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Energy needed in the “warming phase” of the melt process  Qcc = -c i *  w * h m * (Ts - Tm) Where: Qcc = cold content

Energy needed in the “warming phase” of the melt process  Qcc = -c i *  w * h m * (Ts - Tm) Where: Qcc = cold content of snow (MJ/m 2 ) c i = heat capacity of ice(2102 J/kg- o C)  w = density of water  1 g/cm3 = 1000 k g/m3 h m = water equivalent in the snow pack (m)

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Energy needed in the “warming phase” of the melt process  Qcc = -c i *  w * h m * (Ts - Tm) Where: Qcc = cold content

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  1. 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

  2. Emissivity of atmosphere εat Cloudy sky, no forest canopy: εat = 1.72(ea )1/7 (1+0.22 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

  3. Emissivity of atmosphere εat Cloudy sky, with forest canopy: εat = (1- F) ·1.72·( ea )1/7· (1+0.22 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

  4. Approximate temperature of snow If Ta< 0oC, then Tss Ta – 2.5 If Ta> 0oC, then Tss 0oC When Tss = 0oC, and assuming εss 1, then σ (Tss + 273)4 = 27.3 MJ m-2 day-1 And L = εat σ (Tat + 273)4 - σ (Tss + 273)4 becomes L = εat σ (Tat + 273)4 - 27.3 MJ m-2 day-1

  5. Turbulent Exchange of Sensible Heat H = -ρa ca k2 va (Ta – Ts) [ln{(za-zd)/zo}]2 ρa = density of air ca = heat capacity of air k = von Karman’s 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

  6. Turbulent Exchange of Latent Heat LE = - λ 0.622 ρa k2 va (ea – es) P [ln{(za-zd)/zo}]2 λ = heat of vaporization, condensation or sublimation (vaporization+ fusion) ρa = 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

  7. 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

  8. 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

  9. 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

  10. Temperature Index Method of estimating snow melt Dw = M (Ta – Tm), if Ta>Tm Dw = 0, if Ta<Tm 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

  11. 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

  12. Saturated or frozen

  13. Rate of liquid water movement in snow pack • Capillary forces (matric potential) are negligible, so Darcy’s 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) = K*h [(w - ret)/( - ret)]c • K*h = 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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