implying the solution must have the form

-1

-1

1

( x + *c*1 )

( t + *c*2 )

ηcf

ηcf

ηcf

.

(7.15)

Co-ordinates are then normalized by defining

χ = *x *+ *c*1

(7.16)

τ = *t *+ *c*2

(7.17)

thus giving

1

⎛ χ ⎞ ηcf

⎜κ τ ⎟

.

(7.18)

⎟

⎝ cf ⎠

In the normalized co-ordinates ( χ ,τ ), each particular solution represents a wave of

volume flux with its origin at the point ( χ = 0,τ = 0 ). In other words, each differential

element of mobile water, *dA*m, generates its own particular solution where τ = 0 occurs at

the time the water volume becomes mobile within the snow (surface melt or rainfall), and

χ = 0 occurs at the vertical height of the snowpack at the time the water volume becomes

mobile.

To find the depth of penetration of a wave as a function of time is simply a matter of

integrating saturation from the point of generation of the wave to whatever depth

volume of the wave. For an input volume per unit area of *A *(*cm*) over an area σwa, if the

depth of penetration of the resulting flux wave is denoted as *x*U, then at any point τ in

time

χ = *x*U

∫χ

(7.19)

=0

where *A *is the volume/unit area runoff in *cm *and σwa is the area. Therefore we have

χ = *x*U

∫χ =0

.

(7.20)

φ (1 - *S*wi )

Using *S*e from Equation (7.4) and *U *from Equation (7.18), substituting into Equation

(7.20), then solving the integral yields the equation for the depth of penetration of a wave

as a function of time

η

2

ηcf -ηcf

⎛

⎞ ⎛ ρwkw0 g ⎞

(

κ cf τ ) .

1-ηcf

⎜ η φ (1 - *S *) ⎟ ⎜ η

(7.21)

⎟

⎟

⎠

⎝ cf

Next, consider the time it will take a wave of volume flux to penetrate to a given depth

(most useful in calculating the delay between influx and outflux). In order for the water

mass to balance in time, the volume of water passing by a given depth *D *must equal the

initial mobile water volume

τ =∞

∫τ τ U ( *D*,τ )*d*τ = *A*

.

(7.22)

=

0

Substituting from Equation (7.15) yields

-1

-1

1

τ =∞

ηcf

ηcf

ηcf

∫τ τ

κ cf D τ

(7.23)

=

0

where *D *is the depth of penetration at time τ0.

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