D. I. Norum, Don M. Gray, D. H. Male
D i v i s i o n of Hydrology Col lege of Engineer ing
U n i v e r s i t y o f Saskatchewan Saskatoon , Saskatchewan
' p a p e r No. H68 p r e s e n t e d a t t h e F a l l Annual Meeting of t h e A m e r i c a h Geophysical Union, San F r a n c i s c o , C a l i f o r n i a , December 1 0 - 1 3 , 1973.
2 ~ e s e a r c h Paper No. 1 2 , D i v i s i o n o f Hydrology, Col lege o f E n g i n e e r i n g , U n i v e r s i t y of Saskatchewan, Saska toon , Saskatchewan.
INTRODUCTION
T r a d i t i o n a l l y , snow hydrology s t u d i e s have been a s s o c i a t e d w i t h t h e deep packs of mountainous r e g i o n s . These s t u d i e s have g e n e r a l l y been conducted f o r two purposes ; (1 ) t o de te rmine t h e t o t a l volume of w a t e r a v a i l a b l e f o r r u n o f f , and ( 2 ) t o e s t i m a t e t h e peak runoff r a t e . Know- l e d g e of t h e t o t a l volume a v a i l a b l e i s impor tan t from t h e s t a n d p o i n t o f w a t e r supp ly t o major urban a r e a s f o r b o t h domes t ic and i n d u s t r i a l purposes . I n a d d i t i o n , t h i s s o u r c e may a l s o s e r v e t h e i r r i g a t i o n r e q u i r e - ments of a d j a c e n t a r e a s . An e s t i m a t e of t h e peak runof f rate i s obv ious ly n e c e s s a r y f o r f l o o d f o r e c a s t i n g . I t i s n a t u r a l t h e n t h a t s t u d i e s have been c o n c e n t r a t e d on t h e s e two main a s p e c t s , w a t e r supp ly and f l o o d f o r e c a s t i n g .
More r e c e n t l y i t h a s been r e a l i z e d t h a t t h e sha l low packs of t h e p r a i r i e and a r c t i c r e g i o n s are a l s o i m p o r t a n t and worthy of d e t a i l e d s t u d i e s . The snow r e s o u r c e s of t h e s e a r e a s have a g r e a t impact on t h e peop le and environment of t h e a r e a . On t h e p r a i r i e s t h e snow pack i s g e n e r a l l y t h e main s o u r c e of p o t a b l e f r e s h w a t e r supply f o r l o c a l domes t ic use . I n a d d i t i o n , t h e peak runof f from t h e s e sha l low packs can c a u s e l o c a l f l o o d i n g of s i g n i f i c a n t p r o p o r t i o n s a l t h o u g h i n g e n e r a l t h e s e a r e n o t of t h e s p e c t a c u l a r n a t u r e of mountain runof f f l o o d s . From t h e s e s t and- p o i n t s runof f from b o t h mountain packs and p r a i r i e packs have s i m i l a r e f f e c t s on t h e p o p u l a t i o n a l t h o u g h t h e magni tude, i n terms of numbers of peop le a f f e c t e d , is d i f f e r e n t . However t h e snow r e s o u r c e o f t h e p r a i r i e r e g i o n h a s a d d i t i o n a l d i r e c t impact on t h e p o p u l a t i o n ; t h e g r e a t e s t f a c t o r probably b e i n g t h a t of r ep len i shment of s o i l m o i s t u r e f o r c rop p r o d u c t i o n . From t h i s s t a n d p o i n t , shou ld s t u d i e s show t h a t improved management tech- n i q u e s can r e s u l t i n i n c r e a s e d s o i l m o i s t u r e s t o r a g e , t h e r e s u l t i n g i n c r e a s e d c rop p r o d u c t i o n would be o f c o n s i d e r a b l e d i r e c t monetary b e n e f i t t o t h e a r e a . Other examples of problems concerned w i t h snow c o n d i t i o n s which p e r s i s t f o r s e v e r a l months of t h e y e a r r e l a t e t o t r a n s p o r t a t i o n and t h e e f f e c t s of s o i l m o i s t u r e and s o i l t empera tu re regimes on c o n s t r u c t i o n works .
I n g e n e r a l , t h e i n v e s t i g a t i o n s of mountainous snow packs have been d i r e c t e d t o t h e development o f models f o r p r e d i c t i n g o r f o r e c a s t i n g s t reamf low o r d i s c h a r g e . T h e i r u s e i n t h e P r a i r i e Environment i s h i g h l y q u e s t i o n a b l e because of b a s i c d i f f e r e n c e s i n t h e c l i m a t i c , v e g e t a l and t o p o g r a p h i c f e a t u r e s which cause d i f f e r e n c e s i n t h e snow hydrology regime o f t h i s r e g i o n . F u r t h e r , a s sugges ted p r e v i o u s l y , a l though runof f rates and volumes a r e i m p o r t a n t , a P r a i r i e Model must a l s o b e c a p a b l e o f pre- d i c t i n g o t h e r p h y s i c a l pa ramete rs such a s s o i l m o i s t u r e and s o i l t empera tu re . Consequent ly t h e development of a P r a i r i e snowmelt model shou ld be based on a good u n d e r s t a n d i n g of t h e p h y s i c a l p r o c e s s e s involved. For t h i s pur- p o s e , t h e Energy Budget Approach - which simply i n v o l v e s a c c o u n t i n g f o r t h e the rmal energy invo lved - i s a n a p p r o p r i a t e framework w i t h i n which t o deve lop a model. Also i t i s expec ted t h a t such a model may b e t r a n s p o s e d f o r use t o s t u d y snowmelt problems i n o t h e r p a r t s o f t h e c o n t i n e n t hav ing s i m i l a r topograph ic and c l i m a t i c c o n d i t i o n s ( i . e . , A r c t i c ) .
