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-

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.

The water volume flux equation is

Where speed and simplicity are principal con-

-1 ρw kg n

1

1-

= -*n*φ-1(1 - *S*wi )

cerns, SNTHERM may not be suitable.

(25)

w

SNTHERM is applicable to a full range of

where *U *= volume flux of water (cm s1),

meteorological conditions such as snowfall, rain-

fall, freezethaw cycles, and transitions between

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-

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.

w = viscosity of water (g cm1 s1),

The SNAP model (Albert and Krajeski 1998)

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 (*S*wi), 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 *S*wi 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-

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 (*V*av), and used to compute grain

diameter (*d*, cm):

The surface energy balance is equivalent to that

10

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