , 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 0 , 0 .25 and 0 .50 m i n the 2 t / h a p l o t on May 2 7 , 1984. A l s o I n c l u d e d a re the c a l c u l a t e d s o i l s u r f a c e t e m p e r a t u r e s . 90 50 ( J UJ cr < rr UJ a . UJ I -X CJ 35 DEPTH (m) 0.000 * 0.008 A 0.016 o 0.024 • 0.036 o 0.048 v 0.060 • JUNE 14,1984 -20 8 12 16 TIME (h, PST) 20 24 F i g u r e 2 .10 The measured ( I s o l a t e d symbol s ) and c a l c u l a t e d ( c u r v e s ) t e m p e r a t u r e s a t depths of 0 . 0 , 0 . 0 0 8 , 0 . 0 1 6 , 0 . 0 2 4 , 0 . 0 3 6 , 0 .048 and 0 .06 m i n the mulch l a y e r f o r the a p p l i c a t i o n r a t e of 20 t / h a on June 14 , 1984. M u l c h s u r f a c e t e m p e r a t u r e s were measured by the b o l o m e t e r . 91 i n the m o r n i n g . The measured t e m p e r a t u r e g r a d i e n t s are g e n e r a l l y somewhat h i g h e r i n the bot tom h a l f o f the mulch and l o w e r i n the upper h a l f , p o s s i b l y due to c o n v e c t i v e e f f e c t s ( w h i c h s h o u l d be g r e a t e r near the s u r f a c e ) . However p o s i t i o n i n g of the thermometer s was more d i f f i c u l t and i n g r e a t e r e r r o r near the s u r f a c e . The a s s u m p t i o n t h a t the t e m p e r a t u r e s v a r i e d l i n e a r l y w i t h depth i n the mulch used i n the model i s not a n e c e s s a r y one ; a l l t h a t i s used i n the model i s the t o t a l r e s i s t a n c e between the s o i l s u r f a c e and the mulch s u r f a c e . B . Ef f ects_of_Str i_p_s The measured and p r e d i c t e d R n and LE i n the bare and s t r i p p l o t s f o r A p r i l 8 , 1985 are p r e s e n t e d i n F i g . 2 . 1 1 . C a l c u l a t e d R n over the s t r i p s i s a w e i g h t e d average of R p ' s c a l c u l a t e d on the mulch and bare s t r i p s . The w e i g h t i n g i s a c c o r d i n g to the f r a c t i o n a l a r e a o c c u p i e d by bare (1 /4 ) and mulch ( 3 / 4 ) s u r f a c e s . The measured and m o d e l l e d R p agree q u i t e w e l l . Note t h a t the measured midday LE from the bare p l o t was l e s s than the measured v a l u e s from a l l t h r e e s t r i p s . The s o i l i n a l l 4 p l o t s was u n i f o r m l y wet and r |=0 was assumed to be v a l i d everywhere on t h i s d a y . The p r e d i c t e d v a l u e s of LE i n the bare s t r i p s are i n phase w i t h the measured v a l u e s but the magn i tudes a re u n d e r e s t i m a t e d . The p r e d i c t e d t o t a l s a re l e s s by 21.6% f o r the b a r e , 31.0% f o r the N - S , 18.6% f o r the NE-SW and 33.3% f o r the E-W o r i e n t e d p l o t s , r e s p e c t i v e l y (see T a b l e 2 . 6 ) . T h i s i s 92 600 i i 1 1 1 i \ 1 1 1 r -100 0 4 8 12 16 20 24 TIME (h, PST) F i g u r e 2 .11 The measured ( i s o l a t e d symbol s ) R n i n the bare and the NE-SW, and the c a l c u l a t e d R ( c u r v e s ) 1n the b a r e , N - S , NE-SW and E-W n s t r i p p l o t s and the measured ( s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) LE In the b a r e , N - S , NE-SW and E-W s t r i p p l o t s on A p r i l 8 , 1985 93 Table 2.6 Measured and predicted evaporation rates (mm h - 1 ) 1n the bare strip at various directions on April 7 and 8, 1985. April 7, 1985 Strip direction Bare N-S NE-SW E-W Period (PST) Meas. Pred. Meas. Pred. Meas. Pred. Meas. Pred. 6-8 0.034 0.013 0.006 0.013 0.003 0.017 0.053 0.027 8-10 0.056 0.058 0.079 0.066 0.074 0.065 0.104 0.069 10-12 0.233 0.136 0.268 0.147 0.226 0.143 0.298 0.143 12-14 0.213 0.165 0.242 0.185 0.219 0.181 0.262 0.178 14-16 0.277 0.212 0.323 0.232 0.243 0.238 0.301 0.230 16-18 0.211 0.222 0.169 0.189 0.159 0.223 0.192 0.234 Total* 2.048 1.612 2.174 1.664 1.848 1.734 2.420 1.762 April 8, 1985 8-10 0.151 0.124 0.182 0.136 0.155 0.132 0.230 0.143 10-12 0.318 0.276 0.395 0.303 0.352 0.293 0.423 0.293 12-14 0.403 0.336 0.494 0.363 0.437 0.354 0.509 0.345 14-16 0.320 0.193 0.361 0.185 0.273 0.190 0.332 0.180 16-18 0.181 0.145 0.150 0.095 0.126 0.124 0.152 0.135 Total* 2.746 2.148 3.164 2.164 2.686 2.186 3.292 2.192 •Total = 2(hourly rates x time Interval); Meas. = Measured; Pred. • Predicted. Note: Predicted evaporation rates are taken at the middle of the period of consideration. 94 m a i n l y a t t r i b u t e d to m i c r o s c a l e a d v e c t i o n , not i n c l u d e d i n the t h e o r y , o c c u r r i n g between the hot mulch s t r i p and the r e l a t i v e l y c o l d bare s o i l s t r i p . P r e s u m a b l y the w i n d d i r e c t i o n 1n p a r t d e t e r m i n e s the e f f e c t of a d v e c t i o n upon a c o n f i g u r e d s u r f a c e . The wind d i r e c t i o n on A p r i l 8 was p r e d o m i n a n t l y s o u t h w e s t w a r d , i . e . , w ind was b l o w i n g a t 4 5 ° to bo th N-S and E-W o r i e n t e d s t r i p s w h i l e 1t was p a r a l l e l to the NE-SW o r i e n t e d s t r i p s . The e f f e c t s were t h e r e f o r e e x p e c t e d to be more s e v e r e f o r the N-S and E-W o r i e n t e d p l o t s than f o r the NE-SW o r i e n t e d p l o t as i s seen i n the p e r c e n t a g e s g i v e n above . I n c o r p o r a t i n g the a d v e c t i v e energy i n the t h e o r y s h o u l d a l s o improve the agreement i n the r e s u l t s f o r the bare p l o t , w h i c h was i n the n o r t h - e a s t c o r n e r o f the s i t e and was p r e s u m a b l y r e c e i v i n g a d v e c t e d heat from the s t r i p p l o t s . E r r o r s i n p r e d i c t e d LE f o r A p r i l 7 ( T a b l e 2 . 6 ) , a r e l a t i v e l y c o o l e r day due to c l o u d s , a re s l i g h t l y l e s s than t h a t f o r A p r i l 8 but the a d v e c t i o n e f f e c t s appeared to be of s i m i l a r i m p o r t a n c e . In a d d i t i o n to a d v e c t i o n , b u o y a n c y - d r i v e n c o n v e c t i o n r e s u l t i n g from the s i n k i n g m o t i o n o f a i r above the c o l d bare s t r i p and r i s i n g m o t i o n o f a i r above the hot mulch s t r i p may c o n t r i b u t e to the net energy f l o w on a h o r i z o n t a l s c a l e (Tanner e t a l . , 1 9 8 7 ) ; h o w e v e r , compared to a d v e c t i o n , i t i s l i k e l y to be i n s i g n i f i c a n t . On A p r i l 7 , the d a y t i m e c u m u l a t i v e LE from the s o i l 1n the mulch s t r i p was found t o be e i t h e r n e g a t i v e or n e g l i g i b l y s m a l l (see T a b l e 2 . 