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Prediction of snowmelt rates in a deciduous forest
Journal of Hydrology, 101 (1988) 145-157 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
145
[11
PREDICTION OF S N O W M E L T R A T E S IN A D E C I D U O U S FOREST
A.G. PRICE Department of Geography, University of Toronto, Scarborough Campus, 1265 Military Trail, West Hill, Ont. MIC 1A4 (Canada) (Received September 21, 1987; revised and accepted November 18, 1987)
ABSTRACT Price, A.G., 1988. Prediction of snowmelt rates in a deciduous forest. J. Hydrol., 101: 145- 157. A model is presented for the prediction of the energy available for the melting of snow in a deciduous forest. The turbulent exchanges H and LE can be estimated using physically based equations, but are small at the forest floor, accounting typically for less than 10% of daily melt totals, largely because of the much reduced windspeed within the forest. Net allwave radiative energy flux density (Q*) alone is a good predictor of snowmelt rates. Physical modelling of Q* is not operationally possible, and the simple regression of Q* on solar energy flux (KS), a technique used with success in some circumstances, is shown to be inappropriate in a forest during snowmelt. A multivariate model using two commonly available variables: K~ and air temperature in the forest, explains over 3/4 of the variance of Q*.
INTRODUCTION
Forests are one of the major cover types in eastern North America, and are ecosystems of great intrinsic interest. In addition, they are major contributors to the fluxes of water, energy and oxygen into the atmosphere. From an economic viewpoint, forests are of enormous consequence. T h e h y d r o l o g y o f f o r e s t e d b a s i n s is q u i t e d i s t i n c t , a s is t h e n a t u r e o f t h e e n e r g y b a l a n c e d u r i n g s n o w m e l t a t t h e f o r e s t floor. I n m a n y o f t h e m o r e northerly forests, the seasonal melting of the snow releases the single largest w a t e r i n p u t o f t h e y e a r . A l t h o u g h t h e i n t e n s i t y o f s n o w m e l t i n p u t s is m u c h l o w e r t h a n t h a t o f t y p i c a l r a i n f a l l s , t h e t o t a l v o l u m e o f w a t e r r e l e a s e d is q u i t e large - as much as a third of annual water inputs. Infiltrating snowmelt water is t h e m o s t i m p o r t a n t r e c h a r g e e v e n t f o r b o t h s o i l a n d g r o u n d w a t e r , s i n c e t h e s n o w m e l t p u l s e is u n d i m i n i s h e d b y i n t e r c e p t i o n o r e v a p o t r a n s p i r a t i o n . S n o w m e l t w a t e r w h i c h is n o t s t o r e d i n t h e s o i l o r i n t h e g r o u n d w a t e r m o v e s t o s t r e a m s a s t h e ~ ' s n o w m e l t f l o o d " , a n d i t is i n t h i s c o n t e x t t h a t t h e p r e d i c t i o n o f s n o w m e l t r a t e s is m o s t p o i n t e d l y o f i n t e r e s t , s i n c e h y d r o l o g i c a l f o r e c a s t i n g using operational basin streamflow models requires a knowledge of snowmelt rates.
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146 THEORY T h e e n e r g y b a l a n c e of a s n o w p a c k c a n be written: Hm =
Q* + H + L E + Hp + H g -
AHo
(1)
where: H m = e n e r g y flux d e n s i t y a v a i l a b l e for m e l t i n g s n o w Q* = net a l l w a v e r a d i a t i v e e n e r g y flux d e n s i t y H = sensible e n e r g y flux d e n s i t y (Sensible h e a t ) L E = l a t e n t e n e r g y flux d e n s i t y ( L a t e n t h e a t ) Hp = e n e r g y flux d e n s i t y into s n o w p a c k f r o m r a i n H~ = e n e r g y flux d e n s i t y b e t w e e n s n o w a n d g r o u n d A H o = c h a n g e in h e a t deficit or n e g a t i v e h e a t s t o r a g e in the s n o w p a c k (All a b o v e in u n i t s of W m - 2 or M J m - 2 d -1) U n d e r m o s t conditions, eqn. (1) c a n be simplified to: Hm = Q* + H + L E
(2)
T h e flux of w a t e r r e l e a s e d (Vm) is then:
Ym
Hm (flw Lf) -1
=
(3)
w h e r e Pw = d e n s i t y of w a t e r (kg m-3), and Lf = l a t e n t h e a t of fusion of w a t e r ( J k g 1). T h i s flux of w a t e r is r e l e a s e d n e a r the s u r f a c e of t h e s n o w p a c k , and, h a v i n g m o v e d t h r o u g h the s n o w p a c k , b e c o m e s t h e i n p u t to the b a s i n system. All t h r e e t e r m s in eqn. (2) pose p r o b l e m s of e s t i m a t i o n . H a n d LE, w h i c h r e s u l t f r o m t u r b u l e n t airflow a c r o s s the surface, m a y be e x p r e s s e d as: U LE
= Pa Cp D h (Tz - To) f ( R i ) = pa Lv Dw (O'-~-2) (ez - eo) f ( R i )
where: = = T~ = To = = ez = eo = Pa Cp = Lv = P = Dh = Dw -f(Ri) = Uz
Zo
w i n d s p e e d at m e a s u r e m e n t h e i g h t z (m s - l ) r o u g h n e s s l e n g t h of s n o w s u r f a c e (m) a i r t e m p e r a t u r e (°C) s u r f a c e t e m p e r a t u r e (°C) v a p o u r p r e s s u r e of t h e a i r ( k P a ) v a p o u r p r e s s u r e of s u r f a c e ( k P a ) a i r d e n s i t y ( k g m -3) specific h e a t of a i r at c o n s t a n t p r e s s u r e (J k g 1K - 1) l a t e n t h e a t of v a p o r i z a t i o n of w a t e r (J kg-1) p r e s s u r e of the a t m o s p h e r e ( k P a ) e x c h a n g e coefficient for h e a t (m s 1) e x c h a n g e coefficient for w a t e r v a p o u r (m s -~) a f u n c t i o n of R i c h a r d s o n n u m b e r , a s t a b i l i t y m e a s u r e (nd)
(4) (5)
147 The value of the exchange coefficients Dh and Dw can be shown to be: Dh = Dw = k 2 uz [ln2 (Z/Zo)] -1
(6)
The Richardson number, R i , can be written for the whole slab of air between the surface and the measurement height, z, as: Ri
=
(Tz -
To) z g (T~ u2z) ~
(7)
where: g = acceleration due to gravity (ms-2); k = Von Karman constant (nd); T~ = mean temperature of the layer (K); and z = measurement height
(m). For stable conditions, and with all the physical constants or computable variables combined into two constants K' and K", and for a particular z and z0, the equations reduce to: H = LE =
K ' u z (Tz K" uz(ez-
To) (1 + a R i ) -1 e o ) ( 1 + aRi) -~
(8) (9)
