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in the rip currents as seen in the physical model by Haller et al. (1997).
Most of these nearshore circulation models were developed by finite difference
methods, although Wu and Liu use finite element techniques. Some of these models
are being used for engineering work, although it should be pointed out that most of
them are very computer intensive and require very small time steps (on the order of
seconds) to reach steady-state solutions. This often causes problems when trying to
determine the effect of several days’ worth of wave conditions or to predict 1 or 50
years of coastal conditions.
Models typically havebeendevelopedonly formonochromatic (single frequency)
wave trains rather than for directional spectra. There is a definite need for further
development of these models.
5.6 SWASH ZONE DYNAMICS
The swash zone is defined as that region on the beach face delineated at the up-
per limit by the maximum uprush of the waves and at its lower extremity by the
maximum downrush. This portion of the beach face is intermittently affected by suc-
cessive waves that traverse this zone in a zig-zag fashion, and it is an area where very
interesting patterns, called beach cusps, can occur. Finally, substantial quantities of
longshore sediment transport may occur within the swash zone limits. Thus, it is of
interest to understand this phenomenon better and to develop a predictive capability
for this region.
In this section we will examine progressively more complex and realistic cases of
swash zonedynamics by using simpler (analytic)models than those numericalmodels
previously described. As shown in Figure 5.11, friction, gravity, inertia, and pressure
gradients are all potentially significant forces acting on a water element within the
swash zone.
Figure 5.11 Forces acting on elemental swash particle. The coordinate x points onshore
and y alongshore.
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5.6 SWASH ZONE DYNAMICS 115
Figure 5.12 Definition sketch for swash problem.
We will consider the case of a wave approaching the beach face at some angle
θ to the x-axis that points onshore. The leading edge of the wave advances up the
beach face with initial velocity V0 (see Figure 5.12). To render the problem tractable,
it is assumed that it is possible to idealize the water particle as a solid particle that
retains its identity as it moves up and down the beach face. The height of the particle
is h, and its length and width are taken as �s and �n, where s is in the direction
of the particle motion and n is perpendicular. The appropriate equations of motion
are
ρh�s�n
dVx
∂t
= −ρgh�s�n sinβ − ρ f
8
�s�n|V |Vx
ρh�s�n
dVy
∂t
= −ρ f
8
h�s�n|V |Vy, (5.57)
or, after eliminating common terms,
dVx
dt
= −g sinβ − f
8h
|V |Vx
dVy
dt
= − f
8h
|V |Vy, (5.58)
in which β is the bottom slope and f is the Darcy–Weisbach friction factor. All the
terms on the right-hand side (gravity and friction) serve to decelerate the uprushing
particle. Various forms of the bottom friction will be discussed in later sections.
The initial conditions (at x = 0) are
Vx0 = V0 cos θ0
Vy0 = V0 sin θ0 (5.59)
5.6.1 OBLIQUELY INCIDENT WAVES AND NO FRICTION
Equations (5.58) are readily solved for the case of no friction. A simple integration
of these equations yields the following for the two horizontal components of water
particle velocity and displacement (for a particle located initially at the origin):
Vx = V0 cos θ0 − gt sinβ
Vy = V0 sin θ0
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116 WAVES AND WAVE-INDUCED HYDRODYNAMICS
The position of the particle can be found by integrating the velocities with respect to
time:
x(t) = V0 cos θ0t − gt
2
2
sinβ
(5.60)
y(t) = V0 sin θ0t
Several results can be obtained from this simple formulation. For example, if we
consider the case of normally incident waves (θ = 0), the maximum uprush occurs at
the time that the velocity is zero, as given by
tmax = V0g sinβ (5.61)
It is seen that the corresponding value of the maximum uprush is
xmax = V
2
0
2g sinβ
, (5.62)
which is recognized as the familiar result for trajectory of a solid particle shot into
the air vertically with speed V0, however, in this case, the value of gravity has been
reduced by the sine of the beach face slope. The time required for the particle to
return to the starting location (x = 0) is known as the frictionless natural period of
a water particle on a beach face, or the swash period Tn0 , as given by 2tmax:
Tn0 =
2V0
g sinβ
(5.63)
The natural period depends both on the initial shore-normal velocity and the slope
of the beach face.