1)EPTli-DENSITY MEASUEUMENT OE' SHALLOW PACKS
Depth and d e n s i t y measurements o l s h a l l o w p r a i r i e snow packs a r e complicated by t h e e f f e c t s of wind. l'he average wind v e l o c i t y I n t h e p r a i r i e r e g i o n d u r i n g w i n t e r months i s 4 - 6 m sec - I (15 - 25 Km h r - l j w i t h maximums of approximately 20 m sec'l (75 Km h r - l ) . When s n o w f a l l o c c u r s under such c o n d i t i o n s i t i s ex t remely d i f f i c u l t t o o b t a i n mean- i n g f u l measurements w i t h c o n v e n t i o n a l p r e c i p i t a t i o n gauges. Black (1954) , Gray, e t a 1 (1970a) , and Peck (1972) a l l p o i n t o u t t h a t under windy con- d i t i o n s s t a n d a r d p r e c i p i t a t i o n gauges underca tch when compared t o ground survey measurements. Gray, et & found t h a t F i s c h e r and P o r t e r p r e c i p i - t a t i o n gauges on t h e average on ly r e g i s t e r e d 432 o f t h a t measured on t h e ground.
I f ground s u r v e y s a r e used t o e s t a b l i s h t h e w a t e r e q u i v a l e n t of snow packs d i f f i c u l t y i s a l s o encounte red due t o t h e e f f e c t s of wind. On t h e P r a i r i e s , a l though t h e g e n e r a l snowpack dep th i s f a i r l y un i fo rm, l o c a l v a r i a t i o n s i n topography and v e g e t a t i v e cover may c a u s e major d e p a r t u r e s from t h i s average . Each f i e l d t h e r e f o r e , h a s i t s own p e c u l i a r c a t c h and r e t e n t i o n c h a r a c t e r i s t i c s . T h i s f a c t , of c o u r s e , h a s l o n g been recognized by a g r i c u l t u r a l i s t s i n t h a t s t r i p c ropp ing i s used a s much as a m o i s t u r e c o n s e r v a t i o n p r a c t i c e a s a wind e r o s i o n c o n t r o l measure. S t u d i e s have shown t h a t s t u b b l e f i e l d s r e t a i n a s s o i l m o i s t u r e , on t h e a v e r a g e , an amount of w a t e r e q u i v a l e n t t o approx imate ly 37% of t h a t of t h e average o v e r w i n t e r pack whereas f a l l o w l a n d s r e t a i n on ly about 9%. P a r t of t h i s d i f f e r e n c e can , o f c o u r s e , b e a t t r i b u t e d t o t h e f a c t t h a t t h e s t u b b l e w i l l r e t a i n snow blown from a d j a c e n t f a l l o w s t r i p s which under c e r t a i n c o n d i t i o n s may b e complete ly denuded.
I n g e n e r a l , because o f t h e expanse of areas of f l a t o r g e n t l y r o l l i n g topography, t h e s p a r c i t y o f t a l l , d e n s e v e g e t a t i v e growth and t h e c o n t i n u a l s t r o n g , s u r f a c e winds , s e v e r e d r i f t i n g and r e d i s t r i b u t i o n of t h e snowpack may occur o v e r t h e w i n t e r months.
Gray, e t a 1 (1970b) r e p o r t e d t h a t depending on wind c o n d i t i o n s and t h e t ime of measurement a f t e r s n o w f a l l b e g i n s t h e d e n s i t i e s o f f r e s h l y - f a l l e n snow measured a t t h e ground s u r f a c e v a r y i n t h e range from 0.04 - 0.25. It was a l s o found t h a t because o f wind a b r a s i o n , pack ing and o t h e r f a c t o r s t h a t f r e s h l y - f a l l e n snow q u i c k l y reached a d e n s i t y i n t h e range o f t h e average d e n s i t y o f t h e o v e r w i n t e r pack, 0.25 - 0.30. Only i n c a s e s where s u f f i c i e n t v e g e t a t i o n was p r e s e n t t o s h e l t e r and s u p p o r t t h e snow was t h e d e n s i t y found t o be s i g n i f i c a n t l y l e s s t h a n t h e v a l u e s neasured i n t h e open f i e l d s . S l a u g h t e r , -- e t a1 (1973) r e p o r t e d d r i f t e d snow of d e n s i t y g r e a t e r t h a n 0.50 i n t h e t u n d r a r e g i o n of Alaska. Bi le110 (1966) sugges ted t h a t s e a s o n a l snow cover d e n s i t y could be r e l a t e d t o a i r t e m p e r a t u r e and wind v e l o c i t y w i t h t h e d e n s i t y d e c r e a s i n g w i t h i n c r e a s i n g a v e r a g e s e a s o n a l a i r t empera tu re and i n c r e a s i n g w i t h a v e r a g e wind speed .
The use of twin probe gamma r a d i a t i o n snow gauges on s h a l l o w packs (Gray, -- e t a l , 1970a) does a l l o w t h e pack t o be p r o f i l e d f o r changes i n
d e n s i t y . However, u n l i k e t h e c a s e of deep mountain packs , s i n c e P r a i r i e packs a r e ex t remely he te rogeneous i n t h e i r d i s t r i b u t i o n i t i s n e a r l y i m p o s s i b l e t o e s t a b l i s h " r e p r e s e n t a t i v e " p o i n t sampl ing s i t e s . Consequent ly p o r t a b l e equipment must be used and many s i t e s may have t o be sampled t o o b t a i n a r e p r e s e n t a t i v e d e n s i t y p r o f i l e .
