Energy needed in the

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Energy needed in the warming phase of the melt
process  Qcc -ci ?w hm (Ts -
Tm)   Where Qcc cold content of snow
(MJ/m2) ci heat capacity of ice(2102 J/kg-oC)
?w density of water ? 1 g/cm3 1000 k g/m3 hm
water equivalent in the snow pack (m) Ts
average temperature of the snow Tm melting
point temperature (0oC)   Energy needed to melt
snow  Qm hm ?w lf   Qm energy needed to
melt a snow pack (MJ/m2) hm water equivalent in
the snow pack (m) ?w density of water ? 1 g/cm3
1000 k g/m3 lf latent heat of fusion of water
0.0224 MJ/kg change to 0.334 MJ/kg
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Emissivity of atmosphere eat Cloudy sky, no
forest canopy eat 1.72 ( ea )1/7
(10.22 C2) (Ta 273)1/7 ea
vapor pressure at 2 m height (k Pa) Ta
Temperature at 2 m height (oC) C cloud cover
fraction
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Emissivity of atmosphere eat Cloudy sky, with
forest canopy eat (1- F) 1.72 ( ea
)1/7 (10.22 C2) F
(Ta 273)1/7 ea vapor pressure at 2 m height
(k Pa) Ta Temperature at 2 m height (oC) C
cloud cover fraction F fraction of forest
canopy cover
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Approximate temperature of snow If Talt 0oC, then
Tss ? Ta 2.5 If Tagt 0oC, then Tss ? 0oC When
Tss 0oC, and assuming ess ? 1, then s (Tss
273)4 27.3 MJ m-2 day-1 And L eat s (Tat
273)4 - s (Tss 273)4 becomes L eat s (Tat
273)4 - 27.3 MJ m-2 day-1
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Turbulent Exchange of Sensible Heat H -?a ca
k2 va (Ta Ts) ln(za-zd)/zo2 ?a
density of air ca heat capacity of air k von
Karmans constant 0.4 va velocity of air Ta
temperature of air Ts temperature of snow za
height of temperature and velocity measurement zd
zero plane displacement zo surface roughness
height
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Turbulent Exchange of Latent Heat LE - ? 0.622
?a k2 va (ea es) P ln(za-zd)/zo2 ?
heat of vaporization, condensation or sublimation
(vaporization fusion) ?a density of air k
von Karman constant 0.4 va velocity of
air ea water vapor pressure in air es vapor
pressure of snow za height of temperature and
velocity measurement zd zero plane displacement
zo surface roughness height
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Effect of Forest density on wind velocity vaF
(1-0.8 ? F) ? vaO vaF estimated wind
velocity in the forest F Fractional forest
cover vaO measured wind velocity in the open
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Energy input from rainfall on melting snow  R
r ? cw ? ?w ? (Tr - Tm)   Where R energy from
rainfall (MJ/m2) r quantity of rainfall (m) ci
heat capacity of liquid water (0.00422
MJ/kg-oC) ?w density of water ? 1 g/cm3 1000
kg/m3 Tr temperature of the rain on entry to
snow ? air temperature Tm melting point
temperature (0oC)   Energy input from rainfall
that freezes in the snowpack  R r ? cw ? ?w ?
(Tr - Tm) r ? ?w ? lf   lf latent heat of
fusion of water 0.334 MJ/kg
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Energy Exchange Processes   S K L H LE
R G   S energy gained or lost by snow pack
(MJ/m2) K net short wave radiation (MJ/m2) L
net long wave radiation (MJ/m2) H sensible heat
exchange with atmosphere (MJ/m2) LE latent heat
exchange with the atmosphere (MJ/m2) R heat
input by rain (MJ/m2) G conductive heat
exchange with the ground (MJ/m2)   K and R will
always add energy to a snow pack, but other
process can either add energy to or remove energy
from the snow pack
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Temperature Index Method of estimating snow
melt  Dw M ?(Ta Tm), if TagtTm Dw 0, if
TaltTm   Where, Dw snow melt (depth/time,
mm/day) M melt coefficient, melt factor, or
degree-day factor (mm/oC-day) Ta air
temperature (oC) Tm temperature at which melt
begins (oC) M varies with latitude, slope
inclination and aspect, forest cover, time of
year. Various recommended values and equations
have been developed for different settings, for
example, for eastern US Forests M fF ?(0.7
0.0088 J) ? fsl (eq 5-59) Where fF Forest
type factor J Julian date and fsl ratio of
solar Radiation received on the slope to
radiation on horizontal surface  
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Hybrid Snow Melt Approach Dw (K L) Mr Ta
?w lf   Mr restricted melt coefficient
2.0 mm/oC-day K net shortwave radiation input(
MJ/m2) L net long-wave radiation input or
output ( MJ/m2) Ta air temperature (oC) ?w
density of water ? 1 g/cm3 1000 k g/m3 lf
latent heat of fusion of water 0.334 MJ/kg
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Saturated or frozen
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• Rate of liquid water movement in snow pack
• Capillary forces (matric potential) are
  negligible, so Darcys Law simplifies to
• Vz Kh(?w)
• Vz downward rate of water movement (length per
  unit time)
• Kh(?w) hydraulic conductivity of snow at liquid
  water content of ?w
• Kh(?w) Kh ?(?w - ?ret)/(? - ?ret)c
• Kh saturated hydraulic conductivity of snow,
  function of snow density
• ?ret maximum vol. water content of liquid water
  held against gravity
• porosity (dimensionless)
• C empirical constant ? 3

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