7 ) . On the f o l l o w i n g day the c u m u l a t i v e LE was p o s i t i v e , even though the s o i l s u r f a c e g a i n e d m o i s t u r e 95 Table 2.7 Measured and predicted evaporation rates (mm h"1) 1n the mulch strip at various directions on April 7 and 8, 1985. April 7, 1985 Strip direction N-S NE-SW E-W Period (PST) Meas. Pred. Meas. Pred. Meas. Pred. 6-8 0.002 -0.007 0.004 -0.007 0.020 -0.006 8-10 -0.024 -0.009 -0.009 -0.009 -0.014 -0.009 10-12 -0.036 -0.005 -0.017 -0.005 -0.030 -0.006 12-14 -0.009 -0.000 -0.010 -0.001 -0.014 -0.001 14-16 -0.015 0.008 0.005 0.007 0.007 0.007 16-18 0.012 0.025 0.037 0.024 0.024 0.024 Total* -0.140 0.024 0.020 0.018 -0.014 0.018 April 8, 1985 8-10 -0.009 -0.012 -0.020 -0.013 -0.029 -0.013 10-12 0.008 0.000 0.010 -0.000 0.007 -0.001 12-14 0.016 0.008 0.013 0.007 0.021 0.006 14-16 0.014 0.008 0.015 0.007 0.028 0.006 16-18 0.023 0.014 0.026 0.014 0.021 0.013 Total* 0.104 0.036 0.088 0.030 0.096 0.022 *Total = Z(hourly rates x time interval); Meas. = Measured; Pred. = Predicted. Note: Predicted evaporation rates are taken at the middle of the period of consideration. 96 d u r i n g the e a r l y hours of the d a y . A p p a r e n t l y , the wet m u l c h , f o l l o w i n g heavy r a i n f a l l i n the p r e v i o u s weeks , must have a c t e d as a s o u r c e of p y and r e v e r s e d or r e d u c e d the * m g r a d i e n t i n ( P V S ( T 0 ) ~ P V ) i n the mulch l a y e r r e s u l t i n g i n e i t h e r c o n d e n s a t i o n or v e r y l i t t l e e v a p o r a t i o n . I t can o n l y be e x p l a i n e d by the model i f r m i s d r a s t i c a l l y reduced from 12 -1 -1 1 x 10 s m to < 500 s m to a l l o w e v a p o r a t i o n from the mulch to o c c u r . T h i s c o n t r a d i c t s the o r i g i n a l a s s u m p t i o n t h a t e v a p o r a t i o n f rom the mulch i s n e g l i g i b l e , w h i c h may be t r u e o n l y f o r a d r y m u l c h . Note t h a t an r m o f 300 s m _ 1 i s r e q u i r e d by the model to a p p r o x i m a t e the measured LE on A p r i l 7 ( a l l p r e d i c t e d LE i n T a b l e 2 .7 are c a l c u l a t e d w i t h r m = 300 s rn"1) w h i l e a s l i g h t l y h i g h e r r m (" 500 s m " 1 ) , a f t e r the mulch had d r i e d f o r a d a y , may be a p p r o p r i a t e to match the measured LE on the f o l l o w i n g d a y . The measured and m o d e l l e d t e m p e r a t u r e s a t 0 .005 m depth i n the m i d d l e of the bare s t r i p o r i e n t e d i n N - S , NE-SW and E-W d i r e c t i o n s and f o r the bare p l o t on A p r i l 8 a re p r e s e n t e d i n F i g . 2 . 1 2 . For the mulched p l o t s the model t ends to u n d e r e s t i m a t e these d u r i n g the d a y t i m e w h i l e i t o v e r e s t i m a t e s m a r g i n a l l y f o r r e s t of the d a y . The u n d e r e s t i m a t i o n (and p o s s i b l y the o v e r e s t i m a t i o n as w e l l ) i s a t t r i b u t e d m a i n l y to the m i c r o - s c a l e a d v e c t i o n d i s c u s s e d p r e v i o u s l y t h a t i s not i n c o r p o r a t e d i n the m o d e l . An u n d e r e s t i m a t i o n i n bo th LE and T Q c annot be due to an e r r o r i n z Q or k s , the o t h e r parameter s w h i c h a re somewhat u n c e r t a i n . An i n c r e a s e i n z „ w i l l i n c r e a s e o LE and w i l l d e c r e a s e T„ w h i l e an i n c r e a s e i n k„ w i l l warm up o s r 97 F i g u r e 2 .12 The measured ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s a t 0 .005 m depth i n the b a r e , and i n the bare s t r i p o f the N - S , NE-SW and E-W s t r i p p l o t s on A p r i l 8 , 1985 . 98 the s o i l f a s t e r a t the expense o f reduced L E . S o , o n l y an i m p o r t a t i o n o f e n e r g y , as by a d v e c t i o n to the bare s t r i p can i n c r e a s e b o t h LE and s o i l t e m p e r a t u r e s s i m u l t a n e o u s l y . I t was found t h a t i f the measured a i r t e m p e r a t u r e s f rom 10 to 15 PST on A p r i l 8 a re a r t i f i c i a l l y d o u b l e d , w h i c h a p p r o x i m a t e l y s i m u l a t e s t h i s a d v e c t i o n , a 20% r i s e 1n d a y t i m e c u m u l a t i v e LE and a 4 ° C r i s e 1n T Q a t 13 h compared w i t h the b a s i c run ( T a b l e 2 . 8 ) o c c u r s i n the bare N-S s t r i p ; the measured and p r e d i c t e d v a l u e s a re then i n a g r e e m e n t . T h i s i s , however , i n c o n t r a d i c t i o n to the e f f e c t s o f t e n seen w i t h l o c a l a d v e c t i o n i n w h i c h i n c r e a s e s i n LE o c c u r a t the expense o f c o o l i n g the wet s o i l s u r f a c e ( e . g . , i r r i g a t e d g r a s s ) , i . e . a s o - c a l l e d w e t - b u l b d e p r e s s i o n o c c u r s ( R i d e r e t a l . , 1 9 6 3 ) . Though the model ( w i t h o u t a d v e c t i o n ) u n d e r e s t i m a t e s the a b s o l u t e v a l u e s o f s o i l t e m p e r a t u r e , i t d i d p r e d i c t the r e l a t i v e d i f f e r e n c e s between the t r e a t m e n t s ( o r i e n t a t i o n s ) c o r r e c t l y . Both the model and the measurements d e m o n s t r a t e t h a t e a r l y and l a t e i n the d a y t i m e the bare E-W s t r i p warms up more r a p i d l y than the NE-SW s t r i p , w h i l e the N-S s t r i p warms up the s l o w e s t . T h i s i s a t t r i b u t e d to the s h a d i n g of the bare s t r i p by the m u l c h , w h i c h i s m i n i m a l f o r the E-W s t r i p a t the se t i m e s . Both measured and m o d e l l e d t e m p e r a t u r e s f o r the bare p l o t a re i n good agreement ( the same z Q as f o r June 14 was u s e d ) . S i n c e the p l o t s were wet on A p r i l 8 t h i s s u p p o r t s the i d e a t h a t p a t c h i n e s s l e d to the d i s a g r e e m e n t s i n s o i l t e m p e r a t u r e s p r e v i o u s l y noted f o r June 14. F i g . 2 . 1 3 shows the IRT measured and m o d e l l e d s u r f a c e 99 Table 2.8 Measured and predicted soil temperature and temperature amplitude (°C) at 0.005 m depth In the bare strip at various directions on September 3, 1984 and April 7 and 8, 1985. September 3, 1984 Soil temperature at 0.005 m Temperature amplitude at 13 h (PST) at 0.005 m Strip direction Meas. Pred. Meas. Pred. Bare 35.5 35.2 12.2 11.9 N-S 30.2 30.3 9.8 9.4 NE-SW 31.5 30.0 10.7 9.1 E-W 33.6 29.6 11.6 8.9 April 7, 1985 Bare 18.6 20.4 10.8 9.9 N-S 17.2 16.7 10.1 6.8 NE-SW 17.0 16.4 9.8 6.9 E-W 17.1 16.2 9.8 6.6 April 8, 1985 Bare 26.4 26.5 N-S 25.5 21.2 NE-SW 25.3 20.8 E-W 25.1 20.5 11.3 10.7 10.5 10.5 10.8 7.8 7.5 7.3 Meas. = Measured, Pred. => Predicted. 100 0 '—— 1 1 1 1 1 1 1 i ' 0 4 8 12 16 20 24 TIME (h, PST) F i g u r e 2 . 1 3 The measured ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l s u r f a c e t e m p e r a t u r e s i n the b a r e , and i n the bare s t r i p o f the N - S , NE-SW and E-W s t r i p p l o t s on A p r i l 8 , 1985 . The s u r f a c e t e m p e r a t u r e s were measured by the b o l o m e t e r . 101 t e m p e r a t u r e s i n the bare s t r i p a t d i f f e r e n t o r i e n t a t i o n s and i n the bare p l o t on A p r i l 8 , 1985. IRT measured mulch s u r f a c e t e m p e r a t u r e s a re a l s o i n c l u d e d . I t r e v e a l s t h a t the mulch s t r i p s u r f a c e t e m p e r a t u r e s a re 8-10 °C h i g h e r t h a n the nearby bare s t r i p s u r f a c e t e m p e r a t u r e s w h i c h s u p p o r t s the i d e a t h a t a d v e c t i o n from the mulch s t r i p s to the bare s t r i p s c o u l d have o c c u r r e d . F i g . 