flowing through the snowpack without allowing time for complete drainage or refreezing
between pairs of volume flux waves, however, it is necessary to know the time of
collision of a wave of volume flux with its predecessor, which itself is not traveling
through new snow. Unfortunately, in the event that there are multiple waves flowing
through the snow, no explicit form for the time of collision can be found. Furthermore, no
explicit equation for the time a given volume flux will take to reach any given depth can
be found. This is not the downfall of the model, however, because, while the equations
governing the time of collision between waves are not tractable in general, they are very
well behaved, becoming linear very shortly after the generation of the wave. Therefore
the snow melt module uses a Newton's method approximation to find the time a given
volume flux will take to reach the bottom of the snowpack (and, if necessary, the time of
collision of two waves in the general case). This usually takes four or less iterations of
the approximating loop to come within 0.5 % of the true value.
The process of water volume flux waves absorbing residual mobile water saturation from
the tails of their predecessors (described by Equation [7.36]) , which states that any wave
will eventually overtake any preceding wave given an infinite time and an infinite snow
depth to traverse), although it necessitates the Newton's method approximation above,
also leads to the great strength of this model: its computational simplicity. Because waves
continually overtake each other on the way to the bottom of the pack, there rarely are
very many of them within the snowpack at any given time. The only conditions under
which there will be very many waves within the snowpack is when dAm is low enough,
and snow depth is high enough that melt waves take a long time (compared to the time
scale) to flow out of the pack, and input volume flux is decreasing at a rapid enough rate
that waves are not colliding within the snow. Using an hourly time step, and a surface
energy balance for melt input, it is rare to see more than three waves within the snow at
any one time in the seasonal snow of New England. Any further modeling of flux waves
on their way through the pack would not result in any greater accuracy of outflow
prediction, as only one wave may actually flow out of the bottom of the pack at any time.
7.2 Evolution of Parameters
While the snow property parameters, such as grain size, kw0, Swi, nc, and φe are relatively
stable over the usual lifetime of a wave of volume flux, over the lifetime of a snowcover
they do vary. Here the parameters are considered as constants in a given time step, yet
they may change over the course of the snowmelt season.
7.2.1 Grain Growth
Grain size is currently used only to calculate the permeability of snow. The equations of
Brun (1989) are used to calculate average grain volume. The initial volume of the snow
crystals is currently taken as 0.00005 cm3, which is quickly dwarfed by the growth
equations, especially if the snow is wet. Once the average crystal volume in the wet
portion of the pack and the average crystal volume in the dry portion of the pack are
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