Remote scann ing of n a t u r a l gamma r a d i a t i o n from t h e s o i l th rough t h e snow pack may prove t o be u s e f u l i n overcoming t h e problem of p o i n t sampl ing. B i s s e l l and Peck (1973) moni tored t h e n a t u r a l gamma r a d i a t i o n from t h e s o i l w i t h a d e t e c t o r p l a c e d 2 m above t h e ground s u r f a c e . They found t h a t t h e w a t e r e q u i v a l e n t cou ld b e de te rmined w i t h a s t a n d a r d e r r o r o f 11 mm when t h e pack had a w a t e r e q u i v a l e n t o f 50 t o 400 nun.
Dmi t r i ev , -- e t a 1 (1972) used n a t u r a l gamma r a d i a t i o n methods t o de te rmine t h e b a s i n w a t e r e q u i v a l e n t from a i r c r a f t . For packs w i t h a w a t e r e q u i v a l e n t of 1 0 - 300 mm t h e s t a n d a r d d e v i a t i o n of t h e measurements w a s l e s s t h a n 1 0 mm. They p o i n t o u t t h a t a s t h e r a d i a t i o n i s o r i g i n a t i n g from t h e upper 300 - 400 m of s o i l t h i s p r o c e s s "sees" w a t e r which i s s t o r e d a s i c e l e n s e s on t h e ground s u r f a c e , whereas c o n v e n t i o n a l sampl ing methods g e n e r a l l y do n o t measure t h i s q u a n t i t y .
G r a s t y , -- e t a 1 (1973) r e p o r t t h e u s e o f gamma ray s p e c t r o m e t r y s u r v e y s u s i n g t o t a l r a d i o a c t i v i t y and po tass ium a c t i v i t y from n a t u r a l s o u r c e s t o de te rmine snow w a t e r e q u i v a l e n t . With presnow f l i g h t s t o o b t a i n background counts t h e major s o u r c e s o f e r r o r appear t o be d u p l i c a t i o n o f f l i g h t t r a c k , changes i n s o i l m o i s t u r e and d i f f e r e n c e s i n a tmospher ic p r e s s u r e . For s h a l l o w packs (maximum wate r e q u i v a l e n t of 140 mm) t h e s t a t i s t i c a l e r r o r s due t o low count r a t e were n o t s i g n i f i c a n t . A f t e r c o r r e c t i o n s were made
, f o r s o i l m o i s t u r e changes t h e s t a n d a r d e r r o r f o r t h e potass ium scannery was 1 2 mm.
L i n l o r (1972) h a s shown t h a t a i r b o r n e e l e c t r o m a g n e t i c wave methods can b e used t o de te rmine t h e d e n s i t y p r o f i l e of a snow pack. T h i s would b e a s t e p beyond t h o s e methods t h a t o n l y g i v e a measure o f t h e t o t a l w a t e r e q u i v a l e n t .
No doubt i n t h e f u t u r e snow scann ing by s a t e l l i t e methods w i l l i n c r e a s e i n impor tance , b u t a s y e t t h e most v a l u a b l e u s e of s a t e l l i t e imagery is t o d e f i n e a r e a s covered by snow r a t h e r t h a n p r o v i d e measurements o f d e n s i t y o r w a t e r e q u i v a l e n t . However McGinnis (1972) h a s shown t h a t n e a r - i n f r a r e d d a t a when used i n c o n j u n c t i o n w i t h r e f l e c t e d v i s i b l e r a d i a t i o n a p p e a r s t o be u s e f u l i n d e t e c t i n g m e l t i n g snow and i c e . Under m e l t i n g c o n d i t i o n s t h e n e a r - i n f r a r e d r a d i a t i o n i s s t r o n g l y absorbed w h i l e t h e v i s i b l e r a d i a t i o n i s r e f l e c t e d . L i n l o r s p e c u l a t e d t h a t s a t e l l i t e s might be used f o r t h e e l e c t r o m a g n e t i c wave methods.
For s t u d i e s of sha l low packs i n t h e p r a i r i e r e g i o n t h e u s e of s a t e l l i t e i n f o r m a t i o n i n h y d r o l o g i c s t u d i e s , p a r t i c u l a r l y i n t h e s t u d y of r u n o f f e v e n t s may be l i m i t e d because of t h e s h o r t t ime p e r i o d o v e r which
m e l t o c c u r s . P r e s e n t l y t h e t i m e i n t e r v a l between c o n s e c u t i v e p a s s e s o v e r an a r e a i s s e v e r a l days and t h u s t h e sys tem may m i s s t h e most s i g n i f i c a n t t i m e o f t h e m e l t s eason .
THE ENERGY BUDGET
As s t a t e d p r e v i o u s l y it i s c o n s i d e r e d t h a t t h e most f e a s i b l e method of d e v e l o p i n g a snow m e l t model f o r t h e p r a i r i e r e g i o n i s u s i n g t h e energy budget approach a s i t i s based on t h e a c t u a l p h y s i c a l p r o c e s s e s . C a r l s o n , e t a1 (1972) have p o i n t e d o u t t h e n e c e s s i t y of a model hav ing a s much -- p h y s i c a l r e a l i t y a s p o s s i b l e t o pe rmi t t r a n s p o s i t i o n of t h e r e s u l t s t o o t h e r a r e a s and t o make f u l l u s e of s p a r s e i n p u t d a t a . I n a d d i t i o n they p o i n t o u t t h a t t h e model s h o u l d have a s few e m p i r i c a l l y - d e r i v e d p a r a m e t e r s a s p o s s i b l e t o a l l o w f u l l l a t i t u d e f o r improvement of t h e model.