2 .14 shows the measured and m o d e l l e d s o i l t e m p e r a t u r e s a t 0 .005 m depth i n the bare s t r i p s a t d i f f e r e n t o r i e n t a t i o n s and i n the bare p l o t on September 3 , 1984. At t h a t t ime the mulch had been i n p l a c e f o r o n l y 3 weeks and the s o i l was i n the d r i e s t c o n d i t i o n m e a s u r e d . I t i s c l e a r e r here t h a n f o r A p r i l 8 , 1985 ( F i g s . 2 . 12 and 2 . 1 3 ) t h a t the bare E-W s t r i p warmed up f a s t e s t , f o l l o w e d by the NE-SW s t r i p , w h i l e the N-S s t r i p warmed up the s l o w e s t f o r the rea son d e s c r i b e d p r e v i o u s l y . The measured and m o d e l l e d t e m p e r a t u r e s a re i n b e t t e r agreement f o r September 3 than t h a t o f A p r i l 8 , 1985, e x c e p t f o r the E-W d i r e c t i o n . The wind was b l o w i n g p r e d o m i n a n t l y sou thward a l l d a y , i . e . , p e r p e n d i c u l a r t o the E-W s t r i p s . I t i s presumed t h a t a d v e c t i o n was more i m p o r t a n t f o r the E-W s t r i p s than those o r i e n t e d i n p a r a l l e l o r n e a r l y p a r a l l e l to the w i n d d i r e c t i o n . I t i s a l s o p o s s i b l e t h a t w i t h the g r e a t e r mulch t h i c k n e s s on S e p t . 3 the bare s t r i p c o u l d have been a e r o d y n a m i c a l l y s h e l t e r e d . T h i s c o n t r a d i c t s the a d v e c t i o n t h e o r y and was r u l e d out f o r A p r i l 8 ; l a c k o f measured LE d a t a on September 3 and the d i f f e r e n c e s i n mulch h e i g h t and wind speed p r e c l u d e making such a d e f i n i t i v e 102 0 4 8 12 16 20 24 TIME (h, PST) F i g u r e 2 .14 The measured ( i s o l a t e d s y m b o l s ) and c a l c u l a t e d ( c u r v e s ) s o i l t e m p e r a t u r e s a t 0 .005 m depth i n the b a r e , and i n the bare s t r i p o f the N - S , NE-SW and E-W s t r i p p l o t s on September 3 , 1984. 103 c o n c l u s i o n f o r September 3 . Which mechanism d o m i n a t e s r e q u i r e s f u r t h e r i n v e s t i g a t i o n . 2 . 5 S e n s i t l v i . t y _ A n a l y si.s A s e n s i t i v i t y a n a l y s i s was p e r f o r m e d to d e t e r m i n e the e f f e c t s o f z Q , a t m o s p h e r i c s t a b i l i t y , r ^ , k s , , E f , and f on the e v a p o r a t i v e f l u x e s and s o i l t e m p e r a t u r e s f o r the bare a n d / o r u n i f o r m l y mulched p l o t s on June 14, 1984. In a d d i t i o n , the e f f e c t o f s h a d i n g on the s o i l s u r f a c e t e m p e r a t u r e was examined f o r the s t r i p s on A p r i l 8 , 1985. The p a r a m e t e r s t e s t e d , v a l u e s a s s i g n e d to them and t h e i r e f f e c t on LE ( f o r the measurement p e r i o d ) and the s o i l s u r f a c e maximum t e m p e r a t u r e (T Q m a x ) compared to the b a s i c runs a re p r e s e n t e d i n T a b l e 2 . 9 . -4 -3 C h a n g i n g z Q f rom 5 x 10 m to 2 x 10 m reduced TQ m a x ( o c c u r r e d a t 14 h PST) i n the bare p l o t by 3 .0 ° C . I t had l i t t l e e f f e c t on L E , i n c r e a s i n g i t by o n l y 0 .01 mm (0.6%) over the 12 h p e r i o d . I n c r e a s i n g z Q causes more o f the a v a i l a b l e energy ( R n - L E ) to be d i s s i p a t e d as s e n s i b l e heat energy r e s u l t i n g i n a r e d u c t i o n i n the s o i l heat f l u x d e n s i t y and a d e c r e a s e i n s o i l t e m p e r a t u r e s . S i m i l a r f i n d i n g s have been r e p o r t e d by S t a t h e r s e t a l . ( 1 9 8 5 ) . The e f f e c t f rom the d i a b a t i c i n f l u e n c e f u n c t i o n s i s s i m i l a r to t h a t o f z „ i n o n a t u r e ; s i n c e both a f f e c t the a e r o d y n a m i c r e s i s t a n c e s . A Table 2.9 Effect of Changes In Parameters on the Evaporative Fluxes and Sol i Temperatures. Parameters Mulch appl 1ca< Change in Ident i f ica-tion Changed t lon rate LE (mm) From (basic) To (t/ha) o„m< (°c: 5 x 10" 4 m 2 x 10" 3 m 0 0.01 (0.6) 1 3.0 d Atmospheric S tab i l i t y h Unstable ( r , and v a r* are reduced by 1.5 times) Neutral 20 0.01 ( i . 7 ) 1 1.01 '! 300 s m"1 0 0 1.74(92.1) 1 8.2 d 0 < z < 0.10 m 0.93j z > 0.10 m 1.05 1.87 2.11 0 0.21 (10.9) d 3.0 d h 0.96 1.92 20 0.06 (20.6) d 1.9 d <: 0.067 0 20 0.12 (39.3) d 3.7 d E f 4.5 1.0 20 0.24 (77.4) d 1.5 d fm 0 1.0 20 0.17 (56.1) 1 5.2 1 1 ' i n c r e a s e ; d • decrease; the numbers in parentheses for LE indicate the change in percentage over the measured to ta l . 105 change from u n s t a b l e to the n e u t r a l a t m o s p h e r i c c o n d i t i o n t h a t i n c r e a s e s r!j and rl by a f a c t o r o f 1.5 ( e s t i m a t e d from the a a d i a b a t i c I n f l u e n c e f u n c t i o n s i n C a m p b e l l , 1977) i n c r e a s e d T o max ( ° c c u r r e d a t 1 4 n P S T ) ° y 1 * ° ° c a n d d a y t i m e LE by 1.7% f o r the 20 t / h a p l o t . The p o t e n t i a l e v a p o r a t i o n (LEp) f rom the bare s o i l s u r f a c e f o r the same i n p u t energy o v e r 12 h p e r i o d was 3 .63 mm s -1 w h i l e i n c o r p o r a t i n g r * = 300 s m reduced LE to 1.89 mm ( 0 . 5 2 L E I and i n c r e a s e d T„ m a w f rom 3 2 . 6 t o 4 0 . 8 °C ( o v e r v p ' o max 1 8 ° C ) a t 14 h PST (note t h a t f o r the p o t e n t i a l case T Q m a x o c c u r s a t 13 h P S T ) . W i t h LE a t the p o t e n t i a l r a t e the measured and p r e d i c t e d t e m p e r a t u r e s a re then i n good a g r e e m e n t , e . g . , 3 1 . 7 ° v s . 3 2 . 2 °C a t 0 .005 m depth a t 14 h PST. T h e r e f o r e we s u s p e c t e d t h a t the spot where the t h e r m o c o u p l e s were l o c a t e d was r e l a t i v e l y w e t . On June 14 the p l o t was i n t r a n s i t i o n from wet to d r y , so t h a t p a t c h i n e s s o f the s u r f a c e was e x p e c t e d . In p r a c t i c e C g and k g can be c a l c u l a t e d f a i r l y a c c u r a t e l y from s o i l m o i s t u r e e s t i m a t e s (de V r i e s , 1963) 1f the f r a c t i o n a l d i s t r i b u t i o n of s o i l c o n s t i t u e n t s a re e x a c t l y known. S i n c e q u a r t z c o n t e n t o f the s o i l was not e x p e r i m e n t a l l y d e t e r m i n e d and the v a l u e f o r the t h e r m a l c o n d u c t i v i t y o f the s o i l s o l i d s was adopted on the b a s i s of s o i l t e x t u r e i t was i m p o r t a n t to a s s e s s the e f f e c t o f k g on the s o i l t e m p e r a t u r e d i s t r i b u t i o n . H o l d i n g a l l o t h e r v a r i a b l e s c o n s t a n t and d o u b l i n g the v a l u e of k t h r o u g h o u t the s o i l p r o f i l e reduced the T n by 3 . 