where a is an empirically determined constant whose value is 10.0 (Webb, 1970). The details of the derivation of these equations are given in Price (1977). The assumptions involved in arriving at eqns. (8) and (9) are considerable, but are relatively easy to justify for the stable conditions (Ri > 0) which invariably occur over a melting snowpack (since T~ is usually > To). Equations (6)-(9) are not physically sophisticated, and incorporate a number of broad assumptions, one of them being the equality of the exchange coefficients under varying stabilities. In defence of the equations it should be noted that there is good agreement on the form of the correction for stable conditions, and that the equations are extremely easy to apply operationally, if To and e0 are known. They are relatively undemanding of input data, and in e x t r e m i s can be operated with only uz, Tz and relative humidity. Overall, the equations are physically so superior to the commonly used empirically based "melt equations" (U.S. Army Corps of Engineers, 1956), that their lack of general usage is puzzling. Detailed modelling of the turbulent exchanges H and L E is undertaken exclusively in the open (Granger and Male, 1978). Under these circumstances the only exchanging surface is the snow, and the conditions are the simplest possible. In a forest, not only does moving air exchange energy with the snow surface, but also with the trees. The form of the wind profile in the forest, therefore, is a result of these interactions. Measured profiles in forests are complex, with several reversals, and under these circumstances, orthodox flux-profile relationships are quite inapplicable. Equations (8) and (9) show the turbulent exchanges to be a nonlinear function of uz, the nonlinearity depending on Ri. Since over a melting forest snowpack R i is invariably positive and usually large, then H and L E are much reduced. Small values of Uz imply large values of Ri, and uz at the forest floor is known to be small. Thus, a p r i o r i , one can reason that H and L E will be small at the forest floor during snowmelt. Equations (8) and (9), derived for open
148 conditions, will overestimate H and LE, since some mixing is occurring between trees and air r a t h e r t han snow and air. These equations, with the use of a liberal estimate of surface roughness length (z0) allow estimation of values of H and LE in the forest. The third term in eqn. (2) also differs between the forest and open conditions. Q* may be written: Q* = g~ (1 - ~) + L*
(10)
where: K~ = incoming (solar) shortwave energy flux density; a = surface reflectivity in the shortwave (albedo); and L* = L$ - L T = net longwave energy flux density (all in W m -2 or M J m - 2 d 1). The major energy input to the system is KS, which is easily and routinely measured, and which at least at a scale of tens of kilometres is not significantly spatially variable. The first difference between Q* in the open and the forest is that in the forest, K~ is strongly interfered with by the forest canopy, so that only a proportion reaches the snowpack, the rest being absorbed or reflected. The amount of absorption of K$ by the snowpack will depend on ~, whose values and variability may not differ greatly between forest and open. The second major difference between Q* in forest and open is L* - (L$ - LI"). Under most conditions, L$ and L$ vary in response to similar controls, and L* is therefore a relatively conservative quantity. During forest snowmelt, however, L$ may become large, since the canopy is absorbing a large proportion (about 90%) of K$ incident on it, heating up, and radiating in the longwave (Price and Petzold, 1984). At the same time, LI" from the snow is '~locked" at a maximum value of 316 W m ~, the black-body emittance at 0°C. L* may, then, become much larger in the forest than in the open. The net result of these two differences between the forest and open is t hat although K* at the forest floor may be small, L* may be large, and Q* values in the forest may be as much as twice those in the open (Hendrie and Price, 1979). SITE DESCRIPTIONAND INSTRUMENTATION The measurement site is on the property of Atomic Energy of Canada Limited (AECL), in subbasin 3 of the Perch Lake watershed, Chalk River, Ontario (46°02'N; 77°20'W; see Fig. 1), and is in an extensive deciduous forest of mixed aspens (Populus tremuloides Michx. and P. grandidentata Michx.) which form the dominant canopy at 22m. Birch, (Betula papyrifera Marsh.) forms a secondary canopy at about 15 m, and there is a general cover of beaked hazel (Corylus cornuta Marsh.) at 1-2 m. The radiation data in the present paper were measured using Middleton CN1 net radiometers, one over the canopy, and three at 2 m over the snow surface. A Kipp-Zonen CM-5 solarimeter was used to measure KS above the canopy, and a Middleton solari-albedometer to measure the shortwave fluxes at the snow surface. Windspeed was measured at 1, 2, 5, 7.5, 10 and 12.5m over the snowpack, using Casella sensitive anemometers. Wet and dry-bulb temperatures were measured at each of these levels using aspirated psychrometers.
149
\
\
To D
Fig. 1. Location of the study area.
Water volumes released from the snowpack were measured using a 5 x 5 m lysimeter plot, which intercepted all meltwater reaching the base of the snowpack. DISCUSSION OF DATA In this paper is part of seven years' data collected at the site. Windspeeds were measured continuously at the site in 1978, 1984 and 1985 in order to investigate the sizes of H and LE. Using eqns. (8) and (9), and incorporating assumptions likely to overestimate the values of the turbulent exchanges (logarithmic profile and z0 = 5 mm), these equations were used with observed values of T1.0, el.0 and ul.0 to predict daily melt resulting from H and LE. When Tl.0 and Q* were greater than zero, then To was taken to be 0°C and e0 to be 0.611 kPa. When Q* was negative, and surface melt ceased, it was assumed that the snow surface acted in a fashion analogous to a wet-bulb thermometer, gaining by H what it might lose by LE. For the nocturnal period, then, ( H + L E ) = O.
150
20;3
2113
3"
E
UA U-
rnU.l
2913
WU.I I.UUJ no_
30•3
31 1 3
I
I 14
12
O0
12
214
C30
z_z
i
"
!
II O0
12
O0
12
O0
O0
1'2
O0
1'2
O0
112
O0
TIME E.S.T.
Fig. 2. H o u r l y w i n d s p e e d o v e r the c a n o p y and n e a r t h e forest floor.
W i n d s p e e d (both n e a r t h e forest floor and over the c a n o p y ) s h o w s a very s t r o n g diurnal p e r i o d i c i t y (Fig. 2) w i t h m a x i m u m v a l u e s near t h e forest floor of a b o u t 1 m s -1 o c c u r r i n g near solar n o o n . M i n i m u m w i n d s p e e d s at the forest floor o c c u r at night, w i t h v a l u e s t y p i c a l l y b e t w e e n 0 and 0.3 m s - i. This implies that m o s t wind m o t i o n n e a r the forest floor o c c u r s as a result of c o n v e c t i o n w i t h i n t h e c a n o p y , c a u s e d by c a n o p y h e a t i n g as a result o f a b s o r p t i o n o f solar 1.o. w z a • o •
D~
oo
•
oo o o
IJA CSu-
o• • • °
OoO
~
WW
0.5
~to
•
•
•
1:5
DALLY A V E R A G E W I N B S P E E D
£o
ABOVE THE CANOPY
z:5 (ms-D
Fig. 3. D a i l y a v e r a g e w i n d s p e e d o v e r t h e c a n o p y and n e a r the forest floor.
151 TABLE 1 Observed v a l u e s of Q* a n d c o m p u t e d v a l u e s for H and LE, 1984 Hour
7-8 ~9 ~10 1~11 11-12 1~13 1~14 1~15 1~16 1~17 1718 1~19
29/03/84
10/04/84
12/04/84
H
LE
H
LE
H
LE
1.0 2.8 3.1 8.6 10.7 9.2 10.9 10.9 6.4 5.5 1.3 0.2
-5.4 -3.0 0.3 1.3 2.7 2.1 2.2 1.8 0.9 0.7 0.1 0.0
-
-
0.2 2.4 15.6 11.6 9.8 12.1 10.6 11.4 9.8 7.7 1.9 0.6
0.0 0.8 5.7 4.8 4.5 5.5 5.0 4.9 4.1 2.9 0.7 0.2
1.2 2.8 9.6 12.8 12.4 14.6 11.6 9.7 2.8 0.7 -
0.78 0.50 0.50 1.8 2.2 3.5 2.7 2.0 0.4 0.0 -
Totals for period (Mdm ~) 0.25
0.01
0.28
0.05
0.34
0.14
Q*
Q*
Q*
2.74
6.03
7.74
All v a l u e s in W m -2.