The energy budget approach can be d e s c r i b e d a s e q u a t i n g t h e t ime r a t e of change of t h e energy o f t h e pack t o t h e energy f l u x e s i n t o t h e pack; t h a t i s
where
U = energy of t h e pack, t = t ime ,
= n e t s o l a r r a d i a n t f l u x i n t o t h e pack a t t h e snow-air i n t e r f a c e ,
QSL = n e t l o n g wave r a d i a n t f l u x i n t o t h e pack a t t h e snow- a i r i n t e r f a c e ,
Q~~ = s e n s i b l e h e a t f l u x i n t o t h e pack a t t h e snow-air
i n t e r f a c e ,
SM = n e t energy f l u x i n t o t h e pack due t o mass t r a n s f e r a t
t h e snow-air i n t e r f a c e , = s o l a r r a d i a n t f l u x through t h e pack i n t o t h e ground
benea th , = h e a t f l u x by conduc t ion from t h e pack t o t h e ground
benea th , Q G M = n c t energy f l u x from t h e pack due t o mass t r a n s f e r a t
t h e graund-snow i n t e r f a c e .
From a thermodynamic s t a n d p o i n t Equat ion 1 is n o t complete f o r i t does n o t i n c l u d e terms f o r mechanical work a t t h e snow-air i n t e r f a c e , f o r work a r i s i n g from t h e change i n dep th o f t h e pack , f o r p o t e n t i a l energy o r f o r k i n e t i c energy . Ilowever, i t can be shown t h a t each of t h e s e terms i e n e g l i g i b l e when compared t o the accuracy w i t h which t h e v a r i o u s terms of Lquat ion 1 can be measured.
On t h e P r a i r i e s , t h e g e n e r a l p a t t e r n of snow pack d i s a p p e a r a n c e i s t h a t snow f i r s t d i s a p p e a r s from t h e f a l l o w f i e l d s . T h i s i s fo l lowed by a g r a d u a l s h r i n k a g e of t h e remaining snowcover u n t i l o n l y p a t c h e s o r d r i f t s p e r s i s t i n g u l l i e s and s h e l t e r e d a r e a s . M e l t from t h e s e deep r e s i d u a l d r i f t s f r e q u e n t l y is n o t e v i d e n t a s s u r f a c e runof f u n t i l a f t e r t h e open f i e l d s a r e f r e e of snow. It i s impor tan t t o examine t h e energy budge t o f m e l t i n g snow f o r t h e s e two complete ly d i v e r s e c a s e s ; under complete snowcover and f o r t h e s i t u a t i o n where o n l y i s o l a t e d r e s i d u a l p a t c h e s of snow c o n t r i b u t e t o s t reamflow. The p r a c t i c a l problems invo lved i n such a s t u d y n e c e s s i t a t e t h a t two s e p a r a t e approaches be t a k e n i n t h e energy budge t c a l c u l a t i o n s .
1. When t h e ground is comple te ly snow-covered t h e p r e s e n c e o f a uniform u n d e r l y i n g s u r f a c e a t t h e bot tom of t h e atmosphere s u g g e s t s t h a t t h e u s e o f m e t e o r o l o g i c a l t u r b u l e n c e t h e o r y may b e a p p l i e d t o t h e c a l c u l a t i o n o f QSH and QSM w i t h some v a l i d i t y .
2. When t h e l a n d i s p a r t l y snowfree , t h e a p p l i c a t i o n o f s t a n d a r d m e t e o r o l o g i c a l t e c h n i q u e s f o r c a l c u l a t i o n of Q and Q is probab ly
SM i n v a l i d s i n c e t h e u n d e r l y i n g s u r f a c e i s patchySHand t h e r e f o r e non uni- form and he te rogeneous - w i t h r e s p e c t t o t h e h e a t t r a n s f e r p r o c e s s e s .
S o l a r R a d i a t i o n (QSS, Q ~ ~ )
The incoming s o l a r r a d i a t i o n a t t h e e a r t h ' s s u r f a c e is g e n e r a l l y cons idered t o be a f u n c t i o n of t h e s o l a r a l t i t u d e , t h e e x t e n t , d i s t r i b u t i o n and form of c l o u d i n e s s , t h e a b s o l u t e humidi ty and t h e amount of ozone and d i r t i n t h e atmosphere. I n a d d i t i o n t h e amount of s o l a r r a d i a t i o n r e c e i v e d by a s u r f a c e w i l l depend upon i t s s l o p e and a s p e c t . Most snow m e l t models do n o t t a k e t h e s e f a c t o r s i n t o c o n s i d e r a t i o n . However Gray and O ' N e i l l (1973) have shown t h a t t h i s i s impor tan t d u r i n g t h e m e l t p r o c e s s of sha l low p r a i r i e packs . They i n v e s t i g a t e d t h e energy exchange o f d i f f e r e n t s l o p e s o f a P r a i r i e wa te r shed d u r i n g 6 days o f t h e 1972 snowmelt p e r i o d . T h e i r f i n d i n g s i n d i c a t e d t h a t by s imply a d j u s t i n g t h e d i r e c t beam component of incoming shor twave r a d i a t i o n t h e n e t r a d i a t i o n t o a sou th- fac ing s l o p e was approx imate ly f i v e t i m e s g r e a t e r t h a n t h e amount r e c e i v e d by a s i m i l a r n o r t h - f a c i n g s l o p e . Landals and G i l l (1972) have a l s o shown t h a t a s p e c t and s l o p e a r e i m p o r t a n t f a c t o r s a f f e c t i n g t h e m e l t of s h a l l o w packs . A s o u t h - f a c i n g s l o p e c o n t r i b u t e d e s s e n t i a l l y no runof f when a pack w i t h 46 mm o f w a t e r e q u i v a l e n t was on bedrock. I t appeared t h a t t h e e n t i r e pack was l o s t t o s u b l i m a t i o n and e v a p o r a t i o n . A s i m i l a r n o r t h - f a c i n g s l o p e y i e l d e d 82% r u n o f f . On v e g e t a t e d a r e a s (5 m j a c k p i n e ) approximately 70% runof f o c c u r r e d on b o t h s l o p e s a l t h o u g h t h e sou th- fac ing s l o p e y i e l d e d g r e a t e r peak f lows .
