The solution to this integral gives the time required for a wave of volume flux to

penetrate to depth *D*

ηcf

⎛

⎞ηcf -1

1

1 -ηcf

κ cf ⎞ηcf ⎟

⎛

τ0 = ⎜

*Am *⎜

.

(7.24)

⎟

⎜ ηcf

*D*⎠ ⎟

⎝

⎜

⎟

⎝

⎠

For modeling purposes, note that Equation (7.21) may also be written in the simpler form

1-*n*c

1

⎛ *n *-1 ⎞

*nc*

*xU *= κ cf ⎜ c ⎟ τ nc .

(7.25)

*A* ⎠

⎝

Having depth of penetration as a function of time, and vice versa, requires only that the

effects of preceding flux waves be incorporated in order to construct a working model of

water movement through homogeneous snow. Therefore, consider two flux waves, each

proceeding according to Equation (7.18)

1

⎛χ ⎞

ηcf

*U*1 ( χ1 ,τ1 ) = ⎜ 1 ⎟

(7.26)

⎝ κτ1 ⎠

and

1

⎛ χ2 ⎞

ηcf

*U * 2 ( χ2 ,τ 2 ) = ⎜

⎜κ τ ⎟

.

(7.27)

⎟

⎝ cf 2 ⎠

Assume that χ1 - χ2 = ε > 0 (that the wave *U*1 began at an earlier point in time than the

wave *U*2). Since there is no singularity within the region over which the *U*2 solution

applies, flux will be continuous and differentiable over this region, up to the singularity

caused by the *U*1 *U*2 junction. As the *U*2 wave flows through the snow, Equation (7.25)

must always be met, even as new water volume is absorbed into wave *U*2 from the

residual saturation of wave *U*1. Therefore, to accommodate the new water volume the

second wave has encountered while still maintaining its solution to the conservation and

flux-saturation equations, the volume flux wave *U*2 must travel deeper into the snowpack

than it otherwise would, thereby encountering more residual saturation in the process.

Numerically, the residual saturation encountered will take the form

1

⎡ ηw ⎤

*nc*

1

*S*1 = ⎢

(7.28)

⎥ *U*1

*nc*

⎣ ρwkw0 g ⎦

which, when combined with *U*1 from Equation (7.26), becomes

1

1

⎡ ηw ⎤ ⎛ χ1 ⎞

*nc *-1

*nc*

*S*1 = ⎢

⎜

⎟ ,

(7.29)

ρwkw0 g ⎥ ⎜ κ cf τ1 ⎟

⎣

⎦⎝

⎠

which means that if *x*A2 is the deepest point within the snow at which the *U*2 solution

applies, (the point of the singularity), then the volume of water absorbed by the *U*2 wave

in moving some infinitesimal distance through the snow will be given by

1

1

⎛ *x*A2 ⎞

⎡ ηw ⎤

*nc *-1

*nc*

∂ A2 = φ (1 - *S*wi ) ⎢

∂ xA2 .

⎜

⎜κ τ ⎟

(7.30)

ρwkw0 g ⎥

⎟

⎣

⎦

⎝ cf 1 ⎠

61