radiation, and that nocturnal windspeeds and turbulent exchanges are likely to be extremely small. An exception to this occurs when frontal activity generates regional airflows, as from 16th to 18th March in Fig. 2. During the snowmelt period at Perch Lake, such flows are usually accompanied by elevated nocturnal temperatures, in which case the "melting" boundary conditions can be applied overnight. Data from three years of measurement of windspeed show that ul.0 in the forest is roughly predictable from the windspeed over the canopy, as shown in Fig. 3. Although the scatter is considerable, for daily totals the 1 m windspeed is about 1/3 of that over the canopy. Thus eqns. (8) and (9) can be used with only temperature and vapour pressure data actually measured in the forest. Table 1 shows observations and computations made for three days during the 1984 melt. Using the model outlined above, and for times when Q* and TL0 were positive (and the surface therefore known to be melting), H and L E were computed. The days were selected for their high values of windspeed, temperature, and vapour pressure, in an attempt to define upper limits on the size of the turbulent exchanges. The data show t h a t there is a strong diurnal rhythm to the turbulent exchanges, resulting from the pattern of windspeed shown in Fig. 2. Values of H and L E earlier and later in the day than the hours shown in Table 1 are all close to zero. This makes the application of the model
152
TABLE 2 D a i l y m e l t t o t a l s a t t r i b u t a b l e to t h e t h r e e m a j o r e n e r g y fluxes of eqn. (2) D a t e (04/78)
Observed snowmelt
Melt from
M e l t from
runoff
Q*
H
LE
10 11 12 13 14 15 16 17 18 19 20
0.9 1.8 8.5 7.8 8.2 6.8 13.0 14.4 14.5 5.2 5.6
0.1 1.8 8.1 5.6 9.1 7.4 12.2 13.1 14.9 3.6 4.0
0.00 0.34 1.01 0.56 0.64 0.58 0.26 0.26 0.89 0.40 0.17
0.08 0.21 0.17 0.22 0.00 - 0.02 0.05 0.03 0.05 0.00 0.11
Period t o t a l s
86.7
[79.9
Hm
=
[Q*
+ H
+ LE]
=
+
5.11
+
0.99]
86.0
All m e l t r a t e s in m m of water.
much easier, since during the nocturnal period when the surface temperature is not known, the turbulent exchanges are effectively zero. The second feature shown in Table 1 is that, for the three days shown, (again, selected as days of higher H and L E ) , the total amount of energy transferred to the snowpack by the turbulent exchanges is less than 10% of the radiatively transferred energy, with a maximum hourly contribution of 21.3 W m 2, (7.4% of Q* at that hour). This means that errors in the estimation of H and L E are of little consequence, because of the small size of the turbulent exchanges. On the other hand, the equations can be applied to approximate the exchanges when they are positive (into the snowpack), and of any significant size. Table 2 shows the daily melt totals attributable to the three major energy fluxes of eqn. (2) for a typical period of snowmelt. The data in Table 2 were selected because the period shown was one of rapid snowmelt with no rain, during a period when the runoff plot was known not to be leaking, when no ice-lenses were present in the snowpack, and when all data were complete. For the period shown, H and L E account for 7.1% of total melt. Thus for an isothermal forest snowpack, the energy balance can be written to a good approximation as: Hm = Q*~s
(11)
where Q*~ is net allwave radiative energy flux density at 2m over the snowpack, and so: Hm = K~s (1 - ~a~) + L*a~
(12)
153 where the subscript ~'as" signifies "above the snow surface". The problem of snowmelt forecasting has now become one of predicting Q'as. With a view to synthesizing Q*a~, consider the terms of eqn. (12). KSa~ is a complex, highly spatially and temporally variable quantity generated by the interference of canopy elements with the direct solar beam. For any given forest type for daily totals, however, one may be able to predict KSas from K$~c (ac signifies "above canopy"), particularly during snowmelt when canopy density is not changed by leafing or leaf-fall. For example, at the Perch Lake site for 1979: K$~ = 0.50 KSac
(13)
n = 30, s = 0.06. The two remaining variables in eqn. (12) are much less tractable: ~s being a function of snow depth, snow age, snow surface debris accumulation, crystal size, direct-diffuse ratio and sun angle. L*~ (net longwave) is part of and influenced by the energy balance and radiative characteristics of the snow surface and the canopy. Modelling the variability of ~s and L*~ would be desirable, but is beyond the scope of this paper. Even the direct measurement of Q'as is problematic, since great variability in KSa~ makes point measurements of Q*~ very probably unrepresentative for even a small basin. In any event, what is needed for forecasting snowmelt is a model demanding only routinely measured variables such as air temperature and K$~c. An obvious technique is to use K$~c as a predictor of Q*~. First, it is measured routinely; second, it is spatially relatively invariant; (at the small basin scale and for daily totals) third, it is strongly physically linked with Q*~ and Q*~, and less directly with L*a~. The technique of regressing Q* on KS has been applied in hydrology, forestry and agronomy (Petzold and Wilson, 1974; Rauner, 1961; Szeicz et al., 1969), because of its great power: it works well under many circumstances, and if it does, one can use an available variable (K~) to predict a complex, unavailable one (Q*). The model has the form: Q*
=
a + bKJ.
(14)
Daily value totals of Q* are defined as the sum of hourly values from the first negative of the preceding day to the last positive value of the day in question. Totals of KS are simply the sum of all values of KS for any day. Albedo is defined as the daily sum of KT~s divided by the daily sum of K$~. Mean daily temperature is the arithmetic mean of 24 hourly average temperatures for that day. All radiative values in the following equation are in M J m - 2 d -1, and temperatures are in °C. Taking the data from the same period as those shown in Table 1, and regressing Q*~ on K ~ , the least-squares line is given by: Q*~ = 0.24 + 0.17 K~a~
(15)
n = 12, r 2 = 0.92. Just as a comparison, the most commonly used snowmelt predictor is mean daily temperature T~ - typically applied as the "degree--lay index". For the
154 TABLE 3 P r o p o r t i o n s of v a r i a n c e of Q'as e x p l a i n e d by p r e d i c t o r v a r i a b l e s (all s i g n i f i c a n t at p = 0.05) Year
n
1979 1983 1984 1985 All (1978~1985)
Ta
K~
K~(1
30 28 28 38
0.221 0.244 0.491 0.625
0.307 0.273 0.569 0.407
141
0.369
0.432
a)
K$,T a
K~(1 - ~), T a
0.653 0.630 0.687 0.695
0.682 0.732 0.813 0.910
0.790 0.835 0.792 0.905
0.682
0.769
0.820
s a m e 12 d period for 1978 used to derive eqn. (15), the r e s u l t is: Q'as =
1.6 + 0.56 T a
(16)
n = 12, r 2 = 0.29 (not s i g n i f i c a n t at p = 0.05). I n this case, K ~ a c is a far b e t t e r p r e d i c t o r t h a n T a. On this basis, one m i g h t t e n t a t i v e l y p r o c e e d w i t h the t e c h n i q u e . I f the a n a l y s i s is r e p e a t e d for 1979, h o w e v e r , t h e r e s u l t is: Q'as
=
0.71 + 0.13 K~a c
(17)
n = 28, r 2 = 0.31. F o r 1983, the r e l a t i o n s h i p is: Q'as -
0.68 + 0.12 Klac
(18)