The r e f l e c t e d s o l a r r a d i a t i o n i s r e l a t e d t o t h e incoming r a d i a t i o n by t h e a lbedo f a c t o r . S t u d i e s conducted by O ' N e i l l and Gray (1972a) showed t h a t d u r i n g t h e m e l t - f r e e p e r i o d t h e a l b e d o of p r a i r i e packs ranged from
70 t o 80 pe rcen t depending on snowfa l l cond i t i ons . During t h e m e l t p e r iod t h e time decay of t h e albedo showed an a c c e l e r a t e d r a t e of change wi th t i m e , q u i t e d i s s i m i l a r t o t h e shape of t he r e l a t i o n s h i p u s u a l l y assumed f o r deep packs. It was a l s o found t h a t po in t measurements of a lbedo were i n c l o s e agreement w i th s p a t i a l l y averaged measurements ob t a ined by f l y i n g over t h e snowpack (provided snow remained w i t h i n t h e f i e l d of view of t h e s enso r ) . O ' N e i l l and Gray (1972b) found t h a t t h e a lbedo and t h e e x t i n c t i o n of s o l a r r a d i a t i o n i n snow a r e coupled and l a r g e l y c o n t r o l l e d by t h e p r o p e r t i e s of a t h i n " a c t i v e l aye r " a t t h e snow su r f ace .
So l a r r a d i a t i o n t o t h e ground Q has gene ra l l y been assumed t o be n e g l i g i b l e o r inc luded i n t h e groundGSheat f l u x term (Anderson, 1968; Boyd, e t a l , 1962; U.S. Corps of Engineers , 1956). However O 'Nei l l and Gray (1972b) have shown t h a t t h i s i s o f t e n no t t r u e f o r shal low pack con- d i t i o n s . Their conc lus ion was t h a t t h e r a d i a t i v e h e a t f l u x through snow du r ing t h e melt season may be of s i g n i f i c a n t magnitude f o r snow pack depths up t o 100 mrn. I n a d d i t i o n they found t h a t t he s imple d i f f u s i o n model (Giddings and LaChappelle, 1961) , which d e s c r i b e s r a d i a t i o n p e n e t r a t i o n i n snow, when extended t o t h e multi-wavelength s i t u a t i o n appears t o s e r i o u s l y underes t imate t h e s o l a r r a d i a t i o n p e n e t r a t i o n t o an absorbing s u r f a c e below t h e a c t i v e l a y e r .
N e t Long Wave Radia t ion (Q ) s L
I n t h e absence of s i g n i f i c a n t f o r e s t cover , such a s t hose cond i t i ons encountered on t h e P r a i r i e s , t h e p r i n c i p a l f a c t o r s i n f luenc ing t h e long- wave exchange a r e :
1. The temperature of t h e snow s u r f a c e and a i r l a y e r c l o s e t o t h e ground;
2. The a b s o l u t e a i r humidi ty;
3. The amount and form of c loudiness ; and
4 . The wind v e l o c i t y i n t h e a i r l a y e r c l o s e t o t h e ground.
Winters on t h e p r a i r i e s a r e cha rac t e r i zed by lengthy noc tu rna l per iods of c l o u d l e s s s k i e s . During t h e s e pe r iods , because of t h e low a b s o l u t e humidity of t h e a i r , t h e longwave r a d i a t i o n l o s s i s l a r g e and t h e t o t a l n e t r a d i a t i o n exchange i s nega t ive . That is , t h e outgoing r a d i a t i o n l o s s du r ing t h e evening exceeds t h e g a i n du r ing t h e day. Under t h e s e cond i t i ons t h e temperature of t h e pack i s lowered.
Even dur ing the m e l t p e r i o d , t h e noc tu rna l r a d i a t i o n l o s s e s may be s u f f i c i e n t t o r e f r e e z e a l l o r a p o r t i o n of t he thawed s o i l and t o reduce t h e temperature of t h e s u r f a c e c r u s t below 0°C. Usual ly , however, t h i s l o s s does no t exceed 15% of t h e average d a i l y hea t i npu t . A s t h e n e t long-wave r a d i a t i o n exchange i s dependent on both a i r temperature and
humid i ty , t h i s exchange can be g r e a t l y a l t e r e d by a change i n magni tude o f t h e s e v a r i a b l e s . It i s a g e n e r a l l y recognized f a c t t h a t on t h e P r a i r i e s , a p p r e c i a b l e m e l t i n g o f t h e pack w i l l n o t occur u n t i l t h e mean d a i l y a i r t e m p e r a t u r e s exceed 5°C. It shou ld be p o i n t e d o u t t h a t a t t h e t ime of m e l t t h e r e i s o f t e n a r e v e r s a l i n t h e t i m e o f o c c u r r e n c e o f maximum c loud cover . That i s , t h e days a r e o f t e n c l e a r whereas c louds form i n t h e even ings (prob- a b l y caused by e v a p o r a t i o n d u r i n g t h e day) . The obv ious e f f e c t o f t h e i n c r e a s e d c l o u d i n e s s i s t o reduce t h e n o c t u r n a l r a d i a t i o n l o s s and subsequen t c o o l i n g of t h e snowpack. S i n c e t h e sha l low P r a i r i e pack responds q u i c k l y t o d a i l y t empera tu re changes , t h e s e c o n d i t i o n s a r e conducive t o h i g h m e l t r a t e s .