0 ° and 1.9 °C (a t 14h 106 PST) and LE by "11 and 21% f o r the bare and 20 t / h a p l o t , r e s p e c t i v e l y . A r a t i o n a l approach i s to c o n s i d e r "80% of sand i n the s o i l m a t r i x as q u a r t z f r a c t i o n ( B r a d y , 1984; F i g . 2 . 4 , p - 4 0 ) . In so d o i n g , k 1s found to I n c r e a s e by 20 - 30% o n l y (because o f the r e l a t i v e l y low b u l k d e n s i t y ) o v e r the v a l u e s used 1n the m o d e l . I t t h e r e f o r e r e v e a l s t h a t a change i n s o i l t h e r m a l p r o p e r t i e s w i t h i n r e a l i s t i c p h y s i c a l l i m i t s w i l l have a s m a l l e f f e c t upon m o d e l l e d s o i l t e m p e r a t u r e s . However , 1t a p p e a r s t h a t a combined e f f e c t o f s m a l l e r r^ and g r e a t e r k g , bo th r e s u l t i n g from s o i l wetness i n the v i c i n i t y of the t h e r m o c o u p l e s i n the bare p l o t made the measured s o i l t e m p e r a t u r e s l o w e r than the p r e d i c t e d v a l u e s . The model i s h i g h l y s e n s i t i v e to the r a d i a t i o n l o a d on the s u r f a c e , a change 1n from 0 .067 to 0 .0 t h a t d i m i n i s h e s S d e n t i r e l y d e c r e a s e d TQ m a x by 3 . 7 ° C ( a t 14h PST) and LE by ~40% f o r the 20 t / h a r a t e t r e a t m e n t . E v a p o r a t i o n from a wet s o i l i s g r e a t l y reduced by a mulch ( J a c k s e t a l . , 1955; Army et a l . • , 1 9 6 1 ) , a t t r i b u t e d m a i n l y to the l o w e r P u « . ( T f t ) because o f l o w e r s o i l s u r f a c e t e m p e r a t u r e V w w a n d / o r the g r e a t e r r e s i s t a n c e of the mulch to e v a p o r a t i o n (Tanner et a l . , 1 9 8 7 ) . C o n s e q u e n t l y , most o f the energy t h a t r e a c h e s the s o i l s u r f a c e hea t s the s o i l . The model p r e d i c t s t h a t ( R ° - G Q ) i s e i t h e r n e g a t i v e or m a r g i n a l l y p o s i t i v e i . e . , a s m a l l f r a c t i o n ( i f any) of the r a d i a n t energy i s used f o r e v a p o r a t i o n . I t a l s o p r e d i c t s t h a t the energy u t i l i z e d f o r LE i s t r a n s p o r t e d downward from the top o f the mulch onto the s o i l s u r f a c e m a i n l y by the wind e d d i e s ( p a r a m e t e r i z e d by E- i n 107 the m o d e l ) . E f has a s e v e r e e f f e c t on LE and a r e l a t i v e l y s m a l l e f f e c t on s o i l t e m p e r a t u r e s . A change 1n E f f rom 4 . 5 to 1.0 ( c o n d u c t i o n i s t h e n the o n l y mode o f mass and energy t r a n s f e r t h r o u g h the mulch l a y e r ) f o r the 20 t / h a r a t e reduced LE by 78% ( the measured d a y t i m e t o t a l was 0 .32 mm) and T m a x by 1 . 5 ° C ( a t 14 h P S T ) . I n t e r e s t i n g l y , s i n c e the a d d i t i o n a l energy b r o u g h t down by the enhancement i s p a r t i t i o n e d between LE and G Q , bo th LE and s o i l t e m p e r a t u r e s i n c r e a s e or d e c r e a s e c o n c u r r e n t l y w i t h an i n c r e a s e or d e c r e a s e i n E f . Depending on the type o f the mulch i n u s e , longwave r a d i a t i o n exchange between the s o i l and the mulch s u r f a c e may i n f l u e n c e the s o i l t e m p e r a t u r e s to a g r e a t e x t e n t . W i t h f_ = 1, LE i n c r e a s e d by 50% and the T„ m a „ i n c r e a s e d by 5 .2 °C in u max ( a t 14 h PST) compared to the s i t u a t i o n w i t h f = 0 In the 20 t / h a p l o t . A c c o r d i n g to the m o d e l , s h a d i n g , a l t h o u g h v e r y e f f e c t i v e i n c a u s i n g a b i g t e m p e r a t u r e d i f f e r e n c e a t any i n s t a n t between s u n l i t and shaded a r e a s , does not i n f l u e n c e the s o i l t e m p e r a t u r e s i n the m i d d l e o f the bare s t r i p s i g n i f i c a n t l y because o f i t s s h o r t e x i s t e n c e d u r i n g the d a y t i m e . Row d i r e c t i o n , w h i c h i n p a r t d e t e r m i n e s how l o n g a p o i n t i n the bare s t r i p remains shaded a t d i f f e r e n t hours o f the d a y , a l s o has a s m a l l e f f e c t on the T„ m a „ o f the bare s t r i p ; a maximum o max v d i f f e r e n c e o f 3 .4 °C ( o c c u r r e d a t 12 PST) a t 0 .005 m depth was measured between the N-S and E-W o r i e n t a t i o n s f o r the d r i e s t s o i l c o n d i t i o n (September 3 , 1984) w h i l e the d i f f e r e n c e was < 1 ° C f o r the wet s o i l c o n d i t i o n ( A p r i l 7 or 8 , 1985 ; see 108 T a b l e 2 . 8 ) . The d i f f e r e n c e of 3 .4 °C was m a i n l y the r e s u l t of a d v e c t i o n ( d i s c u s s e d i n the R e s u l t s s e c t i o n ) , and p a r t l y due to the e f f e c t s o f s h a d i n g and row d i r e c t i o n c o m b i n e d . S o i l t e m p e r a t u r e s were a l s o p r e d i c t e d u s i n g one-d i m e n s i o n a l m o d e l s , t o d e t e r m i n e the i n f l u e n c e of the l a t e r a l heat f l o w between the s o i l under the bare s t r i p and t h a t under the mulch s t r i p . T h i s was done w i t h the t w o - d i m e n s i o n a l model by a s s i g n i n g the same but a p p r o p r i a t e p a r a m e t e r s and t h e r m a l p r o p e r t i e s to both the s t r i p s . The s o i l t e m p e r a t u r e s were i n i t i a l i z e d w i t h the t e m p e r a t u r e p r o f i l e from the p r e v i o u s n i g h t c o r r e s p o n d i n g to e i t h e r the m i d d l e of the mulch or the bare s t r i p as the case may b e . The r e s u l t s o b t a i n e d w i t h the 1-d model a re compared w i t h those o b t a i n e d i n the c e n t e r of the mulch and bare s t r i p s w i t h the 2-d model f o r the N-S o r i e n t e d p l o t ( chosen randomly) on A p r i l 8 , 1985 . I t was found t h a t f o r the 1-d s i m u l a t i o n s w i t h the u n i f o r m l y a p p l i e d mulch case the p r e d i c t e d s u r f a c e t e m p e r a t u r e was 0 .4 °C h i g h e r a t 7 h and 0 .4 °C l o w e r a t 17 h than t h a t p r e d i c t e d i n the m i d d l e o f the mulch s t r i p w h i l e w i t h the bare s o i l the s u r f a c e t e m p e r a t u r e was 0 . 8 °C l o w e r a t 3 h and 3 .1 °C h i g h e r a t 17 h t h a n i n the m i d d l e of the bare s t r i p . Note t h a t the 1-d bare s o i l c a s e , f o r w h i c h t h e r e a re no s h a d i n g e f f e c t s , i s compared w i t h the 2-d bare s t r i p case i n w h i c h the s h a d i n g e f f e c t s were not e l i m i n a t e d . That i s , the d i f f e r e n c e o f 3 . 