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FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 5.2 WATER WAVE MECHANICS 97 Table 5.2 Breaking Wave Characteristics and the Surf Similarity Parameter ζ→ ≈0.1 0.5 1.0 2.0 3.0 4.0 5.0 Breaker Collapsing/ None Type Spilling Plunging Surging (Reflection) κ 0.8 1.0 1.1 1.2 N 6–7 2–3 1–2 <1 <1 r 10−3 10−2 0.1 0.4 0.8 κ = breaking index; N = number of waves in surf zone; r = reﬂection from beach. Source: Battjes (1974). deﬁned as the ratio of the beach slope, tanβ, to the square root of the deep water wave steepness, by the following expression: ζ = tanβ/ √ H0/L0 (5.17) His results are shown in Table 5.2, which shows the breaker type, the breaking index, the number of waves in the surf zone, and the reﬂection coefﬁcient from a beach as a function of the surf similarity parameter. At ﬁrst, simple theoreticalmodelswere proposed to predict breaking. Theoretical studies of solitary waves (a single wave of elevation caused, for example, by the displacement of a wavemaker in one direction only) in constant-depth water showed that the wave breaks when its height exceeds approximately 0.78 of the water depth. This led to the widespread use of the so-called spilling breaker assumption that the wave height within the surf zone is a linear function of the local water depth H = κh, where κ , the breaker index, is on the order of 0.8. Later experiments with periodic waves pointed out that the bottom slope was important as well, leading to elaborate empirical models for breaking (e.g., Weggel 1972). The spilling breaker assumption, however, always leads to a linear dependency of wave height with water depth. In the laboratory, thewave height is often seen to decreasemore rapidly at the breaking line than farther landward. In the ﬁeld, on the other hand, Thornton and Guza (1982), showed that the root-mean-square wave height in the surf zone (on their mildly sloping beach) was reasonably represented by Hrms = 0.42h. Dally, Dean, and Dalrymple (1985) developed a wave-breaking model based on the concept of a stable wave height within the surf zone for a given water depth. This model has two height thresholds, each of which depends on the water depth. As waves shoal up to the highest threshold (a breaking criteria), breaking com- mences. Breaking continues until the wave height decreases to the lower threshold (a stable wave height). This stable wave height concept appears in experiments by Horikawa and Kuo (1966), which involved creating a breaking wave on a slope. This breaking wave then propagates into a constant depth region. Measurements of the wave height along the wave tank showed that the waves approach a stable (broken) wave height of H = �h, where � is about 0.35–0.40 in the constant depth region. P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 98 WAVES AND WAVE-INDUCED HYDRODYNAMICS The Dally et al. model, valid landward of the location of initial breaking, is ex- pressed in terms of the conservation of energy equation ∂ECg ∂x = −K h [ECg − (ECg)s], (5.18) where K is an empirical constant equal to approximately 0.17. For shallow water, the energy ﬂux can be reduced to ECg = 18ρgH 2 √ gh, and thus the preceding equation relates the wave height H to the water depth h. For planar beaches, where the depth is given by h = mx , an analytic solution can be obtained, H Hb = √√√√[( h Hb )( K m − 12 ) (1+ α)− α ( h Hb )2] , (5.19) where m is the beach slope and α = K� 2 m ( 5 2 − Km ) (Hb Hb )2 For the special case of K/m = 5/2, a different solution is necessary: H Hb = ( h Hb )√[ 1− β ln ( h Hb )] (5.20) Here, β = 5 2 �2 ( Hb Hb )2 The range of solutions to Eqs. (5.19) and (5.20) are shown in Figure 5.3. Note that for K/m > 3, the wave heights can be much less than predicted by a spilling wave assumption (which coincides approximately with the K/m = 3 curve), whereas for Figure 5.3 Wave height variation predicted across a planar beach (Dally et al. 1985, copyright by the American Geophysical Union). P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 5.2 WATER WAVE MECHANICS 99 Figure 5.4 Comparisons of analytic solution of Dally et al. (1985, copyright by the American Geophysical Union) with laboratory data of Horikawa and Kuo (1966). steeper beaches, K/m < 3, the wave heights are larger. Comparisons with Horikawa and Kuo data for waves breaking on a planar beach are shown in Figure 5.4. For more complicated beach proﬁles, numerical solutions to Eq. (5.18) are used. Dally (1990) extended this model to include a realistic surf zone by shoaling a distri- bution of wave heights rather than a monochromatic wave train. Wave breaking is one of the most difﬁcult hydrodynamics problems. The highly nonlinear and turbulent nature of the ﬂow ﬁeld has prevented the development of a detailed model of wave breakers, which has spurred the development of macroscale models. Peregrine and Svendsen (1978) developed a wake model for turbulent bores, arguing that, from a frame of referencemovingwith thewave, the turbulence spreads from the toe of the bore into the region beneath the wave like a wake develops. A wave-breaking model for realistic wave ﬁelds was proposed by Battjes and Janssen (1978), who also utilized the conservation of energy equation, but the loss of wave energy in the surf zone was represented by the analogy of a turbulent hydraulic jump for the wave bore in the surf zone. Further, the random nature of the wave ﬁeld was incorporated by breaking only the largest waves in the distribution of wave heights at a point. In the ﬁeld, thismodelwas extended byThornton andGuza (1983), who were able to predict the root-mean-square wave height from the shoaling zone to inside the surf zone to within 9 percent. Svendsen (1984)developed the rollermodel,which is basedon thebore–hydraulic jump model. The roller is a recirculating body of water surﬁng on the front face of P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH CB373-05 CB373 June 27, 2001 11:28 Char Count= 0 100 WAVES AND WAVE-INDUCED HYDRODYNAMICS Figure 5.5 Schematic of vertically descending eddies with the arrow show- ing the direction of breaker travel (from Nadaoka 1986). the wave after breaking is initiated. This roller has mass and momentum that must be accounted for in the governing equations. He showed that agreement between theory and setup measurements was better with the roller model than a breaking- index model. Measurements of breaking waves in the laboratory have provided valuable in- sights into the nature of the breaking process. Nadaoka, Hino, and Koyano (1989) have shown that spilling breakers produce what they refer to as “obliquely descend- ing eddies,” which are near-vertical vortices that remain stationary after the breaker passes. These eddies descend to the bottom, pulling bubbles down into the water column. Figure 5.5 shows a schematic of the eddies they observed. The role of the eddies in nearshore mixing processes (both of mass and momentum) is as yet un- known; further, the generation mechanism, which implies the rotation of horizontal vorticity due to the roller on the face of a spilling breaker, into near-vertical vorticity, is at yet unknown. However, Nadaoka, Ueno, and Igarashi (1988) observed these eddies in the ﬁeld and showed that they are an important mechanism for the suspen- sion of sediment and that the bubbles, drawn into the vortices, provide a buoyancy that creates an upwelling of sediment after the wave passage. 5.2.5 MEAN WAVE QUANTITIES Associated with the passage of periodic waves are some useful quantities found by averaging in time over the wave period. For example, there is a mean transport of water toward the shoreline, the mass transport, which is not predicted by the linear Airy theory, which assumes that each water particle under a waveform is traveling in a closed elliptical orbit. We deﬁne the mass transport as M = 1 t2 − t1 ∫ t2 t1 ∫ η −h