Snowmelt → Energy Balance Method

The input variables for the Energy Balance method of estimating runoff due to snowmelt are defined as follows:

QSW = net shortwave radiation [W / m2]
QLW = net longwave radiation [W / m2]
Tair = air temperature [deg C]
Tsurf = surface (snow) temperature [deg C]
RH = relative humidity [none] (in [0,1])
p0 = atmospheric pressure [mbar]
uz = wind velocity at height z [m / s]
z = reference height for wind [m]
z0air = surface roughness height [m]
h0snow = initial snow depth [m]
h0swe = initial depth, snow water equivalent [m]
ρsnow = density of the snow [kg / m3]
csnow = specific heat capacity of snow [J / (kg deg_C)]
ρair = density of the air [kg / m3]
cair = specific heat capacity of air [J / (kg deg_C)]
Lf = latent heat of fusion, water [J / kg] (334000)
Lv = latent heat of vaporization, water [J / kg] (2500000)
eair = air vapor pressure at height z [mbar]
esurf = vapor pressure at the surface [mbar]
g = gravitational constant = 9.81 [m / s2]
κ = von Karman's constant = 0.41 [unitless]

For each variable, you may choose from the droplist of data types. For the "Scalar" data type, enter a numeric value with the units indicated in the dialog. For the other data types, enter a filename. Values in files must also have the indicated units.

Single grids and grid sequences are assumed to be stored as RTG and RTS files, respectively. Time series are assumed to be stored as text files, with one value per line. For a time series or grid sequence, the time between values must coincide with the timestep provided.

Note: If net total radiation has been measured, it can be entered as QSW and then QLW can be set to zero. Any meteorological variables entered here (such as Tair) are automatically shared with other other processes, such as Evapotranspiration and Precipitation.


Equations Used by the Energy-Balance Method

M = (1000 * Qm) / (ρwater * Lf) = meltrate [mm / sec]
Mmax = (1000 * hsnow / dt) * (ρwater / ρsnow) = max possible meltrate [mm / sec]
dhsnow = M * (ρwater / ρsnow) * dt = change in snow depth [m]
Qm = QSW + QLW + Qh + Qe - Qcc = energy flux used to melt snow [W / m2]
Qh = ρair * cair * Dh * (Tair - Tsurf) = sensible heat flux [W / m2]
Qe = ρair * Lv * De * (0.662 / p0) * (eair - esurf) = latent heat flux [W / m2]
Dn = κ2 * uz / LN[(z - hsnow) / z0air]2 = bulk exchange coefficient (neutrally stable conditions) [m / s]
Dh = Dn / [1 + (10 * Ri)],     (Tair > Tsurf) = bulk exchange coefficient for heat [m / s] (stable)
= Dn * [1 - (10 * Ri)],     (Tair < Tsurf) = bulk exchange coefficient for heat [m / s] (unstable)
De = Dh = bulk exchange coefficient for vapor [m / s]
Ri = g * z * (Tair - Tsurf)   /
     [uz2 (Tair + 273.15)] = Richardson's number [unitless]
Qcc = (see note below) = cold content flux [W / m2]
Ecc[0] = h0snow * ρsnow * csnow * (T0 - Tsnow) = initial cold content [J / m2] (T0 = 0 now)
eair = esat(Tair) * RH = vapor pressure of air [mbar]
esurf = esat(Tsurf) = vapor pressure at surface [mbar]
esat = 6.11 * exp[(17.3 * T) / (T + 237.3)] = saturation vapor pressure [mbar, not KPa], Brutsaert (1975)


Notes on the Equations

The cold content of the snow pack, Ecc, represents an energy deficit that must be overcome before snow begins to melt. First, Qnet is computed as the sum of all energy fluxes (the Q's). Wherever (Qnet < 0 and hsnow > 0) the snow cools and the cold content increases. Similarly, wherever (Qnet > 0 and hsnow > 0) the snow warms and the cold content decreases. In both cases the cold content changes according to: Ecc = [Ecc - (Qnet * dt)] and we have M=0 as long as (Ecc > 0). However, if warming continues long enough to consume the cold content (so that Ecc drops to zero), then the snow begins to melt (M > 0). In this case the meltrate is given by M = Qnet / (ρwater * Lf).


References

Brutsaert, W. (1975) On a derivable formula for long-wave radiation from clear skies, Water Resources Research, 11, 742-744.

Dingman, S.L (2002) Physical Hydrology, 2nd ed., Prentice Hall, New Jersey. (see Chapter 7, pp. 285-299)

Schlicting, H. (1960) Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 647 pp.

Zhang, Z., D.L. Kane and L.D. Hinzman (2000) Development and application of a spatially-distributed Arctic hydrological and thermal process model (ARHYTHM), Hydrological Processes, 14, 1017-1044.