1 ° C a t the m i d d l e o f the bare s t r i p has not r e s u l t e d from the e f f e c t o f l a t e r a l heat f l o w o n l y but a l s o from the e f f e c t o f s h a d i n g by the m u l c h . However , when the e f f e c t of s h a d i n g was m i n i m a l , e . g . , a t 13 h , the t e m p e r a t u r e d i f f e r e n c e was ~ 1.2 ° C . T h i s d e m o n s t r a t e s t h a t the l a t e r a l heat f l o w i s damped out by the t i m e i t t r a v e l s about 0 .15 m i n t o the s o i l under the mulch s t r i p . For d r y s o i l c o n d i t i o n s , h o r i z o n t a l t e m p e r a t u r e d i f f e r e n c e s may be g r e a t e r ; however , g r e a t l y reduced k g s h o u l d r e t a r d heat t r a n s f e r . 2 . 6 C o n c l u s i o n s A model has been d e v e l o p e d to s t u d y the energy b a l a n c e of the a t m o s p h e r e - m u l c h - s o i l sys tem and to p r e d i c t e v a p o r a t i o n from and t e m p e r a t u r e s i n s o i l s c o v e r e d w i t h m u l c h , e i t h e r u n i f o r m l y or i n s t r i p s o r i e n t e d i n d i f f e r e n t d i r e c t i o n s . The model has been t e s t e d on s o i l s c o v e r e d w i t h s t r a w m u l c h , e i t h e r u n i f o r m l y a t r a t e s o f 2 , 10 , o r 20 t / h a or i n s t r i p s w i t h 0 .30 m wide (a t the r a t e of 10 t / h a ) a l t e r n a t i n g w i t h 0 .10 m wide bare s t r i p s o r i e n t e d i n N - S , NE-SW and E-W d i r e c t i o n s . The model p r e d i c t e d s u r f a c e and s u b s u r f a c e s o i l t e m p e r a t u r e s a c c u r a t e l y but u n d e r e s t i m a t e d e v a p o r a t i o n (by 28%) from the s o i l (when wet) f o r the u n i f o r m l y a p p l i e d mulch c a s e , and u n d e r e s t i m a t e d both e v a p o r a t i o n (by 33%) and s o i l s u r f a c e t e m p e r a t u r e s (by ~ 4 ° C ) i n the bare s t r i p f o r the s t r i p c a s e . The d i f f e r e n c e s are r e s p e c t i v e l y a t t r i b u t e d to the p e n e t r a t i o n of wind g u s t s i n t o the s t r a w m u l c h , and the m i c r o s c a l e a d v e c t i o n from the warm mulch s t r i p s onto the 110 r e l a t i v e l y c o l d bare s t r i p s . S e n s i t i v i t y a n a l y s e s showed t h a t the s o i l t h e r m a l p r o p e r t i e s and the a e r o d y n a m i c roughness of the s u r f a c e can be e s t i m a t e d w i t h s u f f i c i e n t a c c u r a c y from r e l a t i o n s h i p s i n the l i t e r a t u r e . They a l s o r e v e a l e d t h a t both LE and s o i l t e m p e r a t u r e s c o u l d be changed s i g n i f i c a n t l y w i t h changes i n s o l a r energy i n p u t , t h a t LE under the mulch was dependent l a r g e l y on the p e n e t r a t i o n of wind e d d i e s , t h a t both LE and s o i l s u r f a c e t e m p e r a t u r e c o u l d be i n c r e a s e d s i g n i f i c a n t l y by r a d i a t i v e l y c o u p l i n g the s o i l s u r f a c e w i t h the mulch s u r f a c e , t h a t n e i t h e r LE nor s o i l s u r f a c e t e m p e r a t u r e under the mulch was i n f l u e n c e d much by the a t m o s p h e r i c s t a b i l i t y c o r r e c t i o n f a c t o r s , and t h a t the s u r f a c e t e m p e r a t u r e s i n the bare s o i l s t r i p d u r i n g the peak hours of the day were not g r e a t l y a f f e c t e d by the s h a d i n g due to the mulch s t r i p . 2 .7 R e f e r e n c e s 1. Army, T . J . , A . F . W i e s e , and R . J . 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M o n t e i t h ( e d . ) , V e g e t a t i o n and the A t m o s p h e r e , V o l . I , Academic P r e s s , p p . 5 7 - 1 0 9 , 1975. 3 9 . W i l l i a m s , G . D . V . , J . S . M c K e n z i e , and M . I . S h e p p a r d . M e s o s c a l e a g r o c l i m a t i c r e s o u r c e mapping by c o m p u t e r , an example f o r the Peace R i v e r r e g i o n . A g r i c . M e t e o r o l . 2 1 : 9 3 - 1 0 9 , 1980. CONCLUDING REMARKS LE under the mulch i s g r e a t l y dependent on the v e r t i c a l t r a n s p o r t o f energy by wind e d d i e s . C o n s e q u e n t l y a p r e c i s e p r o c e d u r e t h a t r e l a t e s the t r a n s p o r t o f energy t h r o u g h the mulch w i t h wind e d d i e s r a t h e r than the e m p i r i c a l a p p r o x i m a t i o n (by the enhancement f a c t o r ) used h e r e i n i s r e q u i r e d . For a d r y i n g m u l c h , the m o i s t u r e g r a d i e n t i n the mulch l a y e r c o u l d be s i g n i f i c a n t and a p p l i c a t i o n of b u l k t r a n s f e r e q u a t i o n s may not be a p p r o p r i a t e . For such a case the mulch l a y e r must be d i v i d e d i n t o s u b - l a y e r s and an e v a l u a t i o n o f the energy b a l a n c e f o r each s u b - l a y e r w i l l t h e n t a k e i n t o a c c o u n t the vapour s o u r c e - s i n k s 1n the r e s i d u e t h a t a re n e g l e c t e d 1n the m o d e l . As w e l l a c r i t e r i o n may be d e v e l o p e d based on the w a t e r c o n t e n t o f the mulch l a y e r to d e t e r m i n e the e f f e c t i v e r e s i s t a n c e o f the mulch to e v a p o r a t i o n ; as the a s s u m p t i o n t h a t no e v a p o r a t i o n o c c u r s from the mulch was found to be i n a c c u r a t e . M i c r o s c a l e a d v e c t i o n was found to be s i g n i f i c a n t i n p r e d i c t i n g bo th LE and s o i l t e m p e r a t u r e s c o r r e c t l y i n the bare s t r i p . I n c o r p o r a t i n g a d v e c t i v e energy i n t o f u t u r e t i l l a g e -s e n s i t i v e s o i l t e m p e r a t u r e models i s v i t a l to i n c r e a s e t h e i r p r e d i c t i v e c a p a b i l i t y . S p r i n g t i m e s o i l t e m p e r a t u r e s i n the bare s t r i p s were found to be s i m i l a r to the bare p l o t , a t t r i b u t e d l a r g e l y to the a d v e c t i o n from the warm mulch s t r i p s to the r e l a t i v e l y c o o l e r bare s t r i p s . T h i s s u g g e s t s t h a t the i n f l u e n c e o f the s o i l t h e r m a l reg imes on seed g e r m i n a t i o n i n 115 the bare s t r i p w i l l be s i m i l a r to t h a t i n the bare p l o t . T h e r e f o r e i t appear s p o s s i b l e to have the s o i l m o s t l y c o v e r e d w i t h mulch w i t h the b e n e f i t s f o r e r o s i o n c o n t r o l t h a t f o l l o w t h e r e f r o m , w i t h o u t d e l e t e r i o u s e f f e c t s on the s o i l t h e r m a l r e g i m e s . A l s o , l a t e r 1n the season p l a n t s i n the bare s t r i p s h o u l d b e n e f i t more from the c o n s e r v e d s o i l m o i s t u r e and l e s s s e v e r e t e m p e r a t u r e f l u c t u a t i o n s . F i n a l l y , no p a r t i c u l a r s t r i p o r i e n t a t i o n has any advantage o v e r o t h e r s i n s o i l t h e r m a l and m o i s t u r e reg imes though i t was s p e c u l a t e d i n the I n t r o d u c t i o n t h a t a p a r t i c u l a r o r i e n t a t i o n c o u l d be p r e f e r a b l e . 