n = 28, r 2 = 0.27. Plainly, a n y model l e a v i n g u n e x p l a i n e d 3/4 of the v a r i a n c e of the d e p e n d e n t v a r i a b l e is of little utility. T h e s o u r c e s of this v a r i a b i l i t y a r e ~as a n d L*a~. T h e t e c h n i q u e w o r k s well in c i r c u m s t a n c e s w h e r e ~ and L* o v e r the s u r f a c e a r e r e l a t i v e l y i n v a r i a n t . I f t h e y a r e not, t h e n the m o d e l is not useful. T h e case of forest s n o w m e l t is p e r h a p s the w o r s t c i r c u m s t a n c e in w h i c h to a p p l y the r e g r e s s i o n t e c h n i q u e . O v e r m a n y surfaces, ~ is e s s e n t i a l l y i n v a r i a n t , e x c e p t p e r h a p s for d i u r n a l c h a n g e s due to sun e l e v a t i o n , or s e a s o n a l c h a n g e s due to c r o p g r o w t h . In c o n t r a s t , s n o w s u r f a c e albedo at P e r c h L a k e h a s b e e n o b s e r v e d to c h a n g e f r o m 0.40 to 0.90 o v e r n i g h t as the r e s u l t of a light snowfall. S n o w is p r o b a b l y t h e s u r f a c e w i t h the m o s t c h a n g e a b l e ~. L* is also m u c h m o r e v a r i a b l e o v e r a s n o w s u r f a c e t h a n it is o v e r m a n y o t h e r surfaces, since it c a n b e c o m e r e l a t i v e l y l a r g e w h e n the s n o w is melting, and, as n o t e d above, this is p a r t i c u l a r l y t r u e in t h e forest. T h u s the c o n t r a s t b e t w e e n 1978 and 1979 is n o t entirely surprising. T a b l e 3 s h o w s the p r o p o r t i o n of t h e v a r i a n c e of Q*a~ e x p l a i n e d by s e v e r a l p r e d i c t o r v a r i a b l e s for s e v e r a l y e a r s individually, a n d for t h e w h o l e d a t a s e t (1978-1985). F o r c o m p a r i s o n , t h e first v a r i a b l e used is simply m e a n a i r temp e r a t u r e (Ta), w h i c h e x p l a i n s b e t w e e n 22 a n d 63% of t h e v a r i a n c e of Q*~s for i n d i v i d u a l years, a n d a b o u t 37% for the w h o l e d a t a s e t . Thus, despite the fact
155 t h a t the d e g r e e , l a y index g e n e r a l l y works b e t t e r in forests t h a n in the open, b e c a u s e of the influence of the l o n g w a v e fluxes in d e t e r m i n i n g Q*, Ta still leaves o v e r h a l f the v a r i a n c e of Q*~s unexplained. Used as the only predictor, KSa¢ explains b e t w e e n 27 and 57% of the v a r i a n c e of Q'as for individual years, and 43% for the whole d a t a s e t - - no real improvem e n t on the p e r f o r m a n c e of Ta, and still leaving m u c h v a r i a n c e unexplained. The use of KSac(1 - ~a~) as a p r e d i c t o r explains b e t w e e n 63 and 70% of the v a r i a n c e of Q'as for individual years, and 68% for the whole dataset. Thus, for the whole dataset, K~ac explains a b o u t 40% of Q*~s v a r i a n c e , and K ~ c (1 - a~s) a b o u t 68%, with the rest of the v a r i a n c e g e n e r a t e d by L*~s. Albedo, however, is not a v a r i a b l e c o m m o n l y available for modelling purposes. T h e f o u r t h set of p r e d i c t o r variables shown in Table 3 is KSac and T~. This c o m b i n a t i o n explains from 68 to 91% of Q'as v a r i a n c e for individual years, and 77% for the whole dataset. It is a f e a t u r e of the d a t a s e t t h a t Ta explains a b o u t 1/3 of the v a r i a n c e of albedo. This m a y seem surprising, but t h e r e are strong c i r c u m s t a n t i a l c o n n e c t i o n s b e t w e e n the two variables. Over the whole s n o w m e l t season, albedo decreases at the same time t h a t a v e r a g e air temp e r a t u r e is (generally) increasing. S h o r t - t e r m (daily) increases of albedo are caused by snowfall g e n e r a t e d d u r i n g the passage of f r o n t a l waves, b o t h at the w a r m and cold fronts. After such snowfalls, air t e m p e r a t u r e falls as the w a r m sector of the d i s t u r b a n c e passes. Air t e m p e r a t u r e is also linked to some parts of the l o n g w a v e balance, so t h a t T a used as a p r e d i c t o r has some c o m m o n v a r i a n c e with b o t h K* and L*. T h e final p r e d i c t o r g r o u p i n g is K$~c(1 - a s s ) , Ta-again only for comparison. This g r o u p i n g explains b e t w e e n 79 and 90% of Q*~ v a r i a n c e for individual years, and 82% for the whole d a t a s e t - only 5% more t h a n explained by KSac, T~ - - the addition of albedo improves e x p l a n a t i o n only by a small amount. F o r the whole dataset, then, Q*~s can be modelled in t h r e e ways using two r o u t i n e l y available predictors; KSac and Ta: Q'as
2.6 + 0.30 Ta
=
(19)
(B) n =
141, r 2 =
Q'as =
0.37, s t a n d a r d e r r o r B
= 0.033.
- 0 . 0 9 + 0.19 K~a c
(20)
(B) n
=
141, r 2 = 0.43, s t a n d a r d e r r o r B
Q'as =
- 0 . 1 5 + 0.19 K$~c + 0.28 T~ (B1)
=
0.019. (21)
(B2)
n = 141, R 2 = 0.77, s t a n d a r d errors: B1 = 0.013, B2 = 0.021. The s t a n d a r d e r r o r s of the coefficients are included to emphasize t h a t these
156
relationships are statistically defined, and that the relative influence of the predictor variables will change from season to season. CONCLUSIONS
Analysis of several years' data collected during snowmelt in a deciduous forest shows that the energy balance is dominated by the radiative exchanges, net allwave (Q*) typically accounting for more than 90% of observed snowmelt. The turbulent exchanges H and L E are small as a result of low windspeeds near the forest floor, and because of damping of turbulent motion caused by the great stability of the air over the snowpack. The net longwave radiative flux (L*) is more influential on Q* in the forest than it is in the open because of the upper limit on LT dictated by the fixed temperature of the melting snow surface, and because of large values of L$ generated by the canopy as a result of absorption of shortwave radiation. The shortwave balance at the snow surface is highly variable because of changes in surface reflectivity, which can change over a season from initial values of 0.8 down to final values of 0.15 as the pack thins, and dustfall and forest debris concentrate at the surface. Albedo can change dramatically in the short term (e.g. from 0.40 to 0.90 overnight) as a result of a few millimeters of snowfall. Given the above, it is not surprising that the regression of Q* on KS, implicitly assuming relative invariance of a and L*, is much less successful in the forest during snowmelt than it is in other environments. In fact, the technique is quite inappropriate in the context of forest snowmelt. The modelling of changes in ~ and L* is very difficult because of the complexity of the physical processes involved, so the explicit incorporation of these two variables is not possible for operational modelling. A model using KSac and mean daily air temperature (Ta) is shown to explain 77% of the variance in Q*a~. Because of the general availability of these two variables, this model can be easily applied to operational snowmelt modelling. ACKNOWLEDGEMENTS
The work was carried out with granting from the Natural Sciences and Engineering Research Council (Canada), and with support from the Environmental Research Branch, Chalk River Nuclear Laboratories. Particular thanks go to Dr. P.J. Barry of AECL, who has been a staunch supporter of this and associated projects from the outset. L:K. Hendrie helped with data collection and analysis. REFERENCES Granger, R.J. and Male, D.H., 1978. Melting of a prairie snowpack. J. Appl. Meteorol., 17 (12): 1833-1842. Hendrie, L.K. and Price, A.G., 1979. Energy balance and snowmelt in a deciduous forest. In: S. Colbeck and M. Ray (Editors), Proc. Modeling of Snowcover Runoff. U.S. Army Cold Regions Res. Engl. Lab., Hanover, N.H., pp. 211-221.
157 Petzold, D.E. and Wilson, R.G, 1974. Solar and net radiation over melting snow in subarctic woodlands. Proc. 31st East. Snow Conf., Ottawa, Ont., pp. 51-57. Price, A.G., 1977. Snowmelt runoff processes in a subarctic area. Climatol. Res. Ser~ 10, McGill Subarctic Res. Pap., 29: xxx. Price, A.G. and Petzold, D.E., 1984. Surface emissivities in a boreal forest during snowmelt. Arc. Alp. Res., 16 (1): 45-51. Rauner, Yu, L., 1961. On the heat budget of a deciduous forest in winter. Izo. Abad. Vau. SSSR (Ser. Geogr.), 4:83 90. Can. Dep. Transp. Meteorol. Branch. Meteorol. Transl., 11, 6(~77. Szeicz, G., Endrodi, G. and Tajchman, S., 1969. Aerodynamic and surface factors in evaporation. Water Resour. Res., 5 (2): 380-394. United States Army Corps of Engineers, 1956. Snow Hydrology: Summary Report of snow investigations. North Pac. Div., Portland Oreg., 437 pp. Webb, E.K., 1970. Profile relationships: the log~linear range and extension to strong stability. Q. J. R. Meteorol. Soc., 96: 67-90.
