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Cap5_COASTAL PROCESSES_ DEAN_e_Dalrymple

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104 WAVES AND WAVE-INDUCED HYDRODYNAMICS
and um is the maximum orbital velocity of the waves, um = (ηC/h)max = HC/2h →
κ
√
g(h + η)/2 ≈ 0.4C in the surf zone. This form of the bottom shear stress results
from linearizing the originally nonlinear shear stress term.
Substituting for the bottom shear stress and solving Eq. (5.34) yields the first
solution of Longuet-Higgins,
V (x) = 5πgκm
∗(h + η)
2 f
(
sin θ
C
)
, (5.36)
where m∗ is the modified slope, m∗ = m/(1+ 3κ2/8). This equation shows that the
longshore velocity increases with water depth and with incident wave angle but
decreases with bottom friction factor. The equation applies within the surf zone,
resulting in an increasing velocity out to the breaker line, and then the velocity is
zero offshore. Equation (5.36) is valid for any monotonic beach profile when the
lateral shear stress terms are negligible.
One contribution missing in the preceding equation is the influence of the lateral
shear stresses, which in this case, would tend to smooth out the velocity profile given
above, particularly the breaker line discontinuity, which is not evident in laboratory
experiments (Galvin and Eagleson 1965). Longuet-Higgin’s second model included
a lateral shear stress term τxy of the form
τxy = ρνe ∂V
∂x
The eddy viscosity, which has the dimensions of length2/time, was assumed to be
νe = N x
√
gh, where N is a constant and the “mixing” length scale is the distance
offshore, x . His analysis yielded the following form of the longshore current, which
is nondimensionalized by the values at the breaker line, X = x/xb and V0 = V (xb),
given by Eq. (5.36):
V/V0 =
{
B1X p1 + AX, for 0 < X < 1
B2X p2 , for X > 1,
(5.37)
where the coefficients and powers are
B1 =
(
p2 − 1
p1 − p2
)
A (5.38)
B2 =
(
p1 − 1
p1 − p2
)
A (5.39)
p1 = −34 +
√(
9
16
+ 1
P
)
(5.40)
p2 = −34 −
√(
9
16
+ 1
P
)
(5.41)
A = 1(
1− 52 P
) (5.42)
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5.4 LOW-FREQUENCY MOTIONS AT THE SHORELINE 105
Figure 5.6 Longshore currents on a planar beach (from Longuet-Higgins 1970b, copyright by
the American Geophysical Union).
The variable P is the ratio of the eddy viscosity to the bottom friction,
P = 8πm N
κ f .
There is an additional solution for the case when P is 2/5; the reader is referred to
Longuet-Higgins (1970b) for this case. The velocity profiles for this longshore current
model are shown in Figure 5.6. Comparisons with laboratory data show that values of
P between 0.1 and 0.4 are reasonable. For P = 0, we have the same result as before
(Eq. (5.36)).
The mean two-dimensional cross-shore circulation just offshore of the breaker
line is also interesting. Matsunaga, Takehara, and Awaya (1988) andMatsunaga and
Takehara (1992) showed in the laboratory that there is a train of horizontal vortices
coupled to the undertow that are spawned at the breaker line and propagate offshore.
Figure 5.7 shows a schematic of the vortex system in their wave tank. They explain
the presence of these coherent eddies as an instability between the mean shoreward
bottom flows and the offshore undertow. Li and Dalrymple (1998) provided a linear
model for this instability based on an instability of the undertow. For very small
waves, the vortices do not appear.
5.4 LOW-FREQUENCY MOTIONS AT THE SHORELINE
Wave staffs or current meters placed in the surf zone often measure extremely en-
ergetic motions at frequencies much lower than those of the incident waves. Yet,
Figure 5.7 Schematic of horizontal vortex sys-
tem (from Matsunaga et al. 1988).
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106 WAVES AND WAVE-INDUCED HYDRODYNAMICS
offshore these low-frequency motions either do not exist or are a very small portion
of the total wave field. These low-frequency waves are surf beat, edge waves, and
shear waves.