116 A2£end2x_I A n a l y t i c a l S o l u t i o n to the O n e - D i m e n s i o n a l Heat T r a n s f e r P r o b l e m w i t h Upper Boundary C o n d i t i o n o f t h e 3 r d k i n d and Inhomogeneous Thermal P r o p e r t i e s £ £ 0 b l e m _ D e f 2 n l t i . on The o n e - d i m e n s i o n a l heat t r a n s f e r e q u a t i o n i s where z 1s the d e p t h , t i s the t i m e , T ( z , t ) i s the t e m p e r a t u r e , C 1s the v o l u m e t r i c heat c a p a c i t y , and k 1s the t h e r m a l c o n d u c t i v i t y . C and k a re g i v e n b y : C & -v at az ( i ) c 6 c ( z + h) (2a) k 6 k ( z + h) f o r 0 < z < d and C C CO (2b) k k CO f o r z > d . The boundary c o n d i t i o n a t z = 0 i s a G ( 0 , t ) + B T ( 0 , t ) = f ( t ) , (3) 117 where a and B a r e c o n s t a n t s , f ( t ) 1s a known f u n c t i o n , and 6 ( z , t ) i s the heat f l u x d e n s i t y g i v e n by F o u r i e r ' s law as G ( z , t ) = - k ( z ) § J • (4) As w e l l Lim |I = 0 Z-*CD T ( Z , t ) = T ( Z ) , (5) where T ( Z ) i s a known c o n s t a n t w i t h the " b a r " r e f e r r i n g to the t i m e - a v e r a g e o v e r the p e r i o d (P) o f I n t e r e s t . O n l y s o l u t i o n s p e r i o d i c i n t w i t h p e r i o d P are c o n s i d e r e d . Bo th T ( z , t ) and G ( z , t ) a r e c o n t i n u o u s a t z=d. S o l u t i o n T ( z , t ) i s r e p r e s e n t e d as a complex F o u r i e r s e r i e s , I . e . , T ( z . t ) = ? c j ( z ) e 1 n w t , (6) n = - c o where u = 2 i r / P . S u b s t i t u t i o n I n t o (4) shows t h a t G ( z , t ) 1s r e p r e s e n t e d by G ( z , t ) = ? c j ( z ) e i n w t , (7) n = - co 118 where c j ( z ) = - k ( z ) d c j ( z ) / d z . A l s o f ( t ) 1s g i v e n by f ( t ) = ? c ?e 1 n w t , (8) n = -co where the c ^ ' s a re known ( d e t e r m i n e d from the i n v e r s e t r a n s f o r m ) . S i n c e each F o u r i e r c o e f f i c i e n t f o r - n i s the complex c o n j u g a t e o f t h a t f o r n , o n l y n > 0 need be c o n s i d e r e d . As shown by Novak (1986) s u b s t i t u t i o n o f (6) i n t o (1) y i e l d s f o r n > 0 and 0 < z < d an e q u a t i o n o f the B e s s e l type w i t h s o l u t i o n ( l - Y k ) / 2 c j ( z ) = (z+h) C A n H ^ > ( 1 3 / 2 v z ) + B n H j 2 ) ( 1 3 / 2 v 2 ) ] , (9) where H* 7 and ' a re the Hankel f u n c t i o n s o f o r d e r p o f the 1s t and 2nd k i n d s , r e s p e c t i v e l y , P = ± (1 - Y k ) / ( 2 - Y k + Y c ) , . ( 2 - Y k + Y c ) / 2 v z = v / n c o 6 c / 6 k [ 2 / ( 2 - v k + Y c ) ] ( z + h ) , and A„ and B„ a re unknown c o n s t a n t s . The s o l u t i o n i s v a l i d f o r n n (2 - v k + Y c ) > 0 and the s i g n i n the e x p r e s s i o n f o r p can a l w a y s be chosen so t h a t p > 0 . For n > 0 and z > d , c^ i s g i v e n by 119 c j ( z ) = D n e x p [ - ( 1 + 1) ^ n w C ' / k ^ z - d ) ] (10) where D n 1s an unknown c o n s t a n t . A c c o r d i n g t o (10) c j ( z ) s a t i s f i e s ( 5 a ) . S u b s t i t u t i n g ( 6 ) , ( 7 ) , and (8) i n t o (3) and the c o n t i n u i t y c o n d i t i o n s a t z = d y i e l d s ( a f t e r r e p l a c i n g c ^ ( z ) and I t s d e r i v a t i v e 1n (3) by (9) and i t s d e r i v a t i v e ) 3 l i n e a r e q u a t i o n s f o r A n , B n , and D n , w h i c h a re e a s i l y s o l v e d by m a t r i x I n v e r s i o n . For n=0 and f o r a l l z f rom (1) w i t h (6) and ( 7 ) , c5<z> " ct >z k l l j + T < Z >* ( " ) where c Q (= c o n s t a n t ) i s d e t e r m i n e d from ( 3 ) , i . e . , a 4 + >o iih + T<z>i - cl- <i2> E q u a t i o n (12) i s a l i n e a r e q u a t i o n f o r c £ , w h i c h c o m p l e t e s the s o l u t i o n . R e f e r e n c e Novak , M . D . T h e o r e t i c a l v a l u e s o f d a i l y a t m o s p h e r i c and s o i l t h e r m a l a d m i t t a n c e s . Boundary L a y e r M e t e o r o l o g y 34: 1 7 - 3 4 , 1986 . 120 A £ £ 1 N D I X _ H A n a l y t i c a l S o l u t i o n t o the Two-Dimens1ona l H o r i z o n t a l l y P e r i o d i c Heat T r a n s f e r P r o b l e m w i t h Upper Boundary C o n d i t i o n o f the 3 rd K i n d . P r o b l e m _ D e f i n i t i o n The t w o - d i m e n s i o n a l heat t r a n s f e r e q u a t i o n 1s c | i . k ( s f i + a £ i >, ( i , 0 1 8x^ 9z^ where z 1s the d e p t h , x 1s the h o r i z o n t a l c o o r d i n a t e , t 1s the t i m e , T ( z , x , t ) i s the t e m p e r a t u r e , C i s the v o l u m e t r i c heat c a p a c i t y , and k i s the t h e r m a l c o n d u c t i v i t y . Bo th C and k are c o n s t a n t . The boundary c o n d i t i o n s a re g i v e n by a G z ( 0 , x , t ) + B T ( 0 , x , t ) = f ( x , t ) T ( x , z , t ) = T (x + X , z , t ) (2) l i m | I = 0 ; 11m §1 =0 z -*co 8 x z+a> 9 t T ( x , Z , t ) = T ( Z ) , where a and B are c o n s t a n t s , f ( x , t ) i s a known f u n c t i o n , 6 ( z , x , t ) 1s the component o f the heat f l u x d e n s i t y i n the z - d i r e c t i o n , T ( Z ) 1s a known c o n s t a n t , and Z 1s l a r g e enough t h a t T " ( Z , x , t ) ^ 0 . The d o u b l e b a r r e f e r s t o an a v e r a g e o v e r x f rom 0 to X and an average o v e r t f rom 0 to P . O n l y s o l u t i o n s p e r i o d i c 1n t w i t h p e r i o d P a re c o n s i d e r e d . 121 S o l u t i o n T ( z , x , t ) 1s r e p r e s e n t e d as a d o u b l e complex F o u r i e r s e r i e s , 1 . e . , T(z.x.t) = S S c n T m ( z ) tH***+n»t) , (3) n = -oo m=-oo where A = 2TT/X and w = 2 n / P . A l s o f ( x , t ) i s g i v e n by f(x.t) - f ? c f , e ^ m A x + n w t ) , (4) n = -co m=-co f where the c n m ' s a re known ( d e t e r m i n e d from the I n v e r s e t r a n s f o r m ) . E v i d e n t l y (3) and (4) s a t i s f y ( 2 b ) . S u b s t i t u t i n g (3) i n t o (1) y i e l d s f o r a l l m and n e x c e p t m=n = 0 d 2 T n m / m 2 , 2 ^ inwC N T n , _ x The s o l u t i o n t o (5) i s g i v e n by cL<z) • A nm e Y f i m Z + B n m e " Y n m Z (6) nmv ' nm nm v 1 2 2 2 where Y „ m = m A + i n w C / k and A n m and B „ m a r e unknown nm nm nm c o n s t a n t s . To s a t i s f y ( 2 c ) , A n m = 0 . Bnm 1s d e t e r m i n e d from 1 ' nm nm (2a) w i t h G ( z , x , t ) g i v e n by F o u r i e r ' s Law, i . e . , G z ( z , x , t ) ' - - k | | , (7) and f ( x , t ) g i v e n by ( 4 ) . T h i s y i e l d s c nm B nm = 6 + " a k Y n " * < 8 ) nm For m=n=0 the c o m p l e t e s o l u t i o n t o (5) i s g i v e n by C J 0 ( Z ) = A 0 0 z + B 0 0 « < 9 ' where AQ Q and BQ Q a re unknown c o n s t a n t s . From (2d) A 0 0 Z + B 0 0 = T ( Z ) ^ 1 0 a ) and from (2a) • a k A 0 0 + B B 0 Q = cJQ (10b) f rom w h i c h A Q 0 and B Q 0 a re d e t e r m i n e d , w h i c h c o m p l e t e s the s o l u t i o n . APPENDIX I I I I n p u t P a r a m e t e r s Used i n the S i m u l a t i o n S tudy on May 2 7 , 1984 P a r a m e t e r s M u l c h a p p l i e d @ 2 t / h a " (See N o t a t i o n f o r u n i t s ) T ( 0 . 8 5 , t ) 11 .54 C s 2 .32 x 1 0 6 k $ 1.08 a * 0 .1 £g 0 .96 T 18 .0 z u 1.0 z Q 2 x 1 0 " 3 d & 1.22 A t m o s p h e r i c s t a b i l i t y 1.5 c o r r e c t i o n f a c t o r s H 0 .008 m m a ! 