With r e s p e c t t o r a d i a t i v e components i t i s cons idered t h a t t h e immediate needs concern s t u d y of t h e development o f methodology, t e c h n i q u e s and p rocedures which w i l l e n a b l e :
I-. E x t r a p o l a t i o n o f p o i n t r a d i a t i o n measurements i n s p a c e and a d j u s t i n g t h e s e a c c o r d i n g t o a "gross" topograph ic and l andscape model.
2. Accura te p a r t i t i o n i n g and e v a l u a t i o n of t h e r a d i a t i v e terms a s t h e y c o n t r i b u t e t o snowmelt - p a r t i c u l a r l y under c o n d i t i o n s o f p a t c h y snowcover.
Energy T r a n s f e r by Mass F lux a t Snow-Air I n t e r f a c e (Q ) S M
The energy t r a n s f e r a t t h e snow-air i n t e r f a c e due t o mass f l u x i s
where
MSL, MSV = l i q u i d and vapor mass f l u x e s r e s p e c t i v e l y a t t h e snow-air i n t e r f a c e ,
hSL, hSV = l i q u i d and vapor e n t h a l p i e s , r e s p e c t i v e l y .
The l i q u i d f l u x i n t o t h e pack, MSL, can b e r a i n f a l l o r condensate . The v a p o r f l u x , MSV ( u s u a l l y n e g a t i v e ) , i n c l u d e s a l l forms of e v a p o r a t i o n , s u b l i m a t i o n , and vapor t r a n s f e r . The term M h is commonly r e f e r r e d t o a s t h e l a t e n t h e a t t r a n s f e r . T h i s term iSVbesY d i s c u s s e d i n c o n j u n c t i o n w i t h t h e s e n s i b l e h e a t t r a n s f e r p rocess .
"Sens ib le" and "Latent" Heat T r a n s f e r (Q M h ) SH' SV SV
The s e n s i b l e and l a t e n t h e a t t r a n s f e r f l u x e s a r e g e n e r a l l y c a l c u l a t e d from t h e r e s p e c t i v e t e m p e r a t u r e and vapor p r e s s u r e g r a d i e n t s . Var ious aerodynamic formulae such a s t h o s e of Sverdrup (19361, U.S. Army Corps of Engineers (1956) , Thornthwai te and Holzman (1939) , Dyer (1965) , and Bowen (1926) can b e used f o r t h e s e c a l c u l a t i o n s .
Diamond (1953) h a s p o i n t e d o u t t h e misconcep t ion t h a t a p p r e c i a b l e e v a p o r a t i o n t a k e s p l a c e when a warm d r y a i r p a s s e s over a snow s u r f a c e . S i n c e e v a p o r a t i o n can on ly t a k e p l a c e when a vapor p r e s s u r e g r a d i e n t e x i s t s ( e x c e p t f o r m o l e c u l a r d i f f u s i o n ) and s i n c e t h e maximum t e m p e r a t u r e a snow s u r f a c e can r e a c h is 0°C t h e r e l a t i v e humidi ty of t h e warm a i r w i l l have t o be a p p r e c i a b l y less t h a n 100% i f e v a p o r a t i o n is t o t a k e p l a c e . I n a d d i t i o n a h e a t supp ly must be a v a i l a b l e f o r e v a p o r a t i o n o r t h e t e m p e r a t u r e of t h e snow s u r f a c e w i l l b e lowered c a u s i n g a subsequent r e d u c t i o n i n t h e vapor p r e s s u r e a t t h e s u r f a c e and p o s s i b l y condensa t ion .
A p o r t i o n of t h e Sou thern Canadian P r a i r i e s r e c e i v e warm d r y winds of r e l a t i v e l y h i g h v e l o c i t y s e v e r a l t i m e s d u r i n g t h e w i n t e r . These winds , known a s Chinooks, may v a r y i n d u r a t i o n from a . f e w h o u r s t o s e v e r a l days and a r e u s u a l l y accompanied by a b r u p t t empera tu re changes of a s much a s 3 0 ° C . These winds may cause a p p r e c i a b l e m e l t i n g of t h e pack due t o t u r - b u l e n t t r a n s f e r p r o c e s s e s a s s e n s i b l e h e a t t r a n s f e r , however i t is d o u b t f u l t h a t they w i l l cause d i r e c t e v a p o r a t i o n a n d / o r s u b l i m a t i o n . The d i sappear - ance of t h e pack under such c o n d i t i o n s may be a t t r i b u t e d t o m e l t i n g accompanied by i n f i l t r a t i o n and e v a p o r a t i o n from a f r e e w a t e r s u r f a c e .
Gray and O ' N e i l l (1973) have shown t h a t under complete snow cover c o n d i t i o n s on t h e P r a i r i e s n e t r a d i a t i o n accounted f o r 93% of t h e t o t a l energy supp ly w h i l e s e n s i b l e h e a t c o n t r i b u t e d 7%. However on an i s o l a t e d snow p a t c h n e t r a d i a t i o n c o n t r i b u t e d 56% and s e n s i b l e h e a t t r a n s f e r s u p p l i e d 44%, t h u s s u p p o r t i n g t h e argument t h a t a s t h e snow pack m e l t s and becomes pa tchy s i g n i f i c a n t amounts of h e a t a r e advec ted from snow f r e e a r e a s and a r e used t o m e l t t h e snow on t h e a d j a c e n t snow covered a r e a s .