5.6.2 NONDIMENSIONAL EQUATIONS
At this stage it is convenient to express Eqs. (5.58) in nondimensional form. We will
choose Tn0 and V0, as the reference quantities, that is,
t ′ = t
Tn0
V ′ = V
V0
, (5.64)
which transforms Eq. (5.58) into
dV ′x
dt ′
= −2− γ |V ′|V ′x
dV ′y
dt ′
= −γ |V ′|V ′y, (5.65)
where γ is a friction parameter defined as
γ = f V
2
0
4gh sinβ
(5.66)
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5.6 SWASH ZONE DYNAMICS 117
5.6.3 OBLIQUELY INCIDENT WAVES WITH LINEAR FRICTION
Although it is impossible to develop simple solutions to the full swash equations
owing to their nonlinearity, the linearized swash equations can be solved.
The equations are
dV ′x
dt ′
= −2− γLV ′x
(5.67)
dV ′y
dt ′
= −γLV ′y,
where γL is a linearized friction term. Solving,
V ′x(t
′) = 2
γL
(
e−γL t
′ − 1)+ V ′x0e−γL t ′
V ′y(t
′) = V ′y0e−γL t
′
(5.68)
x ′(t ′) =
(
2
γ 2
L
+ V
′
x0
γL
) (
1− e−γL t ′)− 2
γL
t ′
y′(t ′) = V
′
y0
γL
(
1− e−γL t ′) (5.69)
It is of interest to compare the characteristics of the preceding swash solution with
those for frictionless swash. We will consider the case of small γL (i.e., small
friction).
The maximum value of uprush occurs when V ′x = 0, which, for small γL , yields
(using e−γL t
′ ≈ 1− γL t ′ + · · ·)
t ′max =
V ′x0
2+ γLV ′x0
≈ t
′
max0(
1+ γL2 cos θ0
) , (5.70)
in which t ′max denotes the uprush time without friction. Thus, the effect of friction is
to reduce the uprush time. The maximum uprush x ′max is
x ′max = x ′max0
(
1− γL cos θ0
2
)
(5.71)
That is, in accordance with intuition, the effect of friction is to reduce the maximum
uprush.
The maximum backrush velocity can be evaluated by first solving for t ′0, where
x ′(t ′0) = 0 and substituting this value of t ′0 in Eq. (5.68a). Solving for t ′0
t ′0 =
V ′x0
1+ γL cos θ02
, (5.72)
we see that the natural period is reduced. The maximum backrush velocity [V ′B =
V ′(t ′0)] is
V ′x(t
′
0) = −V ′x0
(
1− γL cos θ02
1+ γL cos θ02
)
≈ −Vx0 (1− γL cos θ0) (5.73)
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118 WAVES AND WAVE-INDUCED HYDRODYNAMICS
It is clear that the particle has imparted a net shoreward impulse on the beach
face because its upward momentum is greater than the downward momentum. The
net upward impulse I per unit length of the beach face is
Ix = ρ�xh [Vx(0)+ Vx(Tn)] (5.74)
such that for a frictionless system Vx(Tn) = −Vx(0), the net impulse is zero. The
average force F¯ x per unit length is
F¯ x = IxT =
ρ�xh
T
[Vx(0)+ Vx(Tn)] (5.75)
For the case of linearized friction just considered,
F¯ = ρ�xh
T
(
γLV ′x0
) = ρ�xh
T
γL cos
2 θ0
= ρ�xh
T
V0[cos2 θ0 − cos θ0 cos θF + γL cos2 θ0 cos θF ], (5.76)
where θF is the angle of the return flow.
Figure 5.13 presents a comparison of the nondimensional idealized trajectories
for frictionless and frictional systems. The initial angle of obliquity is 40◦, and the
value of γL is 5.0 for the frictional system. Of particular interest is the saw-toothed
shape of the trajectory for γL = 5.0, whereas the frictionless trajectory is symmetrical
about the peak. A field method to estimate γL would be useful. Because it can be
shown that the return angle is much less than the initial angle, this