An Analytical Solution for Nearshore Circulation Driven by Focused/Defocused Waves

An offshore shoal or bar refracts ocean surface waves and causes wave focusing/defocusing on the adjacent beach. Wave focal patterns characterized by alongshore variations in wave height, wave angle, and breaking location induce alongshore non-uniformities of wave setup and nearshore circulation, e.g., rip currents and alongshore currents, in the surfzone. A simplified analytic model for nearshore circulation generated by focused/defocused waves on a planar beach is developed and theoretical solutions are obtained using transport stream function and perturbations in alongshore distributions of wave height and wave angle at the breaker line. The analytic model suggests that alongshore currents are strongly affected by a pair of counter-rotating vortices generated shoreward of the wave focal zone. The vortices are persistent, and their strengths depend on the amplitudes of alongshore variations in wave height and wave angle. The alongshore gradient in wave height tends to intensify the vortices while the convergence of wave angle tends to weaken the vortices. Divergent flows associated with the vortices in the surfzone are intense, strengthening alongshore currents in the downstream of the wave focal zone and weakening alongshore currents or causing flows reversal in the upstream. Alongshore currents are modulated by rip currents associated with the wave focusing/defocusing patterns.


Introduction
Nearshore wave climate can be strongly affected by offshore bathymetric features, such as a shoal or a bar, which usually appears offshore from an inlet or irregular shoreline. Owing to the refraction over the offshore bathymetric features, nearshore waves on a beach have large alongshore variations in wave height, angle, and wave breaking location, which leads to alongshore variable currents, sediment transport and shoreline changes.
Wave refraction over a bar or a canyon produces a region of wave focusing or defocusing and corresponding high and low breakers along the curved breaker line. A recent study on modeling of waves and circulation in San Francisco Bight (Shi et al., 2007;Eshleman et al., 2007;Hansen et al., 2014), including the entire San Francisco Bar and Ocean Beach, revealed that waves are focused in a narrow region at Ocean Beach. It was also found that the alongshore currents are strongly affected by the wave focal pattern and exhibit non-uniformities and vortices in the surfzone. Long and Özkan-Haller (2005) examined rip currents correlated with small-scale wave variations due to wave refraction over undulations in the offshore bathymetry. Apotsos et al. (2008) used field observations collected onshore of a submarine canyon to investigate alongshore currents driven by alongshore pressure gradients. Hansen et al. (2015) used a numerical model to investigate alongshore circulation and force balances onshore of a submarine canyon.
Alongshore variations in wave height and the associated variations in mean water surface elevation (wave setup) are thought to be major mechanisms to generate rip currents on beaches. Long and Özkan-Haller (2005) reviewed previous investigations on the mechanisms which can be responsible for alongshore gradients in wave height and mean water surface elevation. Those mechanisms include wave refraction over offshore submarine canyons proposed by Shepard and Inman (1950), standing edge waves by Bowen (1969) and Bowen and Inman (1969), intersecting wave O(100 m) trains of identical frequency by Dalrymple (1975), and nearshore rip channels by many researchers (e.g., Hamm, 1992;Aagaard et al., 1997;Haller et al., 2002). With taking into account the instability mechanism in a wave-current interaction model, Dalrymple and Lozano (1978) imposed alongshore wave height variations to demonstrate generation of steady alongshore periodic circulation on alongshore homogeneous beaches. In Long and Özkan-Haller (2005), Shepard and Inman (1950) hypothesis was adopted to examine small-scale ( ) rip currents correlated with small scale wave variations due to wave refraction over undulations in the offshore bathymetry.
In contrast to a submarine canyon that produces a region of wave divergence, an offshore shoal or bar creates wave convergence through focusing. According to wave simulations at Ocean Beach (Eshleman et al., 2007;Shi et al., 2011), the alongshore wave variations associated with wave focal patterns involve broad regions that can have length scales of up to . The alongshore variations in wave height and wave angle depend on incident wave fields. Typically, the alongshore variation in wave height can be 10% of mean wave height at the breaker line and the alongshore variation in wave angle is typically between and .
In this study, we developed an analytic model to describe alongshore currents affected by offshore wave focal (or defocal) patterns. An idealized case was set up to demonstrate nearshore circulation driven by focused waves breaking on a planar beach. In contrast with previous studies on rip current mechanisms such as Bowen (1969), perturbations are introduced not only in wave height but also in wave angle along a wave breaker line in order to present wave focal patterns. Transport stream function was used to obtain theoretical solutions.

