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2001 11:28 Char Count= 0 104 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) P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 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). P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 106 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) P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 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