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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. P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 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 P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 116 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) P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 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) P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 118 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