b

g

(6)

$

For the numbers in the previous example, and with a lateral extension of 100 m,

we have τ = (100 m x 5 m x 100 m)/(100,000 m3/year) = 0.5 year. This appears to be

a reasonable time scale for representing the motion of an organized sediment body.

For this value of τ, and with *x*o = 0 without loss of generality, Eq. 5 is plotted in

Fig. 4 as the line labeled "Constant depth" for constant depth of active movement.

As seen in Eq. 5, the rate of spit growth decreases exponentially as the spit

approaches the channel.

100

80

60

Constant depth

40

Increasing depth into channel

20

0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 4. Prediction of spit elongation by analytical model.

This situation is the same as in the previous example, but is made more realistic

by accounting for the depth of the channel. That is, the depth of active movement is

assumed to increase with approach of the spit to the channel, i.e., the inlet channel

thalweg is deeper than the ambient nearshore contours. For simplicity in obtaining

an analytical solution, the depth is taken to increase linearly with fixed slope from

some depth *D*o distant from the channel (at *x*o). So,

(7)

where *S *= *dD/dx = *slope into the channel. The governing equation (Eq. 1), modified

to include the *x*-dependence of *D*, becomes

b

g

1

=

b

g

(8)

Kraus

7

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