Analytic model
We consider a case of focused waves approaching a straight shoreline with an alongshore uniform bathymetry as illustrated in Fig. 1. is the across-shore coordinate, pointing seaward, is in the alongshore direction, and is the total wave-averaged water depth taking into account the wave-induced setup and setdown. High and low breakers associated with the focal pattern make a curved breaker line described by . We make a general assumption that waves and currents are steady in and near the surfzone. There are no time dependent terms when the forcing function is itself steady. The wave-averaged mass continuity and momentum equations can be written as: where is the wave-averaged velocity vector, is the mean surface elevation, and represents the bottom is the short wave force that is traditionally expressed by the gradients of the radiation stresses defined by Longuet-Higgins andStewart (1962, 1964). Here we use the CL-vortex form of force which has been confirmed to be consistent with the radiation stress expression by Smith (2006) and .
Q w ρ in which is the Stokes drift, is the water density, and k which acts dynamically like a pressure term as pointed out by Smith (2006). represents the wave number vector, and its irrotational property can be expressed by D w is the rate of wave action dissipation in the wave action conservation equation: In Eqs. (4) and (6), and are the wave phase velocity and the wave group velocity, respectively, represents the wave energy, and represents the intrinsic wave frequency. Only considering wave inducing current and ignoring the effect of current on wave, the wave action conservation Eq. (6) becomes The wave force Eq. (3) can be simplified as: Eq. (8) is the same form as in Dingemans et al. (1987),  who explicitly split the wave force into a rotational part and an irrotational part in the case without current refraction. It was pointed out that the irrotational part only generates geostrophic pressure gradients and can be assumed to have a negligible effect on currents.
Some advantages can be found in using the CL-vortex form Eq. (8) to describe the zero-net force outside the surfzone under spatially inhomogeneous wave conditions. Bowen (1969) confirmed that for normally incident waves with alongshore inhomogeneous heights, the alongshore pressure gradient associated with wave setdown is balanced by the alongshore gradient of radiation stresses in the shoaling zone. In a recent discussion of wave setup and setdown generated by obliquely incident waves Shi and Kirby, 2008), Bowen (1969)'s case was extended to more general scenarios with oblique wave incidence. It was verified that, for general obliquely incident waves, the alongshore gradient of pressure is also balanced by the alongshore gradient of radiation stresses defined by Longuet-Higgins andStewart (1962, 1964) outside the surfzone. An easier solution can be obtained in terms of the CL-vortex form because, from Eq. (2), the momentum balance outside the surfzone where can be written as: Eq. (9) indicates that the two gradients are balanced in all directions without a cross-shore or alongshore preference. This is more explicit than the solution in terms of radiation stresses given by Shi and Kirby (2008).
A vorticity equation can be obtained by taking the curl of Eq. (2) ω where represents the vorticity defined by ω = ∇ × u.
(11) According to Eq. (8) A linear bottom friction is adopted as: where is a bottom friction coefficient. Introduce a transport stream function, where Substituting Eqs. (12), (13), and (14) into Eq. (10) yields and is a slope with , that should include the beach slope and the gradient of wave setup (Bowen, 1969).
We make the following coordinate transformation where ( ) presents the image domain and ( ) presents the physical domain.
, and represents alongshore mean breaking location. In comparison with Bowen (1969) and Mei (1992) who made coordinate transformations at the shore line (the mean point of maximum set-up) for rip current generation we made coordinate transformation at the breaker line with the reference point at the mean point of the breaker line.
ξ, η Eqs. (15), (7) and (5) in the new coordinates ( ) can be written as: ∂ ∂ξ A scaling analysis can be carried out by introducing the following scale arguments. We use as the horizontal length scale in both alongshore and across-shore directions, as the vertical length scale. The variation has a scale of , where . Then the nondimensional quantities identified by the circumflexes are where is the breaking index in the assumption that the height of the broken wave remains an approximately constant proportion of the mean water depth The nondimensional forms of Eqs. (17), (18) and (19) can be written as: . We assume that the alongshore variation of the breaker line is small compared with the horizontal length scale, i.e., . In the typical case, we choose , and and then . The perturbations in alongshore distributions of variables are assumed as: represents alongshore uniform variables. Because is a small value, a reasonable approximation for the total water depth is Based on Eq. (20) k C g η and are also -independent according to Eq. (24).

Zero-order solution
Collecting the leading terms of Eqs. (21), (22) and (23) yields a set of the zero-order equations in a dimensional form: The leading order dissipation rate can be obtained using Eqs. (27) and (25) It should be mentioned that, in the derivation of Eq. (29), we assume that following Longuet-Higgins (1970) approximation in the derivation of longshore currents generated by obliquely incident waves. Although a more exact result can be obtained using Taylor expansion to and Snell's law Eq. (28), i.e., where is the phase speed at breaking points, and we neglected in the following derivation for simplicity.
The derivative of with respect to can be written as: By substituting Eq. (31) into Eq. (26) and using Eq.
Eq. (34) is the solution for the alongshore current distribution on a planar beach. This is the same solution that could alternatively be derived by balancing the alongshore gradient of radiation stresses and linear bottom friction.