5.4.1 SURF BEAT
Surf beat was first described by Munk (1949) and Tucker (1950). Their explanation
was that waves moving toward shore are often modulated into groups that force the
generation ofmeanwater level changes (Longuet-Higgins and Stewart 1963).Where
the waves in a group are large, the water level is depressed (set down), whereas it is
raised where the waves are smaller. As these wave groups enter the surf zone, the
low-frequency forced water level variations are released, reflect from the beach, and
travel offshore as free waves. This mechanism can be combined with the more likely
mechanism that the largewaveswithin thewave groups generate a larger setup on the
beach, which must then decrease when the smaller waves of the group come ashore,
causing an offshore radiation of low-frequency motion at the frequency associated
with the wave group.
The low-frequency surf beat motion can be described by the linear shallow water
wave equations, which are the equations of motion in the onshore and alongshore
directions and the conservation of mass equation:
∂u
∂t
= −g ∂η
∂x
(5.43)
∂v
∂t
= −g ∂η
∂y
(5.44)
∂η
∂t
+ ∂uh
∂x
+ ∂vh
∂y
= 0 (5.45)
Eliminating the velocities in the continuity equation by differentiating with respect
to time and then substituting for the velocities, we find the linear shallow waterwave
equation for variable depth:
∂2η
∂t2
= ∂
∂x
(
gh
∂η
∂x
)
+ ∂
∂y
(
gh
∂η
∂y
)
(5.46)
First, we examine the case in which the shoreline is straight and the contours are
parallel to the shoreline; there is noalongshore variation indepth. Further,weassume
for now that there is no variation in the long wave motion in the longshore direction
and thus that the last term on the right-hand side of the equation is zero.
For a planar beach, h = mx , and if a wave motion that is periodic in time with
angular frequency σ is assumed, Eq. (5.46) reduces to
∂2η
∂x2
+ 1
x
∂η
∂x
+ σ
2
gmx
η = 0 (5.47)
A standing-wave solution to this equation is (Lamb 1945)
η(x, t) = AJ0 (2kx) cos σ t, where k = σ√gh , (5.48)
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5.4 LOW-FREQUENCY MOTIONS AT THE SHORELINE 107
A is the amplitude of the motion at the shoreline, and k is a cross-shore varying
wave number. The offshore dependency of the zeroth-orderBessel function, J0(2kx),
determines the behavior of this wave. This solution is not directly forced by thewaves
but is simply a possible response to forcing. A nonlinear version of this motion is
given by Carrier and Greenspan (1957).
Scha¨ffer, Jonsson, and Svendsen (1990) included the effect of wave groups by
adding forcing due to the radiation stress gradient:
∂2η
∂t2
= ∂
∂x
(
gh
∂η
∂x
)
+ 1
ρ
∂2Sxx
∂x2
(5.49)
This equation, with assumptions for the variation of Sxx within the surf zone, was
solved numerically. Scha¨ffer and Jonsson (1990) found reasonable agreement with
the laboratory work of Kostense (1984).
5.4.2 EDGE WAVES
The edge wave presents a more complicated problem. These waves are motions that
exist only near the shoreline and propagate along it. They can be described using
shallow-water wave theory very simply from the work of Eckart (1951), although
models for edge waves have been known since Stokes (1846).
Startingwith the shallow-water wave equation, Eq. (5.46), substituting in a planar
sloping beach, h = mx , and assuming a separable solution that is periodic in time and
in the alongshore direction,
η(x, y, t) = A f (x) cos λy cos σ t,
allows us to obtain
∂2 f
∂x2
+ 1
x
∂ f
∂x
+
(
σ 2
gmx
− λ2
)
f = 0 (5.50)
Theonly solutionof this equation that is boundedat the shoreline anddecaysoffshore
(remember that it is a trapped wave solution) is
η(x, y, t) = Ae−λx Ln(2λx) cos λy cos σ t (5.51)
The function Ln(2λx) is a Laguerre polynomial, which has the following expansion
as a function of z and n:
L0(z) = 1 (5.52)
L1(z) = 1− z (5.53)
L2(z) = 1− 2z + 12 z
2 (5.54)
Ln(z) =
n∑
k=0
(−1)kn!zk
(k!)2(n − k)! (5.55)
An instantaneous snapshot of the first four solutions, which are