145
[11
PREDICTION OF S N O W M E L T R A T E S IN A D E C I D U O U S FOREST
A.G. PRICE Department of Geography, University of Toronto, Scarborough Campus, 1265 Military Trail, West Hill, Ont. MIC 1A4 (Canada) (Received September 21, 1987; revised and accepted November 18, 1987)
ABSTRACT Price, A.G., 1988. Prediction of snowmelt rates in a deciduous forest. J. Hydrol., 101: 145- 157. A model is presented for the prediction of the energy available for the melting of snow in a deciduous forest. The turbulent exchanges H and LE can be estimated using physically based equations, but are small at the forest floor, accounting typically for less than 10% of daily melt totals, largely because of the much reduced windspeed within the forest. Net allwave radiative energy flux density (Q*) alone is a good predictor of snowmelt rates. Physical modelling of Q* is not operationally possible, and the simple regression of Q* on solar energy flux (KS), a technique used with success in some circumstances, is shown to be inappropriate in a forest during snowmelt. A multivariate model using two commonly available variables: K~ and air temperature in the forest, explains over 3/4 of the variance of Q*.
INTRODUCTION
Forests are one of the major cover types in eastern North America, and are ecosystems of great intrinsic interest. In addition, they are major contributors to the fluxes of water, energy and oxygen into the atmosphere. From an economic viewpoint, forests are of enormous consequence. T h e h y d r o l o g y o f f o r e s t e d b a s i n s is q u i t e d i s t i n c t , a s is t h e n a t u r e o f t h e e n e r g y b a l a n c e d u r i n g s n o w m e l t a t t h e f o r e s t floor. I n m a n y o f t h e m o r e northerly forests, the seasonal melting of the snow releases the single largest w a t e r i n p u t o f t h e y e a r . A l t h o u g h t h e i n t e n s i t y o f s n o w m e l t i n p u t s is m u c h l o w e r t h a n t h a t o f t y p i c a l r a i n f a l l s , t h e t o t a l v o l u m e o f w a t e r r e l e a s e d is q u i t e large - as much as a third of annual water inputs. Infiltrating snowmelt water is t h e m o s t i m p o r t a n t r e c h a r g e e v e n t f o r b o t h s o i l a n d g r o u n d w a t e r , s i n c e t h e s n o w m e l t p u l s e is u n d i m i n i s h e d b y i n t e r c e p t i o n o r e v a p o t r a n s p i r a t i o n . S n o w m e l t w a t e r w h i c h is n o t s t o r e d i n t h e s o i l o r i n t h e g r o u n d w a t e r m o v e s t o s t r e a m s a s t h e ~ ' s n o w m e l t f l o o d " , a n d i t is i n t h i s c o n t e x t t h a t t h e p r e d i c t i o n o f s n o w m e l t r a t e s is m o s t p o i n t e d l y o f i n t e r e s t , s i n c e h y d r o l o g i c a l f o r e c a s t i n g using operational basin streamflow models requires a knowledge of snowmelt rates.
0022-1694/88/$03.50
© 1988 Elsevier Science Publishers B.V.
146 THEORY T h e e n e r g y b a l a n c e of a s n o w p a c k c a n be written: Hm =
Q* + H + L E + Hp + H g -
AHo
(1)
where: H m = e n e r g y flux d e n s i t y a v a i l a b l e for m e l t i n g s n o w Q* = net a l l w a v e r a d i a t i v e e n e r g y flux d e n s i t y H = sensible e n e r g y flux d e n s i t y (Sensible h e a t ) L E = l a t e n t e n e r g y flux d e n s i t y ( L a t e n t h e a t ) Hp = e n e r g y flux d e n s i t y into s n o w p a c k f r o m r a i n H~ = e n e r g y flux d e n s i t y b e t w e e n s n o w a n d g r o u n d A H o = c h a n g e in h e a t deficit or n e g a t i v e h e a t s t o r a g e in the s n o w p a c k (All a b o v e in u n i t s of W m - 2 or M J m - 2 d -1) U n d e r m o s t conditions, eqn. (1) c a n be simplified to: Hm = Q* + H + L E
(2)
T h e flux of w a t e r r e l e a s e d (Vm) is then:
Ym
Hm (flw Lf) -1
=
(3)
w h e r e Pw = d e n s i t y of w a t e r (kg m-3), and Lf = l a t e n t h e a t of fusion of w a t e r ( J k g 1). T h i s flux of w a t e r is r e l e a s e d n e a r the s u r f a c e of t h e s n o w p a c k , and, h a v i n g m o v e d t h r o u g h the s n o w p a c k , b e c o m e s t h e i n p u t to the b a s i n system. All t h r e e t e r m s in eqn. (2) pose p r o b l e m s of e s t i m a t i o n . H a n d LE, w h i c h r e s u l t f r o m t u r b u l e n t airflow a c r o s s the surface, m a y be e x p r e s s e d as: U LE
= Pa Cp D h (Tz - To) f ( R i ) = pa Lv Dw (O'-~-2) (ez - eo) f ( R i )
where: = = T~ = To = = ez = eo = Pa Cp = Lv = P = Dh = Dw -f(Ri) = Uz
Zo
w i n d s p e e d at m e a s u r e m e n t h e i g h t z (m s - l ) r o u g h n e s s l e n g t h of s n o w s u r f a c e (m) a i r t e m p e r a t u r e (°C) s u r f a c e t e m p e r a t u r e (°C) v a p o u r p r e s s u r e of t h e a i r ( k P a ) v a p o u r p r e s s u r e of s u r f a c e ( k P a ) a i r d e n s i t y ( k g m -3) specific h e a t of a i r at c o n s t a n t p r e s s u r e (J k g 1K - 1) l a t e n t h e a t of v a p o r i z a t i o n of w a t e r (J kg-1) p r e s s u r e of the a t m o s p h e r e ( k P a ) e x c h a n g e coefficient for h e a t (m s 1) e x c h a n g e coefficient for w a t e r v a p o u r (m s -~) a f u n c t i o n of R i c h a r d s o n n u m b e r , a s t a b i l i t y m e a s u r e (nd)
(4) (5)
147 The value of the exchange coefficients Dh and Dw can be shown to be: Dh = Dw = k 2 uz [ln2 (Z/Zo)] -1
(6)
The Richardson number, R i , can be written for the whole slab of air between the surface and the measurement height, z, as: Ri
=
(Tz -
To) z g (T~ u2z) ~
(7)
where: g = acceleration due to gravity (ms-2); k = Von Karman constant (nd); T~ = mean temperature of the layer (K); and z = measurement height
(m). For stable conditions, and with all the physical constants or computable variables combined into two constants K' and K", and for a particular z and z0, the equations reduce to: H = LE =
K ' u z (Tz K" uz(ez-
To) (1 + a R i ) -1 e o ) ( 1 + aRi) -~
(8) (9)
