U ′ + φ (1 - Swi )Se = 0
(7.5)
where φ is the total pore volume as a function of the total volume. Changes in Swi occur
on a much larger time scale than changes in effective saturation, hence it is sufficient to
consider Swi constant over an individual time step (although it can be altered through the
course of the snowmelt season). To obtain an analytical expression that can easily be
employed in a computer model, the solution of these equations differs from earlier
solutions (Colbeck 1972, Tucker and Colbeck 1977) in that Equation (7.4) is used to find
Se as a function of U, which is then substituted into Equation (7.5) above, to give the
governing equation for water volume flux
1
⎛
1⎞
-1 ⎛ ρwkw0 g ⎞
⎜1- ⎟
nc
U′.
U = -ncφ (1 - Swi ) ⎜
-1
⎝ nc ⎠
(7.6)
⎟ U
⎝ ηw ⎠
In order to make the problem tractable, nc, φ, Swi, and kw0 are assumed to be constant over
each time step. These variables change slowly compared to the time scale with which
mobile water moves through the snow. By simplifying notation as follows
1
⎛ ρwkw0 g ⎞
nc
κ cf = ncφ -1 (1 - Swi )-1 ⎜
(7.7)
⎟
ηw ⎠
⎝
and
ηcf = 1 - n1 ,
(7.8)
c
the flux governing equation becomes
U = -κ cf U cf U ′ .
η
(7.9)
This equation is solved using separation of variables. The general solutions to the volume
flux equation are therefore sums in space and time of particular solutions governed by
boundary conditions while maintaining the restrictions indicated above. It is not
necessary to completely solve the volume flux equation, as a temporal expansion of
particular solutions will tend to approximate the exact solution if the particular solutions
are generated near enough to each other in time. With each separate particular solution
being generated by meteorological data in separate time steps, the accuracy of the
solution will be dependent on the time step of the available meteorological data, with
hourly data proving accurate enough to give promising results, and finer resolution data
resulting in more accurate approximations of the exact solution.
Equation (7.9) is solved by assuming the solutions take the form
U ( x, t ) = B ( x + c1 ) cc (t + c2 )
α
β
(7.10)
where substitution back into Equation (7.9) and re-grouping gives
β ( x + c2 ) cc (t + c2 ) = -κ cf B cf α cc ( x + c1 ) cc cf cc (t + c2 ) cf .
α
β -1
α η +α -1
βη +β
η
(7.11)
For this to hold
αcc = η1f
(7.12)
c
β = - η1
(7.13)
cf
1
-
ηcf
B = κ cf
(7.14)
59
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