γk = bulk density of each i, l, v, and a con-
of SNTHERM (Jordan 1991), as described above.
stituent.
The exchange of energy with the ground is con-
sidered insignificant.
All four snow constituents are assumed to be in
The water equivalent of melt, equal to the avail-
local thermal equilibrium within the snow
able energy from the surface energy balance, is
medium. The basic set of equations developed in
added to rainfall and routed through the snow-
the model can simulate a full variety of snow
pack.
types because this mixture theory is used consis-
tently throughout the model.
Flow through a one-layer snowpack
The water volume flux equation is
Limitations
Where speed and simplicity are principal con-
i
-1 ρw kg n
1
1-
U
U
= -nφ-1(1 - Swi )
cerns, SNTHERM may not be suitable.
U n
(25)
w
t
x
Strengths
SNTHERM is applicable to a full range of
where U = volume flux of water (cm s1),
meteorological conditions such as snowfall, rain-
t = time (s),
fall, freezethaw cycles, and transitions between
n = dimensionless effective saturation (S)
bare and snow-covered ground. This algorithm
exponent,
φ = dimensionless porosity of snow,
provides much useful information about the
snowpack condition that would be useful for run-
Swi = irreducible water saturation of snow
off forecasting and has already been used in dis-
(% of total volume),
ρw = density of water (g cm3),
tributed format. Slope and aspect are model in-
put parameters.
k = absolute permeability of snow (cm2),
w = viscosity of water (g cm1 s1),
SNAP
The SNAP model (Albert and Krajeski 1998)
x = vertical spatial coordinate (cm).
uses a full surface energy balance to estimate melt
Equation 25 assumes the effective saturation
water input to a one-layer snowpack. This model
exponent (n), effective porosity (φ), irreducible
includes a new mathematical solution to the flow
water saturation (Swi), and permeability (k) are
of water through the snow that is more physically
constant over each time step, but may vary over
based than current operational models, yet
the melt season. The variation of n and Swi over
computationally efficient. The mathematical solu-
time are not well understood, and in the present
tion begins with the simplified form of Darcy's
model version are held constant at default values
equation as set forth by Colbeck (1972) in which
of 3.3 and 3%, respectively. Melt volumes are
capillary flow in snow is considered negligible
assumed to travel as waves through the entire
compared to gravity flow. Albert's method then
depth of a single-layer snowpack. The method
diverges from earlier mathematical approaches
allows for volume flux waves to absorb the resid-
(Colbeck 1972, Tucker and Colbeck 1977) in that
ual mobile water from preceding waves and to
it derives an analytical expression for water vol-
determine when the combined meltwater flux
ume flux, and then evaluates the expression
wave will reach the bottom of the pack.
using a Newton's method approximation.
SNAP should provide more accurate predic-
Grain growth and permeability (k)
tion of the magnitude and timing of snowmelt
Grain growth occurs over the melt season
than current operational models, none of which
increasing permeability, and the rate of melt
attempt to physically model the flow of water
infiltration. Conceptually the snowpack is one
through the snowpack. Because SNAP solves an
bulk layer with a wet portion (in which the irre-
analytical expression for water volume flux
ducible water saturation has been met) and a dry
through a bulk layer snowpack, it is expected to
portion (which has either not yet been wetted, or
be more computationally efficient than the multi-
has refrozen). The weighted averages of the two
layered SNTHERM model.
parts are taken as the average crystal volume
within the pack (Vav), and used to compute grain
Surface energy balance
diameter (d, cm):
The surface energy balance is equivalent to that
10
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