where a is an empirically determined constant whose value is 10.0 (Webb, 1970). The details of the derivation of these equations are given in Price (1977). The assumptions involved in arriving at eqns. (8) and (9) are considerable, but are relatively easy to justify for the stable conditions (Ri > 0) which invariably occur over a melting snowpack (since T~ is usually > To). Equations (6)-(9) are not physically sophisticated, and incorporate a number of broad assumptions, one of them being the equality of the exchange coefficients under varying stabilities. In defence of the equations it should be noted that there is good agreement on the form of the correction for stable conditions, and that the equations are extremely easy to apply operationally, if To and e0 are known. They are relatively undemanding of input data, and in e x t r e m i s can be operated with only uz, Tz and relative humidity. Overall, the equations are physically so superior to the commonly used empirically based "melt equations" (U.S. Army Corps of Engineers, 1956), that their lack of general usage is puzzling. Detailed modelling of the turbulent exchanges H and L E is undertaken exclusively in the open (Granger and Male, 1978). Under these circumstances the only exchanging surface is the snow, and the conditions are the simplest possible. In a forest, not only does moving air exchange energy with the snow surface, but also with the trees. The form of the wind profile in the forest, therefore, is a result of these interactions. Measured profiles in forests are complex, with several reversals, and under these circumstances, orthodox flux-profile relationships are quite inapplicable. Equations (8) and (9) show the turbulent exchanges to be a nonlinear function of uz, the nonlinearity depending on Ri. Since over a melting forest snowpack R i is invariably positive and usually large, then H and L E are much reduced. Small values of Uz imply large values of Ri, and uz at the forest floor is known to be small. Thus, a p r i o r i , one can reason that H and L E will be small at the forest floor during snowmelt. Equations (8) and (9), derived for open
148 conditions, will overestimate H and LE, since some mixing is occurring between trees and air r a t h e r t han snow and air. These equations, with the use of a liberal estimate of surface roughness length (z0) allow estimation of values of H and LE in the forest. The third term in eqn. (2) also differs between the forest and open conditions. Q* may be written: Q* = g~ (1 - ~) + L*
(10)
where: K~ = incoming (solar) shortwave energy flux density; a = surface reflectivity in the shortwave (albedo); and L* = L$ - L T = net longwave energy flux density (all in W m -2 or M J m - 2 d 1). The major energy input to the system is KS, which is easily and routinely measured, and which at least at a scale of tens of kilometres is not significantly spatially variable. The first difference between Q* in the open and the forest is that in the forest, K~ is strongly interfered with by the forest canopy, so that only a proportion reaches the snowpack, the rest being absorbed or reflected. The amount of absorption of K$ by the snowpack will depend on ~, whose values and variability may not differ greatly between forest and open. The second major difference between Q* in forest and open is L* - (L$ - LI"). Under most conditions, L$ and L$ vary in response to similar controls, and L* is therefore a relatively conservative quantity. During forest snowmelt, however, L$ may become large, since the canopy is absorbing a large proportion (about 90%) of K$ incident on it, heating up, and radiating in the longwave (Price and Petzold, 1984). At the same time, LI" from the snow is '~locked" at a maximum value of 316 W m ~, the black-body emittance at 0°C. L* may, then, become much larger in the forest than in the open. The net result of these two differences between the forest and open is t hat although K* at the forest floor may be small, L* may be large, and Q* values in the forest may be as much as twice those in the open (Hendrie and Price, 1979). SITE DESCRIPTIONAND INSTRUMENTATION The measurement site is on the property of Atomic Energy of Canada Limited (AECL), in subbasin 3 of the Perch Lake watershed, Chalk River, Ontario (46°02'N; 77°20'W; see Fig. 1), and is in an extensive deciduous forest of mixed aspens (Populus tremuloides Michx. and P. grandidentata Michx.) which form the dominant canopy at 22m. Birch, (Betula papyrifera Marsh.) forms a secondary canopy at about 15 m, and there is a general cover of beaked hazel (Corylus cornuta Marsh.) at 1-2 m. The radiation data in the present paper were measured using Middleton CN1 net radiometers, one over the canopy, and three at 2 m over the snow surface. A Kipp-Zonen CM-5 solarimeter was used to measure KS above the canopy, and a Middleton solari-albedometer to measure the shortwave fluxes at the snow surface. Windspeed was measured at 1, 2, 5, 7.5, 10 and 12.5m over the snowpack, using Casella sensitive anemometers. Wet and dry-bulb temperatures were measured at each of these levels using aspirated psychrometers.
149
\
\
To D
Fig. 1. Location of the study area.
Water volumes released from the snowpack were measured using a 5 x 5 m lysimeter plot, which intercepted all meltwater reaching the base of the snowpack. DISCUSSION OF DATA In this paper is part of seven years' data collected at the site. Windspeeds were measured continuously at the site in 1978, 1984 and 1985 in order to investigate the sizes of H and LE. Using eqns. (8) and (9), and incorporating assumptions likely to overestimate the values of the turbulent exchanges (logarithmic profile and z0 = 5 mm), these equations were used with observed values of T1.0, el.0 and ul.0 to predict daily melt resulting from H and LE. When Tl.0 and Q* were greater than zero, then To was taken to be 0°C and e0 to be 0.611 kPa. When Q* was negative, and surface melt ceased, it was assumed that the snow surface acted in a fashion analogous to a wet-bulb thermometer, gaining by H what it might lose by LE. For the nocturnal period, then, ( H + L E ) = O.
150
20;3
2113
3"
E
UA U-
rnU.l
2913
WU.I I.UUJ no_
30•3
31 1 3
I
I 14
12
O0
12
214
C30
z_z
i
"
!
II O0
12
O0
12
O0
O0
1'2
O0
1'2
O0
112
O0
TIME E.S.T.
Fig. 2. H o u r l y w i n d s p e e d o v e r the c a n o p y and n e a r t h e forest floor.
W i n d s p e e d (both n e a r t h e forest floor and over the c a n o p y ) s h o w s a very s t r o n g diurnal p e r i o d i c i t y (Fig. 2) w i t h m a x i m u m v a l u e s near t h e forest floor of a b o u t 1 m s -1 o c c u r r i n g near solar n o o n . M i n i m u m w i n d s p e e d s at the forest floor o c c u r at night, w i t h v a l u e s t y p i c a l l y b e t w e e n 0 and 0.3 m s - i. This implies that m o s t wind m o t i o n n e a r the forest floor o c c u r s as a result of c o n v e c t i o n w i t h i n t h e c a n o p y , c a u s e d by c a n o p y h e a t i n g as a result o f a b s o r p t i o n o f solar 1.o. w z a • o •
D~
oo
•
oo o o
IJA CSu-
o• • • °
OoO
~
WW
0.5
~to
•
•
•
1:5
DALLY A V E R A G E W I N B S P E E D
£o
ABOVE THE CANOPY
z:5 (ms-D
Fig. 3. D a i l y a v e r a g e w i n d s p e e d o v e r t h e c a n o p y and n e a r the forest floor.
151 TABLE 1 Observed v a l u e s of Q* a n d c o m p u t e d v a l u e s for H and LE, 1984 Hour
7-8 ~9 ~10 1~11 11-12 1~13 1~14 1~15 1~16 1~17 1718 1~19
29/03/84
10/04/84
12/04/84
H
LE
H
LE
H
LE
1.0 2.8 3.1 8.6 10.7 9.2 10.9 10.9 6.4 5.5 1.3 0.2
-5.4 -3.0 0.3 1.3 2.7 2.1 2.2 1.8 0.9 0.7 0.1 0.0
-
-
0.2 2.4 15.6 11.6 9.8 12.1 10.6 11.4 9.8 7.7 1.9 0.6
0.0 0.8 5.7 4.8 4.5 5.5 5.0 4.9 4.1 2.9 0.7 0.2
1.2 2.8 9.6 12.8 12.4 14.6 11.6 9.7 2.8 0.7 -
0.78 0.50 0.50 1.8 2.2 3.5 2.7 2.0 0.4 0.0 -
Totals for period (Mdm ~) 0.25
0.01
0.28
0.05
0.34
0.14
Q*
Q*
Q*
2.74
6.03
7.74
All v a l u e s in W m -2.