0 .18 m r m 1 x 1 0 1 2 9 w m 0 .0075 vm E f 4 . 5 f m 0 .5 m A i r t e m p e r a t u r e d a t a were measured a t the V a n c o u v e r a i r p o r t . APPENDIX IV Climatological Input Parameters on May 27, 1984. Time Incoming A i r Tern- Vapour Wind (PST) Solar Radn perature Density Speed (°C) (xlO 3 (W/m ) kg/m ) (m/s) 0 0.0 10.00 8.15 1.65 1 0.50 10.20 8.25 1.67 2 0.60 9.80 8.05 1.48 3 0.50 9.10 7.70 1.47 4 1.20 9.30 . 8.33 1.17 5 21.10 8.90 7.60 0.92 6 107.39 10.40 8.36 0.98 7 246.80 11.80 8.60 1.62 8 392.60 12.50 8.44 1.96 9 634.60 14.00 9.25 2.10 10 784.50 15.50 8.95 1.85 11 874.40 15.70 9.06 1.27 12 911.40 14.80 9.72 1.45 13 915.40 15.40 10.08 1.62 14 874.40 16.50 10.07 1.64 15 784.40 16.20 9.89 1.37 16 654.80 17.20 8.60 1.37 17 489.90 17.00 7.93 0.98 18 311.70 17.00 8.50 0.56 19 160.30 16.40 9.45 0.43 20 34 .78 13.60 7.86 0.22 21 0.90 14.20 7.66 0.59 22 0.60 14.10 7.13 0.32 23 1.20 12.00 7.54 0.24 24 1.50 12.30 7.68 0.30 A i r temperature and vapour density data were measured at the Vancouver a i r p o r t . APPENDIX V 125 S o i l Temperature (°C) I n i t i a l i z a t i o n P r o f i l e f o r May 27, 1984. Depth Mulch a p p l i e d (m) @ 2 t/ha 0.005 10 .45 0.01 10.76 0 .025 11.49 0.05 12.67 0.10 13.96 0.25 13.55 0.50 11.92 0.85 11.54 126 APPENDIX VI I n p u t P a r a m e t e r s Used i n the S i m u l a t i o n S tudy on June 14, 1984 P a r a m e t e r s (See N o t a t i o n f o r u n i t s ) M u l c h a p p l i c a t i o n r a t e ( t / h a ) Bare 2 10 20 T ( 0 . 8 5 , t ) 13 .32 13 .32 11 .87 11.27 r f 0 < z < 0 .10 m L s l z > 0 .10 m 1.76x10*1 2 . 0 1 x 1 0 ° 2 . 1 8 x l 06 2 . 2 9 x l 0 6 2 .29x10 k jO < z < 0 .10 m s z > 0 .10 m 0 . 9 3 1.05 1.06 1.07 1.07 0 .16 0 .10 0 .10 0 .10 • i 0 .96 0 .96 0 .96 0 .96 300 0 0 0 T 2 2 . 0 2 2 . 0 18 .0 16 .0 Z u 10 .0 10 .0 10 .0 10 .0 z o 5 x l 0 "4 2 x l 0 " 3 2 x l 0 " 3 2 x l 0 " 3 d a 1.22 1.22 1.22 1.22 A t m o s p h e r i c s t a b i l i t y c o r r e c t i o n f a c t o r s 1.5 1.5 1.5 1.5 Hm 5 x l 0 "5 0 .008 0 .04 0 .06 m 1.0 0 .30 0 .093 0 .067 < 0 .0 0 .18 0 .26 0 .29 l x l O 1 2 l x l O 1 2 l x l O 1 2 l x l O 1 2 6 v m 0 .0 0 .0027 0 .005 0 .008 E f 100 .0 4 . 5 4 . 5 4 . 5 1.0 0 .5 0 . 0 0 .0 A i r t e m p e r a t u r e d a t a were measured a t the V a n c o u v e r a i r p o r t and w i n d s p e e d d a t a were measured a t the UBC P l a n t S c i e n c e Research S t a t i o n . APPENDIX VII C l i m a t o l o g i c a l Input Parameters on June 14, 1984. 127 Time Incoming A i r Tem- Vapour Wind (PST) S o l a r Radn p e r a t u r e D e n s i t y Speed 0 (xl0" 3 3 (W/m ) ( C) kg/m ) (m/s) 0 0.0 14.80 10.35 0.83 1 0.30 14.40 10.84 0.56 2 0.70 13.20 10.75 ' 1.39 3 0.70 12.70 9.76 0.83 4 1.50 12.80 9.82 1.67 5 36.30 13.00 9.94 2.22 6 123.60 13.50 10.26 1.67 7 315.00 15.20 9.96 0.83 8 325.50 15.10 9.90 0.83 9 648.00 16.90 10.31 1.67 10 681.00 18.50 10.08 1.67 11 769.90 19.00 11.03 1.67 12 860.90 19.70 10.82 2.78 13 763.10 17.30 12.03 1.67 14 826.90 18.00 11.78 2.22 15 749.20 18.80 11.71 2.22 16 436.50 18.00 11.78 1.67 17 257.20 17.60 11.50 1.39 18 126.90 17.20 11.23 0 .83 19 49.81 16.70 10 .90 0 .56 20 23.65 17.00 11.10 1 .39 21 3.20 16.80 11.68 0.83 22 1.70 .16.20 11.26 1.67 23 1.40 14.40 9.48 1.67 24 1.20 13.60 9.03 2.22 A i r temperature data were measured at the Vancouver a i r p o r t and windspeed data were measured at the U.B.C. Pl a n t Science Research S t a t i o n . APPENDIX V I I I 128 o S o i l Temperature ( C) I n i t i a l i z a t i o n P r o f i l e f o r June 14, 1984. Depth (m) Mulch a p p l i c a t i o n r a t e (t/ha) Bare 10 20 0.005 13.03 14.57 14 .27 0.01 12.95 14.84 14.54 0.025 14.67 15.32 14 .75 0.05 16.39 15.96 15.04 0.10 18.58 16.38 15.19 0.25 18.43 15.15 14.01 0 .50 14.97 13.01 12.22 0.85 13.32 11.87 11.27 129 APPENDIX IX I n p u t P a r a m e t e r s Used i n the S i m u l a t i o n S tudy on J u l y 6 , 1984 P a r a m e t e r s (See N o t a t i o n f o r u n i t s ) M u l c h a p p l i c a t i o n r a t e ( t / h a ) Bare 2 10 20 T ( 0 . 8 5 , t ) 14 .80 14 .80 13 .22 12 .73 r f 0 < z < 0 .10 m L s l z > 0 . 1 0 m 2 .10x10* ! 2 . 1 3 x 1 0 ° 2 . 18X106 2 . 44X10 6 2 .44x10 v f 0 < z < 0 .10 m *s l z > 0 .10 m 1.04 1.09 1.06 1.12 1.12 «; 0 .16 0 .10 0 .10 0 .10 0 .96 0 .96 0 .96 0 .96 300 0 0 0 T 2 2 . 0 2 2 . 0 18 .0 17 .0 z u 1.0 1.0 1.0 1.0 z o 5 x l 0 "4 2 x l 0 " 3 2 x l 0 ~ 3 2 x l 0 ~ 3 d a 1.22 1.22 1.22 1.22 A t m o s p h e r i c s t a b i l i t y c o r r e c t i o n f a c t o r s 1.5 1.5 1.5 1.5 Hm 5 x l 0 "5 0 .008 0 .04 0 .06 m 1.0 0 .30 0 .093 0 .067 m 0 .0 0 .18 0 .26 0 .30 r s l x l O1 2 l x l O 1 2 l x l O 1 2 l x l O 1 2 8 v m 0 . 0 0 .002 0 .0027 0.0047 E f 100 .0 4 . 5 4 . 5 4 . 5 1.0 0 .5 0 .0 0 .0 6 A i r t e m p e r a t u r e d a t a were measured a t the V a n c o u v e r a i r p o r t . APPENDIX X Climatological Input Parameters on July 6, 1984. Time Incoming A i r Tem- Vapour Wind (PST) Solar Radn perature Density Speed o (xl0" 3 3 (W/m ) ( C) kg/m ) (m/s) 0 0.30 14 .50 9.42 0.47 1 0.50 14.40 9.48 2.07 2 0.40 14.30 9.30 2.77 3 0.10 14.00 9.13 2.44 4 0.0 13.60 8.91 2.30 5 12.20 13.60 8.91 1.74 6 63.64 13.80 9.14 1.75 7 110.46 13.90 8.60 1.60 8 212 .40 14 .70 8.40 1.03 9 395.10 14.60 8.35 1.00 10 712.00 15.40 8 .25 1.49 11 804.60 15.90 8.50 2 .14 12 954.30 16.10 7.51 1.78 13 941.30 17.00 7.49 1.76 14 907.30 17.60 7.32 1.81 15 822.80 17.50 6.68 1.92 16 699.90 18.10 7 .54 1.73 17 505.20 17.50 6.68 2.05 18 334 .10 17.50 6.68 1.74 19 148.10 17.20 7.58 1.14 20 44 .54 16.10 6.96 0.22 21 4 .70 15.30 7.68 0.12 22 2.30 14 .60 7.85 0.28 23 3.40 13.90 7.52 0.45 24 3.60 13.90 9.20 0.67 A i r temperature and vapour density data were measured at the Vancouver a i r p o r t . APPENDIX XI 131 S o i l Temperature (°C) I n i t i a l i z a t i o n P r o f i l e f o r J u l y 6, 1984. Depth (m) Mulch a p p l i c a t i o n r a t e (t/ha) Bare ' 10 20 0.005 12.17 14.73 14.79 0.01 12.31 14.93 15.04 0.025 13.84 15.38 15.25 0.05 15.65 16.16 15.66 0.10 18.28 16.78 15.98 0.25 18.90 16.06 15.30 0.50 16.29 14.44 13.89 0.85 14 .80 13.22 12.73 132 APPENDIX X I I I n p u t P a r a m e t e r s Used i n the S i m u l a t i o n S tudy on S e p t . 3 , 1984 P a r a m e t e r s (See N o t a t i o n f o r u n i t s ) S t r i p D i r e c t i o n Bare N-S NE-SW E-W T ( 0 . 5 0 , x , t ) 17 .96 15 .93 15 .97 16.17 0< z < 0 .02 m C {0 .02 < 2 < 0 .10 m 5 z > 0 .10 m 1.26x10*! 1 . 7 1 x 1 0 ° . 2 . 0 5 x 1 0 ° 1.26x10*! 1 . 7 1 x 1 0 ° 2 . 0 5 x 1 0 ° 1.26x10*! 1 . 7 1 x 1 0 ° 2 . 0 5 x 1 0 ° 1.26x10 1.71x10 2 .05x10 0 < z < 0 .02 m k {0 .02 < 2 < 0 .10 m 5 z > 0 .10 m 0 .77 0 .90 1.04 0 .77 0 .90 1.04 0 .77 0 .90 1.04 0 .77 0 .90 1.04 < 0 .19 0 .19 0 .19 0 .19 0 .93 0 . 9 3 0 . 9 3 0 .93 2000 2000 2000 2000 T 2 3 . 3 2 1 . 0 2 1 . 0 2 2 . 