First-order solution
Notice that the first-order equations can be written as: ∂ψ The perturbation in alongshore gradient of wave breaking position can be expressed as a Fourier series where k represents an integer number where, represents the complex conjugate, is a parameter, and in which is the length of the periodic interval, and could be much larger than the length of the focal zone. The corresponding alongshore variation in wave breaking position can be written as: Similarly the effect of wave angle perturbation through at wave breaking locations can be expressed as: The wave height variation can be represented by ψ 1 For simplification, we neglect the nonlinear advection terms (the first two terms in Eq. (36)). Using Eq. (43) and substituting Eqs. (31) and (37) into Eq. (36) yield a summation form of : where The solution for the non-homogeneous Bessel differential equation of general order Eq. (45) is obtained using a fractional-calculus approach (Lin et al., 2005;Wang et al., 2007). The detailed solution can be found in Appendix B. Inside the surfzone, can be written as: in which the boundary condition at is applied. In Eq. (47), is a constant to be determined. and are constants given in Appendix B. The corresponding alongshore component of the velocity is given by Outside the surfzone, all the forcing terms on the RHS of Eq. (45) are zero. However, the solution outside the surfzone must have the same alongshore variation to guarantee the continuity at the breaker line. The solution with the boundary condition as is given by ψ k and The constants and are determined by the conditions that and must be continuous at the breaker line, and (See Appendix C).

Solution for an individual mode k = ±1
A simple approximation for alongshore variations in wave height and wave angle can be made by adopting a single mode, , for example, in Eqs. (42) and (41). They can be expressed as: δ and (sin θ Here, we assume that the wave angle variation at the breaker line is out of phase with the wave height variation according to wave refraction patterns from numerical simulations in the next section. and in Eqs. (51) and (52)  The solution can be obtained using Eqs. (47) and (49).
Inside the surfzone: and outside the surfzone: (54) Fig. 2 shows a typical solution, in which, , , , , (we denote , thereafter), , and . The solution is obtained by superimposing the leading order solution, i.e., alongshore uniform currents, on the first-order solution, i.e., circulation cells as shown in Fig. 3. The first order solution shown in Fig. 3 can also present a case with a normal wave incidence, i.e., . Fig. 3 shows that the circulation cells are counter-rotating vortices generated shoreward of the wave focal zone with an alongshore-divergent flow pattern inside the surfzone. The contour-lines of the stream function in Fig. 3 indicate that the flows inside the surfzone are more intense compared with the flows outside the surfzone.
As the leading order solution gives zero current velocities outside the surfzone, the nearshore circulations are essentially confined to the surfzone. The alongshore currents are strengthened in the downstream of the wave focal zone and weakened in the upstream by the divergent current component from the first-order solution. Obvious flow reversal is generated in the upstream as shown in Fig. 2.
Outside the surfzone, circulation is weak and is not driven because of the zero-net force as discussed in the theory. The circulation outside the surfzone is basically from the first-order solution in which continuous boundary conditions of and are imposed at the breaker line.
The force terms on the RHS of Eq. (45) imply that the analytical solutions are dependent on the configuration of wave focal patterns which can be represented by alongshore variations in wave height and wave angle. Fig. 4 compares the magnitudes of between and . As expected, increasing the amplitude of alongshore variation in wave height induces an increase in the amplitude of transport stream function. In this comparison, doubling wave height variation results in an increase to 2.15 times the transport stream function. In contrast to the effect of wave height variation, increasing the magnitude of alongshore variation in wave angle causes a decrease in transport stream function as demonstrated in Fig. 5. Fig. 5 shows a comparison between the cases with and . The magnitude of transport steam function decreases by a factor of 0.75 with a doubling . The changes in stream function magnitude, corresponding to variations in wave height and wave angle, suggest that the circulation cells tend to be intensified by increasing the alongshore variation in wave height or decreasing the alongshore variation in wave angle. Consequently, the circulation cells associated with the stream function should become more intense with an in- DING Yu-mei, SHI Fengyan China Ocean Eng., 2019, Vol. 33, No. 5, P. 544-553 549 sin θ 1 δ ′ δ ′ sin θ 1 crease in wave height variation, or a decrease in wave angle variation. The magnitude of the stream function is more sensitive to the wave height variation rather than to the wave angle variation. The theoretical explanations for the two opposite effects can be found in the first-order governing equation (36), where the two major force terms containing and have different signs, representing a divergent force and a convergent force, respectively. The forcing term associated with wave height variation ( ) is divergent while the forcing associated with wave angle variation ( ) is convergent. θ 0 (θ 0 ) b = 30 • As the incident wave angle increases, the alongshore current component from the leading-order solution becomes a dominant component. Fig. 6 shows the results from , in which the flow reversal is not obvious. It is shown that the divergent flow component from the first-order solution basically causes a meandering pattern of alongshore currents.