radiation, and that nocturnal windspeeds and turbulent exchanges are likely to be extremely small. An exception to this occurs when frontal activity generates regional airflows, as from 16th to 18th March in Fig. 2. During the snowmelt period at Perch Lake, such flows are usually accompanied by elevated nocturnal temperatures, in which case the "melting" boundary conditions can be applied overnight. Data from three years of measurement of windspeed show that ul.0 in the forest is roughly predictable from the windspeed over the canopy, as shown in Fig. 3. Although the scatter is considerable, for daily totals the 1 m windspeed is about 1/3 of that over the canopy. Thus eqns. (8) and (9) can be used with only temperature and vapour pressure data actually measured in the forest. Table 1 shows observations and computations made for three days during the 1984 melt. Using the model outlined above, and for times when Q* and TL0 were positive (and the surface therefore known to be melting), H and L E were computed. The days were selected for their high values of windspeed, temperature, and vapour pressure, in an attempt to define upper limits on the size of the turbulent exchanges. The data show t h a t there is a strong diurnal rhythm to the turbulent exchanges, resulting from the pattern of windspeed shown in Fig. 2. Values of H and L E earlier and later in the day than the hours shown in Table 1 are all close to zero. This makes the application of the model
152
TABLE 2 D a i l y m e l t t o t a l s a t t r i b u t a b l e to t h e t h r e e m a j o r e n e r g y fluxes of eqn. (2) D a t e (04/78)
Observed snowmelt
Melt from
M e l t from
runoff
Q*
H
LE
10 11 12 13 14 15 16 17 18 19 20
0.9 1.8 8.5 7.8 8.2 6.8 13.0 14.4 14.5 5.2 5.6
0.1 1.8 8.1 5.6 9.1 7.4 12.2 13.1 14.9 3.6 4.0
0.00 0.34 1.01 0.56 0.64 0.58 0.26 0.26 0.89 0.40 0.17
0.08 0.21 0.17 0.22 0.00 - 0.02 0.05 0.03 0.05 0.00 0.11
Period t o t a l s
86.7
[79.9
Hm
=
[Q*
+ H
+ LE]
=
+
5.11
+
0.99]
86.0
All m e l t r a t e s in m m of water.
much easier, since during the nocturnal period when the surface temperature is not known, the turbulent exchanges are effectively zero. The second feature shown in Table 1 is that, for the three days shown, (again, selected as days of higher H and L E ) , the total amount of energy transferred to the snowpack by the turbulent exchanges is less than 10% of the radiatively transferred energy, with a maximum hourly contribution of 21.3 W m 2, (7.4% of Q* at that hour). This means that errors in the estimation of H and L E are of little consequence, because of the small size of the turbulent exchanges. On the other hand, the equations can be applied to approximate the exchanges when they are positive (into the snowpack), and of any significant size. Table 2 shows the daily melt totals attributable to the three major energy fluxes of eqn. (2) for a typical period of snowmelt. The data in Table 2 were selected because the period shown was one of rapid snowmelt with no rain, during a period when the runoff plot was known not to be leaking, when no ice-lenses were present in the snowpack, and when all data were complete. For the period shown, H and L E account for 7.1% of total melt. Thus for an isothermal forest snowpack, the energy balance can be written to a good approximation as: Hm = Q*~s
(11)
where Q*~ is net allwave radiative energy flux density at 2m over the snowpack, and so: Hm = K~s (1 - ~a~) + L*a~
(12)
153 where the subscript ~'as" signifies "above the snow surface". The problem of snowmelt forecasting has now become one of predicting Q'as. With a view to synthesizing Q*a~, consider the terms of eqn. (12). KSa~ is a complex, highly spatially and temporally variable quantity generated by the interference of canopy elements with the direct solar beam. For any given forest type for daily totals, however, one may be able to predict KSas from K$~c (ac signifies "above canopy"), particularly during snowmelt when canopy density is not changed by leafing or leaf-fall. For example, at the Perch Lake site for 1979: K$~ = 0.50 KSac
(13)
n = 30, s = 0.06. The two remaining variables in eqn. (12) are much less tractable: ~s being a function of snow depth, snow age, snow surface debris accumulation, crystal size, direct-diffuse ratio and sun angle. L*~ (net longwave) is part of and influenced by the energy balance and radiative characteristics of the snow surface and the canopy. Modelling the variability of ~s and L*~ would be desirable, but is beyond the scope of this paper. Even the direct measurement of Q'as is problematic, since great variability in KSa~ makes point measurements of Q*~ very probably unrepresentative for even a small basin. In any event, what is needed for forecasting snowmelt is a model demanding only routinely measured variables such as air temperature and K$~c. An obvious technique is to use K$~c as a predictor of Q*~. First, it is measured routinely; second, it is spatially relatively invariant; (at the small basin scale and for daily totals) third, it is strongly physically linked with Q*~ and Q*~, and less directly with L*a~. The technique of regressing Q* on KS has been applied in hydrology, forestry and agronomy (Petzold and Wilson, 1974; Rauner, 1961; Szeicz et al., 1969), because of its great power: it works well under many circumstances, and if it does, one can use an available variable (K~) to predict a complex, unavailable one (Q*). The model has the form: Q*
=
a + bKJ.
(14)
Daily value totals of Q* are defined as the sum of hourly values from the first negative of the preceding day to the last positive value of the day in question. Totals of KS are simply the sum of all values of KS for any day. Albedo is defined as the daily sum of KT~s divided by the daily sum of K$~. Mean daily temperature is the arithmetic mean of 24 hourly average temperatures for that day. All radiative values in the following equation are in M J m - 2 d -1, and temperatures are in °C. Taking the data from the same period as those shown in Table 1, and regressing Q*~ on K ~ , the least-squares line is given by: Q*~ = 0.24 + 0.17 K~a~
(15)
n = 12, r 2 = 0.92. Just as a comparison, the most commonly used snowmelt predictor is mean daily temperature T~ - typically applied as the "degree--lay index". For the
154 TABLE 3 P r o p o r t i o n s of v a r i a n c e of Q'as e x p l a i n e d by p r e d i c t o r v a r i a b l e s (all s i g n i f i c a n t at p = 0.05) Year
n
1979 1983 1984 1985 All (1978~1985)
Ta
K~
K~(1
30 28 28 38
0.221 0.244 0.491 0.625
0.307 0.273 0.569 0.407
141
0.369
0.432
a)
K$,T a
K~(1 - ~), T a
0.653 0.630 0.687 0.695
0.682 0.732 0.813 0.910
0.790 0.835 0.792 0.905
0.682
0.769
0.820
s a m e 12 d period for 1978 used to derive eqn. (15), the r e s u l t is: Q'as =
1.6 + 0.56 T a
(16)
n = 12, r 2 = 0.29 (not s i g n i f i c a n t at p = 0.05). I n this case, K ~ a c is a far b e t t e r p r e d i c t o r t h a n T a. On this basis, one m i g h t t e n t a t i v e l y p r o c e e d w i t h the t e c h n i q u e . I f the a n a l y s i s is r e p e a t e d for 1979, h o w e v e r , t h e r e s u l t is: Q'as
=
0.71 + 0.13 K~a c
(17)
n = 28, r 2 = 0.31. F o r 1983, the r e l a t i o n s h i p is: Q'as -
0.68 + 0.12 Klac
(18)