0 z u 1.0 1.0 1.0 1.0 z o 5 x l 0 "4 2 x l 0 " 3 2 x l 0 " 3 2 x l 0 " 3 d a 1.0 1.0 1.0 1.0 A t m o s p h e r i c s t a b i l i t y c o r r e c t i o n f a c t o r s 1.5 1.5 1.5 1.5 Hm 5 x l 0 "5 0 .06 0 .06 0 .06 m 1.0 0 .093 0 .093 0 .093 < 0 0 .30 0 .30 0 .30 l x l O 1 2 l x l O 1 2 l x l O 1 2 l x l O 1 2 e vm 0 .0 0 .005 0 .005 0 .005 100 .0 4 . 5 f o r the mulch 100 .0 f o r the bare s t r i p s t r i p 1.0 0 0 0 APPENDIX XIII 133 C l i m a t o l o g i c a l Input Parameters on September 3 , 1 9 8 4 . Time Incoming A i r Tem- Vapour Wind Wind (PST) S o l a r Radn p e r a t u r e D e n s i t y Speed d i r e c -- 3 t i o n o (X10 3 (W/m ) ( C) kg/m ) (m/s) 0 0 . 4 0 14 .54 9.71 1.35 S 1 0 . 5 0 12.91 9.23 0.45 S 2 0 . 5 0 13.83 8 . 5 5 0.63 S 3 0 . 8 0 13.98 8.53 0.37 S 4 0 . 8 0 13.27 8.82 0.72 S 5 0 . 5 0 14.39 8.67 0.80 s 6 8 . 0 9 13.83 7.81 0.95 SE 7 80 .08 15.20 7.36 1.37 SE 8 224 .20 16.37 9.10 1.45 SE 9 369.50 16.98 8.80 2.42 S 10 475.70 19.02 8.39 2.63 SE 11 634.10 20.04 8.87 2.95 S 12 736.90 21.26 8.21 3.03 S 13 686.80 22.53 8.78 3.15 S 14 551.30 22.38 8.47 2.79 S 15 482 .00 21.36 9.30 2.86 S 16 286.90 20.39 8 .26 2.30 S 17 148.30 21.11 8.60 1 .88 S 18 63.48 18.51 10.07 0.73 SE 19 12.57 18.41 8.63 1.25 S 20 0.60 17.95 8.26 1.22 SE 21 0 .20 17.90 7.81 1.59 SE 22 0.0 18.05 7.67 1.31 SE 23 0.50 16 .88 7.84 1.60 SE 24 0.90 17.80 7.62 1 .81 SE APPENDIX XIV 134 o S o i l Temperature ( C) I n i t i a l i z a t i o n P r o f i l e f o r September 3, 1984. Depth S t r i p d i r e c t i o n (m) Bare N-S NE-SW E-W 0 .15 0 .28 0 .35 0 .15 0 .28 0 .35 0 .15 0 .28 0 .35 0 .005 13 .04 14 .27 13 .29 11 .89 14 .21 13 .43 11 .75 14 .52 13 .32 12 .09 0 .02 14 .18 14 .54 13 .71 12 .62 14 .48 13 .74 12 .47 14 .82 13 .69 12 .68 0 .10 17 .63 15 .52 15 .18 14 .96 15 .64 15 .34 15 .19 15 .85 15 .43 15 .21 0 .25 18 .53 15 .84 15 .96 16 .07 0 .50 17 .94 15 .97 15 .98 16 .17 135 APPENDIX XV I n p u t P a r a m e t e r s Used 1n the S i m u l a t i o n S tudy on A p r i l 7 , 1985 P a r a m e t e r s S t r i p d i r e c t i o n (See N o t a t i o n f o r u n i t s ) Bare ~ N-S ~ NE-SW ~ E-W T ( 0 . 5 0 , x , t ) 8 . 2 0 7 .80 7 .79 7 .70 2 . 1 3 x l 0 6 2 . 2 8 x l 0 6 2 . 3 8 x l 0 6 2 . 4 9 x l O S 1.00 1.05 1.08 1.11 «: 0 . 1 3 0 . 1 3 0 . 1 3 0 .13 0 .96 0 .96 0 .96 0 .96 0 0 0 0 T 14 .0 14 .0 14 .0 14 .0 z u 1.0 1.0 1.0 1.0 z o 5 x l 0 "4 2 x l 0 " 3 2 x l 0 " 3 2 x l 0 " 3 d a 1.0 1.0 1.0 1.0 A t m o s p h e r i c s t a b i l i t y c o r r e c t i o n f a c t o r s 1.5 1.5 1.5 1.5 5 x l 0 " 5 0 .04 0 .04 0 .04 m 1.0 0 .062 0 .062 0 .062 < 0 0 .21 0 .21 0 .21 l x l O 1 2 300 300 300 0 vm 0 .0 0 .0078 0 .0078 0 .0078 E f 100 .0 4 . 5 f o r the mulch 100.0 f o r the bare s t r i p s t r i p f m 1.0 0 0 0 APPENDIX XVI 136 C l i m a t e - l o g i c a l Input Parameters on A p r i l 7 , 1985. Time Incoming A i r Tem- Vapour Wind Wind (PST) S o l a r Radn p e r a t u r e D e n s i t y Speed d i r e c -_3 t i o n 2 0 (X10 3 (W/m ) ( C) kg/m ) (m/s) 0 0.0 6.45 6.37 0.56 SE 1 0.40 7.06 6.26 0.88 E 2 0.90 7.83 6.13 0.77 E 3 1.10 7 .27 5.92 0.67 E 4 0.70 7.06 5.88 0.89 NE 5 1.20 6.86 5.69 0.12 -6 4.90 6.66 5.55 0.12 -7 79.91 9.40 6.65 0.41 SW 8 243.30 11.34 6.10 0.94 w 9 256.60 10.88 6.22 1.12 s 10 273.60 12.77 6.87 0.36 sw 11 450.30 12.41 7.75 0.88 sw 12 379.70 10.37 7 .05 1.18 sw 13 382.80 11.75 6.42 0.88 w 14 544.40 15.21 4.65 0 .94 w 15 564.60 14 .50 4 .64 1.03 sw 16 403.90 16.74 4 .53 0 .77 sw 17 365.50 16.13 4 .78 1.36 sw 18 179.50 14 .04 6.13 1.46 s 19 38.10 10 .73 6.87 1.33 s 20 0.80 9.66 6.73 0.83 SE 21 0.20 8 .74 6.87 . 0.15 SE 22 0.50 9.30 6.75 0.55 E 23 0.70 8.49 6.50 1.24 E 24 1.00 7.47 6.60 0 .54 E A P P E N D I X X V I I 137 S o i l T e m p e r a t u r e ( C) I n i t i a l i z a t i o n P r o f i l e f o r A p r i l 7 , 1 9 8 5 . D e p t h S t r i p d i r e c t i o n (m) B a r e N - S N E - S W E-W 0 . 1 5 0 . 2 8 0 . 3 5 0 . 1 5 0 . 2 8 0 . 3 5 0 . 1 5 0 . 2 8 0 . 3 5 0 . 0 0 5 4 . 5 9 8 . 0 5 6 . 8 9 4 . 4 1 8 . 2 6 7 . 2 1 4 . 5 3 8 . 2 8 7 . 0 8 4 . 7 0 0 . 0 2 6 . 0 9 8 . 4 4 7 . 3 8 5 . 6 4 8 . 5 6 7 . 6 5 5 . 6 8 8 . 6 5 7 . 6 9 5 . 8 8 0 . 1 0 9 . 3 9 9 . 4 2 9 . 0 3 8 . 7 6 9 . 5 2 9 . 1 7 9 . 0 2 9 . 71 9 . 2 9 8 . 9 7 0 . 2 5 9 . 7 9 8 . 9 6 8 . 7 3 8 . 9 2 0 . 5 0 8 . 0 2 7 . 6 8 7 . 6 2 7 . 6 0 138 APPENDIX X V I I I I n p u t P a r a m e t e r s Used 1n the S i m u l a t i o n S tudy on A p r i l 8 , 1985 P a r a m e t e r s (See N o t a t i o n f o r u n i t s ) S t r i p d i r e c t i o n " Bare N-S NE-SW E-W T ( 0 . 5 0 , x , t ) 8 . 7 5 8 .30 8 .36 8 .11 C S 2 . 1 3 x l 06 2 . 2 8 x l O S 2 . 3 8 x l 0 6 2 .49x10 "s 1.00 1.05 1.08 1.11 «: 0 .13 0 .13 0 . 1 3 0 .13 0 .96 0 .96 0 .96 0 .96 0 0 0 0 T 15 .0 15 .0 15 .0 15 .0 z u 1.0 1.0 1.0 1.0 z o 5 x l 0 "4 2 x l 0 " 3 2 x l 0 " 3 2 x l 0 ~ 3 d a 1.0 1.0 1.0 1.0 A t m o s p h e r i c s t a b i l i t y c o r r e c t i o n f a c t o r s 1.5 1.5 1.5 1.5 Hm 5 x l 0 "5 0 .04 0 .04 0 .04 m 1.0 0 .062 0 .062 0 .062 < 0 0 .21 0 .21 0 .21 l x l O 1 2 300 300 300 e vm 0 .0 0 .0078 0 .0078 0 .0078 100 .0 4 . 5 f o r the mulch 100 .0 f o r the bare s t r i p s t r i p 1.0 0 0 0 A P P E N D I X X I X 139 C I i m a t o l o g i c a l I n p u t P a r a m e t e r s on A p r i l 8 , 1 9 8 5 . T i m e I n c o m i n g A i r T e m - V a p o u r W i n d W i n d (PST) S o l a r R a d n p e r a t u r e D e n s i t y . S p e e d d i r e c --•3 t i o n o ( X 1 0 3 (W/m ) ( C) k g / m ) (m/s ) 0 0 . 6 0 7 . 4 7 6 . 6 0 0 . 5 4 E 1 0 . 4 0 6 . 3 0 6 . 2 1 0 . 5 2 E 2 0 . 5 0 6 . 4 0 6 . 1 6 0 . 8 8 E 3 0 . 6 0 6 . 1 5 5 . 8 5 0 . 5 5 SE 4 0 . 5 0 8 . 2 3 6 . 2 2 0 . 9 4 SE 5 0 . 6 0 7 . 7 8 6 . 3 5 0 . 9 1 E 6 5 . 5 0 5 . 1 3 6 . 2 6 1 . 2 2 SE 7 7 4 . 5 9 9 . 2 5 6 . 2 7 0 . 6 7 E 8 2 3 4 . 8 0 1 0 . 8 3 7 . 7 7 0 . 4 2 SE 9 4 0 3 . 5 0 1 1 . 1 4 7 . 8 3 1 .04 S 10 5 5 6 . 7 0 1 1 . 4 9 7 . 6 6 1 . 2 4 SW 11 6 7 4 . 5 0 1 1 . 2 9 7 . 4 4 1 . 3 6 SW 12 7 4 6 . 6 0 1 0 . 7 8 7 . 1 6 1 . 5 2 SW 13 7 6 8 . 7 0 1 2 . 4 1 7 . 7 5 1 . 4 9 SW 14 7 3 4 . 6 0 1 2 . 4 6 7 . 4 5 1 .54 SW 15 6 5 4 . 5 0 1 2 . 6 6 9 . 0 7 1 . 5 6 SW 16 2 4 2 . 9 0 1 2 . 2 0 9 . 3 6 0 . 7 0 SE 17 3 4 3 . 7 0 1 3 . 1 7 8 . 6 5 1 . 3 0 W 18 1 9 7 . 0 0 1 1 . 8 0 8 . 5 5 1 . 1 9 W 19 34 . 0 0 1 1 . 1 4 8 . 6 0 1 . 0 0 W 20 0 . 4 0 9 . 2 5 8 . 8 2 0 . 8 6 W 21 0 . 3 0 8 . 5 9 8 . 8 8 0 . 6 0 W 22 0 . 2 0 8 . 6 4 8 . 4 3 0 . 3 6 W 23 0 . 0 7 . 4 2 8 . 5 6 0 . 2 1 SW 24 0 . 2 0 6 . 8 1 8 . 5 5 0 . 3 0 W 140 A P P E N D I X XX o S o i l T e m p e r a t u r e ( C) I n i t i a l i z a t i o n P r o f i l e f o r A p r i l 8 , 1 9 8 5 . D e p t h S t r i p d i r e c t i o n (m) B a r e N - S NE-SW E-W 0 . 15 0 . 2 8 0 . 3 5 0 . 15 0 . 2 8 0 . 35 0 . 1 5 0 . 2 8 0 . 3 5 0 . 0 0 5 5 . 3 7 8 . 63 7 . 6 2 5 . 0 4 8 . 89 7 . 9 8 5 . 23 8 . 6 6 7 . 6 1 5 . 1 7 0 . 0 2 6 . 8 8 9 . 00 8 . 0 4 6 . 2 7 9 . 16 8 . 3 7 6 . 41 9 . 0 3 8 . 2 2 6 . 3 8 0 . 1 0 10 . 1 2 9 . 90 9 . 5 6 9 . 2 4 1 0 . 02 9 . 8 0 9 . 55 10 . 0 4 9 . 6 9 9 . 3 3 0 . 2 5 10 . 1 7 9 . 3 4 9 . 1 9 9 . 2 8 0 . 5 0 8 . 4 2 8 . 0 4 8 . 0 3 7 . 9 0 

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