Remarks
The analytic model derived in this study describes the basic physical concept of nearshore circulations induced by focused or defocused waves on a beach. For given wave conditions at the breaker line, the analytic model can predict alongshore currents influenced by wave focal patterns.
In this study, we proposed an analytic model to examine the wave focusing effects on wave-driven alongshore currents. Theoretical solutions were obtained using transport stream function and perturbations in alongshore distributions of wave height and wave angle on a planar beach. Acceleration terms are omitted and a linear bottom friction is utilized in the analytic model.
The leading-order solution for the problem is the solution for alongshore uniform currents. The first-order solution presents a pair of counter-rotating circulation cells shoreward of the wave focal zone. Inside the surfzone, divergent flows associated with the circulations cells are intense, strongly affecting alongshore current patterns. For a small wave incidence, the divergent flows strengthen alongshore currents in the downstream of the wave focal zone and cause flow reversal in the upstream. For a large wave incidence, the divergent flows basically influence the alongshore uniformity of alongshore currents and generate meandering flow patterns. The effects of wave focusing also depend on the configuration of wave focal patterns offshore. A large alongshore gradient in wave height tends to enhance the effects, and a large convergence of wave angles tends to reduce the effects. The solution of stream function is more responsive to wave height variation rather than wave angle variation at the breaker line.
The analytic model gives a discontinuous distribution of alongshore currents with the maximum at the breaker line because of the adoption of linear bottom friction and neglecting diffusion mechanism in the model equations. The linearization of bottom friction may have effects on an across-shore distribution of alongshore currents as presented in the leading-order solution of Eq.(34). It basically gives alongshore current velocity to be proportional to , and the slope of which seems steeper compared with the solution of Longuet-Higgins (1970), which used a quadratic form of bottom friction and took into account effects of the wave orbital velocities. Neglecting the diffusion mechanism in the analytic model causes the peak of alongshore currents appearing at the breaker line with a discontinuity.
Compared with Bowen (1969)'s solutions for rip currents induced by normally incident waves with alongshore non-uniform wave height, our solutions are based on scaling analysis for general obliquely incident waves with a focal/defocal pattern. Perturbations were made in both wave height and wave angle at the breaker line. Wave refraction was taken into account in the analytic model. 5. Across-shore distributions of the transport stream function with and . sin θ 0 (sin θ 0 ) 2 in which, we assume that is small and can be neglected. The solution of Eq. (A2) can be written as: f 0 and can be obtained using the boundary condition.
Eq. (A5) can be further simplified by taking . For a typical case with , and , the third term containing in Eq. (A2) is small, which can be confirmed by the comparison of the function between and , as shown in Fig. A1. The simplified form of Eq. (A5) can be written as: Substituting Eq. (A7) into Eq. (A1) yields Eq. (43).

Appendix B: Solution for Eq. (45)
Let then Eq. (45) becomes Eq. (B2) is the non-homogeneous Bessel's differential equation of 3/2-order. The regular solution (Struve function solution) for the 3/2-order non-homogeneous Bessel's differential equation requires in the non-homogeneous terms on the RHS. Therefore, the equation may not be solved using the standard Struve function. Recently, Lin et al. (2005) and Wang et al. (2007) used a fractional-calculus approach to solve the Bessel differential equation of general order.
For a non-homogeneous Bessel differential equation in a general form it has a particular solution in the form: ρ β ν where , and are given by and For a given function , represents the general derivative of of order . If is not an integer, is the fractional derivative of of order and is the fractional integral of of order .
function with and without .
The particular solution for Eq. (45) can be written as: , , and . Note that the general solution for the homogeneous Bessel's equation can be written as: where is a constant. The solution subject to the boundary condition at can be written as Eq. (47).

Appendix C: Constants P k and Q k in Eqs. (47) and (49)
The constants and in Eqs. (47) and (49) can be determined by the conditions that and must be continuous at the breaker line . and can be written in compact forms as: where DING Yu-mei, SHI Fengyan China Ocean Eng., 2019, Vol. 33, No. 5, P. 544-553