Heat Flux a t Ground-Snow I n t e r f a c e (Q ) GC
During most w i n t e r s on t h e P r a i r i e s , t h e f low of h e a t w i t h i n t h e ground u n d e r l y i n g t h e snowpack i s toward t h e s o i l s u r f a c e and t h u s t h e r e i s a g r a d u a l lowering of s o i l t empera tu re . I n t h e absence o f t h e o c c u r r e n c e o f a n e a r l y l a r g e s n o w f a l l , d e p t h s o f f r o s t p e n e t r a t i o n of 2 m a r e common. However, because of t h e low the rmal c o n d u c t i v i t i e s o f t h e s o i l ( u s u a l l y a t low m o i s t u r e c o n t e n t s ) , t h e r e l a t i v e l y s m a l l the rmal g r a d i e n t s , and t h e p resence of t h e s o i l snow i n t e r f a c e , i t is q u e s t i o n a b l e whether t h e n e t t r a n s f e r is of s u f f i c i e n t magni tude and r a t e t o c a u s e m e l t i n g of t h e pack. Most l i k e l y t h i s h e a t p a r t i a l l y o f f s e t s t h e n e t l o s s through long-wave r a d i a t i o n and t h e r e b y resists lower ing of t h e t empera tu re o f t h e pack.
Energy T r a n s f e r by Mass Flux a t Ground-Snow I n t e r f a c e (Q ) GM
S i m i l a r t o t h e energy t r a n s f e r a t t h e snow-air i n t e r f a c e , t h e energy t r a n s f e r a t t h e ground-snow i n t e r f a c e due t o mass f l u x can b e w r i t t e n a s
where
MGL,MGV = l i q u i d and vapor mass f l u x e s r e s p e c t i v e l y a t t he ground-snow i n t e r f a c e ,
hGL,hGV = l i q u i d and vapor e n t h a l p i e s , r e s p e c t i v e l y .
The l i q u i d mass f l u x , MGL, a c t u a l l y i nc ludes t h e mass f l u x i n t o t h e s o i l ( i n f i l t r a t i o n ) and t h e mass f l u x l eav ing t h e pack a s runo f f .
Although t h e vapor t r a n s f e r ac ros s t he ground-snow i n t e r f a c e dur ing t h e w in t e r months may be s i g n i f i c a n t , i t s c o n t r i b u t i o n t o energy t r a n s f e r dur ing t h e mel t pe r iod i s probably n e g l i g i b l e compared t o t he c o n t r i b u t i o n of t h e l i q u i d t r a n s f e r .
Energy of t h e Pack (U)
The energy of t h e pack can be w r i t t e n a s
where
L = depth of t h e pack Pi,PQ,pv = mass of i c e , l i q u i d and vapor r e s p e c t i v e l y , i n a u n i t
volume of snow, Ui ,UQ,UV = i c e , l i q u i d and vapor s p e c i f i c e n e r g i e s , r e s p e c t i v e l y .
When cont inuous m e l t i s i n progress a snowpack r a p i d l y reaches an i so thermal condi t ion a t a temperature of 0 ° C (U.S. Corps of Engineers , 1956). For deep packs t h e t o t a l f r e e water con ten t probably becomes reasonably c o n s t a n t , t h e r e f o r e t h e time r a t e of change of t h e energy of t h e pack can be approximated by t h e product of t h e t i m e r a t e of change of depth , t h e dens i ty of t h e i c e phase and t h e energy of t h e i c e phase. In gene ra l t h i s va lue w i l l be sma l l compared t o t h e va r ious f l u x e s .
Under p r a i r i e condi t ions t h e me l t p rocess is g e n e r a l l y no t cont inuous because of t h e shal lowness of t h e pack (small energy s t o r a g e c a p a c i t y ) and the r a d i a n t cool ing a t n i g h t . Thus t h e change i n i n t e r n a l energy becomes an important f a c t o r and has a marked in f luence on t h e d i u r n a l f l u c t u a t i o n i n t h e runoff p a t t e r n . I t is w e l l known t h a t under p r a i r i e snowmelt cond i t i ons peak .mel t and runoff r a t e s occur on days when t h e pack has no t had an oppor tun i ty t o r e f r e e z e dur ing t h e preceding n i g h t . As a genera l r u l e , f looding from t h e s e shal low packs w i l l no t occur u n l e s s t h e overn ight temperatures a r e above - 4°C.
For p r a i r i e cond i t i ons , Equation 4 can b e s t be eva lua ted from snow tempera ture measurements, a measurement of t h e average d e n s i t y of t he
pack and a d e t c ~ l m l n a t i o n of t h e l r e e wa te r c o n t e n t , p robab ly by t h e c a l o r i m e t e r mettlod.
MELT WATER FLOW TIIROUGH THE PACK
The energy budget appr0ac.h i s a u s e f u l concep t which can b e used t o e q u a t e t h e the rmal e n e r g i e s i n v o l v e d i n t h e m e l t p r o c e s s ; however i t d o e s n o t d e s c r i b e t h e l i q u i d f low p r o c e s s t h a t t a k e s p l a c e w i t h i n t h e pack nor t h e t r a n s l a t i o n o r t r a v e l t ime t o r e a c h a gauging s t a t i o n . A s t h e i n i t i a l phase o f w a t e r movement o c c u r s w i t h i n t h e snow pack t h i s t r a n s p o r t phenomenon is i m p o r t a n t a s i t r e l a t e s t o b o t h i n f i l t r a t i o n and runof f p r o c e s s e s .