n = 28, r 2 = 0.27. Plainly, a n y model l e a v i n g u n e x p l a i n e d 3/4 of the v a r i a n c e of the d e p e n d e n t v a r i a b l e is of little utility. T h e s o u r c e s of this v a r i a b i l i t y a r e ~as a n d L*a~. T h e t e c h n i q u e w o r k s well in c i r c u m s t a n c e s w h e r e ~ and L* o v e r the s u r f a c e a r e r e l a t i v e l y i n v a r i a n t . I f t h e y a r e not, t h e n the m o d e l is not useful. T h e case of forest s n o w m e l t is p e r h a p s the w o r s t c i r c u m s t a n c e in w h i c h to a p p l y the r e g r e s s i o n t e c h n i q u e . O v e r m a n y surfaces, ~ is e s s e n t i a l l y i n v a r i a n t , e x c e p t p e r h a p s for d i u r n a l c h a n g e s due to sun e l e v a t i o n , or s e a s o n a l c h a n g e s due to c r o p g r o w t h . In c o n t r a s t , s n o w s u r f a c e albedo at P e r c h L a k e h a s b e e n o b s e r v e d to c h a n g e f r o m 0.40 to 0.90 o v e r n i g h t as the r e s u l t of a light snowfall. S n o w is p r o b a b l y t h e s u r f a c e w i t h the m o s t c h a n g e a b l e ~. L* is also m u c h m o r e v a r i a b l e o v e r a s n o w s u r f a c e t h a n it is o v e r m a n y o t h e r surfaces, since it c a n b e c o m e r e l a t i v e l y l a r g e w h e n the s n o w is melting, and, as n o t e d above, this is p a r t i c u l a r l y t r u e in t h e forest. T h u s the c o n t r a s t b e t w e e n 1978 and 1979 is n o t entirely surprising. T a b l e 3 s h o w s the p r o p o r t i o n of t h e v a r i a n c e of Q*a~ e x p l a i n e d by s e v e r a l p r e d i c t o r v a r i a b l e s for s e v e r a l y e a r s individually, a n d for t h e w h o l e d a t a s e t (1978-1985). F o r c o m p a r i s o n , t h e first v a r i a b l e used is simply m e a n a i r temp e r a t u r e (Ta), w h i c h e x p l a i n s b e t w e e n 22 a n d 63% of t h e v a r i a n c e of Q*~s for i n d i v i d u a l years, a n d a b o u t 37% for the w h o l e d a t a s e t . Thus, despite the fact
155 t h a t the d e g r e e , l a y index g e n e r a l l y works b e t t e r in forests t h a n in the open, b e c a u s e of the influence of the l o n g w a v e fluxes in d e t e r m i n i n g Q*, Ta still leaves o v e r h a l f the v a r i a n c e of Q*~s unexplained. Used as the only predictor, KSa¢ explains b e t w e e n 27 and 57% of the v a r i a n c e of Q'as for individual years, and 43% for the whole d a t a s e t - - no real improvem e n t on the p e r f o r m a n c e of Ta, and still leaving m u c h v a r i a n c e unexplained. The use of KSac(1 - ~a~) as a p r e d i c t o r explains b e t w e e n 63 and 70% of the v a r i a n c e of Q'as for individual years, and 68% for the whole dataset. Thus, for the whole dataset, K~ac explains a b o u t 40% of Q*~s v a r i a n c e , and K ~ c (1 - a~s) a b o u t 68%, with the rest of the v a r i a n c e g e n e r a t e d by L*~s. Albedo, however, is not a v a r i a b l e c o m m o n l y available for modelling purposes. T h e f o u r t h set of p r e d i c t o r variables shown in Table 3 is KSac and T~. This c o m b i n a t i o n explains from 68 to 91% of Q'as v a r i a n c e for individual years, and 77% for the whole dataset. It is a f e a t u r e of the d a t a s e t t h a t Ta explains a b o u t 1/3 of the v a r i a n c e of albedo. This m a y seem surprising, but t h e r e are strong c i r c u m s t a n t i a l c o n n e c t i o n s b e t w e e n the two variables. Over the whole s n o w m e l t season, albedo decreases at the same time t h a t a v e r a g e air temp e r a t u r e is (generally) increasing. S h o r t - t e r m (daily) increases of albedo are caused by snowfall g e n e r a t e d d u r i n g the passage of f r o n t a l waves, b o t h at the w a r m and cold fronts. After such snowfalls, air t e m p e r a t u r e falls as the w a r m sector of the d i s t u r b a n c e passes. Air t e m p e r a t u r e is also linked to some parts of the l o n g w a v e balance, so t h a t T a used as a p r e d i c t o r has some c o m m o n v a r i a n c e with b o t h K* and L*. T h e final p r e d i c t o r g r o u p i n g is K$~c(1 - a s s ) , Ta-again only for comparison. This g r o u p i n g explains b e t w e e n 79 and 90% of Q*~ v a r i a n c e for individual years, and 82% for the whole d a t a s e t - only 5% more t h a n explained by KSac, T~ - - the addition of albedo improves e x p l a n a t i o n only by a small amount. F o r the whole dataset, then, Q*~s can be modelled in t h r e e ways using two r o u t i n e l y available predictors; KSac and Ta: Q'as
2.6 + 0.30 Ta
=
(19)
(B) n =
141, r 2 =
Q'as =
0.37, s t a n d a r d e r r o r B
= 0.033.
- 0 . 0 9 + 0.19 K~a c
(20)
(B) n
=
141, r 2 = 0.43, s t a n d a r d e r r o r B
Q'as =
- 0 . 1 5 + 0.19 K$~c + 0.28 T~ (B1)
=
0.019. (21)
(B2)
n = 141, R 2 = 0.77, s t a n d a r d errors: B1 = 0.013, B2 = 0.021. The s t a n d a r d e r r o r s of the coefficients are included to emphasize t h a t these
156
relationships are statistically defined, and that the relative influence of the predictor variables will change from season to season. CONCLUSIONS
Analysis of several years' data collected during snowmelt in a deciduous forest shows that the energy balance is dominated by the radiative exchanges, net allwave (Q*) typically accounting for more than 90% of observed snowmelt. The turbulent exchanges H and L E are small as a result of low windspeeds near the forest floor, and because of damping of turbulent motion caused by the great stability of the air over the snowpack. The net longwave radiative flux (L*) is more influential on Q* in the forest than it is in the open because of the upper limit on LT dictated by the fixed temperature of the melting snow surface, and because of large values of L$ generated by the canopy as a result of absorption of shortwave radiation. The shortwave balance at the snow surface is highly variable because of changes in surface reflectivity, which can change over a season from initial values of 0.8 down to final values of 0.15 as the pack thins, and dustfall and forest debris concentrate at the surface. Albedo can change dramatically in the short term (e.g. from 0.40 to 0.90 overnight) as a result of a few millimeters of snowfall. Given the above, it is not surprising that the regression of Q* on KS, implicitly assuming relative invariance of a and L*, is much less successful in the forest during snowmelt than it is in other environments. In fact, the technique is quite inappropriate in the context of forest snowmelt. The modelling of changes in ~ and L* is very difficult because of the complexity of the physical processes involved, so the explicit incorporation of these two variables is not possible for operational modelling. A model using KSac and mean daily air temperature (Ta) is shown to explain 77% of the variance in Q*a~. Because of the general availability of these two variables, this model can be easily applied to operational snowmelt modelling. ACKNOWLEDGEMENTS
The work was carried out with granting from the Natural Sciences and Engineering Research Council (Canada), and with support from the Environmental Research Branch, Chalk River Nuclear Laboratories. Particular thanks go to Dr. P.J. Barry of AECL, who has been a staunch supporter of this and associated projects from the outset. L:K. Hendrie helped with data collection and analysis. REFERENCES Granger, R.J. and Male, D.H., 1978. Melting of a prairie snowpack. J. Appl. Meteorol., 17 (12): 1833-1842. Hendrie, L.K. and Price, A.G., 1979. Energy balance and snowmelt in a deciduous forest. In: S. Colbeck and M. Ray (Editors), Proc. Modeling of Snowcover Runoff. U.S. Army Cold Regions Res. Engl. Lab., Hanover, N.H., pp. 211-221.
157 Petzold, D.E. and Wilson, R.G, 1974. Solar and net radiation over melting snow in subarctic woodlands. Proc. 31st East. Snow Conf., Ottawa, Ont., pp. 51-57. Price, A.G., 1977. Snowmelt runoff processes in a subarctic area. Climatol. Res. Ser~ 10, McGill Subarctic Res. Pap., 29: xxx. Price, A.G. and Petzold, D.E., 1984. Surface emissivities in a boreal forest during snowmelt. Arc. Alp. Res., 16 (1): 45-51. Rauner, Yu, L., 1961. On the heat budget of a deciduous forest in winter. Izo. Abad. Vau. SSSR (Ser. Geogr.), 4:83 90. Can. Dep. Transp. Meteorol. Branch. Meteorol. Transl., 11, 6(~77. Szeicz, G., Endrodi, G. and Tajchman, S., 1969. Aerodynamic and surface factors in evaporation. Water Resour. Res., 5 (2): 380-394. United States Army Corps of Engineers, 1956. Snow Hydrology: Summary Report of snow investigations. North Pac. Div., Portland Oreg., 437 pp. Webb, E.K., 1970. Profile relationships: the log~linear range and extension to strong stability. Q. J. R. Meteorol. Soc., 96: 67-90.