L. Erikson et al. / Coastal Engineering 52 (2005) 285302

289

mu=yhulsq and mb=yhblsq, and the momentums are

with the mean up-rush or back-wash velocity from

Eq. (2). For swash conditions where Hb0.8, the

Mu=muusu(t) and Mb=mbusb(t), for the up-rush and

bLaw of the WallQ friction factor, Eq. (7), is used and

back-wash, respectively. A new velocity following

collision, ucoll, is determined from the average of the

for conditions where Hz0.8, the sediment-laden

s

friction factor, Eq. (8), is solved iteratively. An

up-rush and back-wash momentums and assuming

iterative approach is necessary due to the shear stress

that momentum is conserved,

term in the denominator which is a function of f as

M coll

ucoll

for M coll N 0

well. The approach is somewhat computationally

x

mb

inefficient in that the swash front velocities are first

calculated using a clear-fluid formulation for the

M coll

friction factor, and if it is found that the value of the

ucoll

for M coll b 0

10

x

mu

Shield's parameter, itself a function of the friction

factor, exceeds 0.8, the swash front velocities and

where

time trajectories are recalculated using the sediment-

Mu Mb

laden friction factor. This approach is used in this

M coll

11

:

2

model to allow for a theoretical estimate of the

friction term, f, as opposed to using an empirical

For the formulation presented here, mb b mu (see

value. The method allows for different up-rush and

Section 5.2) and hence the conditions of negative or

positive values of Mcoll in Eq. (10) were set.

that they are constant during the swash phase in

The new velocity after collision (Eq. (10)) may be

negative and hence the u-subscript is not specified on

question.

ucoll. A negative value would indicate that the up-rush

s

is fully drowned by the returning back-wash.

3.2. Swash interaction

Swash interaction at the SWS is explicitly

The model accounts for two processes in the

accounted for in the model by imposing an exponen-

interaction between successive bores within the swash

tially decreasing velocity of the returning back-wash

zone above the SWS. The first is dcatch-up and

at the SWS. Velocities calculated with Eq. (2) go to

absorptionT where the front of a wave moving

zero after the swash front passes the SWS causing a

discontinuity in the model as the back-wash

landward (up-rush) is passed by a subsequent bore

ddisappearsT at this point. If the velocity is allowed

moving in the same direction (Fig. 1a). The model

to go to zero at the SWS, initial shoreline velocities

simulates the position and velocity of the leading edge

may be overestimated since they are not measured

of the swash (i.e., Eqs. (5) and (2)), and so, in the case

of catch-up and absorption, the model is written to

directly but are calculated with measured wave

follow the faster swash front, effectively drowning the

heights at the SWS (Eq. (3)); any retarding effect

first but slower up-rush. The second process is

that the back-wash may have on the initial up-rush

dcollisionT whereby two separate fronts collide as the

velocity will not be included if the back-wash is

allowed to go to zero at the SWS. The decreasing

back-wash of a preceding swash lens meets the front

velocity at the SWS immediately after the waves pass

of a subsequent swash wave during its up-rush phase

is described by u(t)=uswseat, where t=0 as the swash

as depicted in Fig. 1b. For such a case, a new leading

front passes the SWS, usws is the velocity at the SWS

edge velocity is calculated based on the principals of

at t=0, a=sin(b)/hs and hs is the water depth at the

momentum.

The momentum (mass times velocity) is calculated

boundary of the surf and swash zones (arbitrarily set

for both the up-rush and back-wash at the point (and

at 0.03 m for these simulations).

time) where the fronts meet. To calculate the

3.3. Potential for swash interaction

momentum it is assumed that a fluid element of

length ls at the leading edge of the up-rush collides

The potential for interaction between subsequent

with a fluid element of the same length, at the leading

swash waves on the beach above the SWS may be

edge of the back-wash. The mass of each are

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