Colbeck (1972, 1973) and Colbeck and Davidson (1972) have t r e a t e d t h e snowpack a s a porous medium f o r purposes o f d e s c r i b i n g w a t e r f low i n t h e pack. The snowpack i s c o n s i d e r e d t o b e made up of two l a y e r s , an upper l a y e r of u n s a t u r a t e d snow where in t h e f low i s e s s e n t i a l l y v e r t i c a l , and a lower l a y e r of s a t u r a t i o n i n which t h e f low is b a s i c a l l y h o r i z o n t a l u n l e s s i n f i l t r a t i o n is s i g n i f i c a n t . Co lbeck ' s method r e q u i r e s a measure o f t h e d e n s i t y and l i q u i d w a t e r c o n t e n t a s a f u n c t i o n of d e p t h a t some t i m e , t o p r o v i d e an i n i t i a l c o n d i t i o n f o r c a l c . u l a t i o n of f low th rough t h e u n s a t u r a t e d l a y e r . I n a d d i t i o n t h e w a t e r f l u x a c r o s s t h e s u r f a c e must be known. From t h i s s t a n d p o i n t i t is c l e a r t h a t t h e energy budget and t h e f low p r o c e s s must be coupled m a t h e m a t i c a l l y .
Colbeck (1973) p o i n t s o u t t h a t t h e wave speed f o r f low i n t h e u n s a t u r a t e d l a y e r i s s i g n i f i c a n t l y l e s s than t h a t i n t h e s a t u r a t e d l a y e r . Consequent ly , f o r deep packs t h e u n s a t u r a t e d zone w i l l l a r g e l y de te rmine t h e d e l a y i n r u n o f f . However f o r s h a l l o w packs t h e t i m i n g o f t h e runof f may b e c o n t r o l l e d by t h e s a t u r a t e d l a y e r a t t h e b a s e .
INFILTRATION INTO FROZEN SOIL
For deep mountainous snowpacks i n f i l t r a t i o n i s o f t e n c o n s i d e r e d t o b e u n i m p o r t a n t ; however f o r s h a l l o w p r a i r i e packs a ma jor p o r t i o n of t h e m e l t w a t e r may i n f i l t r a t e i n t o t h e s o i l . The volume i n f i l t r a t e d o b v i o u s l y depends upon t h e m e l t r a t e and t h e s o i l t y p e , b u t t h e major f a c t o r c o n t r o l - l i n g i n f i l t r a t i o n is t h e s o i l m o i s t u r e s t a t u s a t t h e t ime o f f r e e z e u p i n t h e f a l l . Murray and C i l l i e s (1971) found t h a t under P r a i r i e c o n d i t i o n s t h e r e was a l i n e a r d e c r e a s e i n t h e amount of i n f i l t r a t i o n w i t h i n c r e a s i n g s o i l m o i s t u r e c o n t e n t . I n a d d i t i o n , t h e s o i l m o i s t u r e c o n t e n t i n f l u e n c e s t h e s h a p e of t h e i n f i l t r a t i o n r a t e c u r v e s o f a f r o z e n s o i l . These may adop t s e v e r a l d i s t i n c t forms.
1. An i n t a k e r a t e which i s r e a s o n a b l y c o n s t a n t w i t h t ime a t a v e r y low v a l u e - a c o n d i t i o n which would p r e v a i l i f f r o z e n w h i l e a t a h i g h m o i s t u r e c o n t e n t o r a n impervious l a y e r deve lops a t t h e s u r f a c e due t o r e f r e e z i n g of t h e m e l t w a t e r a t t ime o f thaw.
2 . An i n t a k e r a t e which decreases ve ry r a p i d l y wi th t i m e from a reasonably h igh i n i t i a l v a l u e t o near zero - a cond i t i on which may pre- v a i l when a s o i l i s f rozen a t a low moi s tu re con ten t bu t t h e s o i l t empera ture i s below f r eez ing . Meltwater e n t e r i n g t h e s o i l i s f rozen i n t h e pores and movement i s i n h i b i t e d .
3. An i n c r e a s e i n i n f i l t r a t i o n r a t e w i th time - a cond i t i on which may e x i s t when t h e s o i l i s f rozen a t an i n t e rmed ia t e mois ture con ten t . For t h i s ca se , some of t h e meltwater i s a b l e t o p e n e t r a t e t he s o i l and a s t he s o i l warms and more pores m e l t , t h e i n f i l t r a t i o n r a t e i n c r e a s e s .
Obviously the i n f i l t r a t i o n process is coupled t o t h e conduct ion of h e a t i n t h e s o i l p r o f i l e because of t he i n t e r r e l a t i o n s h i p between t h e s o i l temperature and the mass flow p r o p e r t i e s of t h e s o i l . I t i s con- ce ivab le t h a t t o f u l l y understand t h e s i g n i f i c a n c e of t h i s coupl ing i t may be necessary t o compare many s o l u t i o n s of t h e coupled h e a t and mass t r a n s f e r equa t ions f o r f rozen s o i l . Harlan (1973) has developed a numerical t echnique f o r ob t a in ing such s o l u t i o n s .
CONCLUSION
The snowmelt event i s a complex phenomenon involv ing many phys i ca l p rocesses . It is apparent t h a t a snowmelt model f o r t h e p r a i r i e s must be based on an unders tanding of t h e energy t r a n s f e r s involved , t he mass flow p roces s w i t h i n t he pack and t h e h e a t and mass flow i n t h e s o i l beneath t he pack. It i s hoped t h a t an increased knowledge i n each of t h e s e a r e a s may c o n t r i b u t e t o t h e development of a s i m p l i f i e d model r a t h e r than t o i n c r e a s e t h e complexity of e x i s t i n g models.
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