Longshore Current Profiles and Instabilities on Plane Beaches with Mild Slopes

Abstract

The accurate determination of cross-shore longshore current profiles in the surf zone is essentially important in understanding of coastal physical processes and modelling of longshore sediment transport. In this study, a comprehensive laboratory study was undertaken to directly measure longshore current profiles over plane beaches with two mild slopes, 1:40 and 1:100, in a wave basin 55 m long, 34 m wide, and 0.7 m deep. Different wave conditions with an incidence angle of 30° were generated by piston-type wave makers consisting of 70 individual paddles, and two arrays of 29 Acoustic Doppler Velocimeters (ADVs) were used to measure longshore currents in the surf zone. Based on the experimental data collected in this study, three types of cross-shore longshore current profiles were found on the two plane beaches under different testing wave conditions, namely quasi-Rayleigh, quasi-Gaussian, and M-shape profiles. The quasi-Rayleigh profile was found on the beach slope of 1:40, and the other two types were found on the beach slope of 1:100. Analytical formulae were proposed to describe these profiles and agreed well with the laboratory data. The fluctuations of longshore currents observed in this study were attributed to their shear instabilities based on linear instability analysis. The results of the linear instability analysis and the spectra of measured velocities also showed that the three types of velocity profiles led to different instability characteristics.

1. Introduction

The longshore current is one of the wave-driven currents that appears on beaches with obliquely propagating incident waves and is a constituent of nearshore circulation. Its cross-shore profile is important in determining the cross-shore distribution of longshore sediment transport and studying coastal morphology [1]. Several types of longshore current velocity profiles V(x) have been presented in previous studies. Longuet-Higgins [2] established a theoretical longshore current profile. It has a profile shape similar to the Rayleigh distribution of irregular waves, which is hereinafter referred to as the quasi-Rayleigh profile. This type of velocity profile is found to correspond to the common profile observed on plane beaches with relative steep slopes larger than 1:30 [3,4,5,6]. Bowen and Holman [7] and Falques and Iranzo [8] adopted the formula $V(x)=C_{0}x\exp [-{(x/\alpha )}^n]$, where $C_0$ and $\alpha$ are constant and n = 2~5, to fit this type of profile and used it to analyze the linear instability of longshore currents. Allen et al. [9] proposed an artificial longshore current profile in the form $V(x)=C_0{x}^2\exp [-{(x/\alpha )}^n]$ with n = 3 or n = 6. It has a shape similar to the Gaussian distribution of irregular waves, which is hereinafter referred to as the quasi-Gaussian profile. This profile has been used as the background flow for studying weak and strong nonlinear shear instabilities of longshore currents [9,10]. However, the physical background of this velocity profile and its applicability to actual longshore currents has been rarely discussed to date. Apart from the two profiles discussed above, another common longshore current profile has double peaks and usually occurs on barred beaches [5,11], which is hereinafter referred to as the M-shape profile. The possibility of its presence on a very mild plane beach was reported in a recent study [12]. Therefore, various types of longshore current profiles may appear on plane beaches. However, the types of longshore current profiles on plane beaches have not been systematically classified previously.

Several dynamic factors may affect the shape of longshore current profiles. Longuet-Higgins [13] theoretically revealed that if a constant bottom friction coefficient is assumed and lateral mixing is neglected, the profile is a right triangle with its maximum located at the wave breaking point when a constant ratio of breaking wave height to local depth is applied. Inclusion of lateral mixing leads to transformation of the triangle profile to the quasi-Rayleigh profile. More general velocity profiles were presented by applying related mathematical models considering more controlling factors in an accurate and comprehensive manner [14,15]. The controlling factors considered include shear dispersion, wave breaking dissipation, surface roller, etc. The incorporation of the roller effect can improve the prediction of the longshore current velocity profile by shifting the location of the velocity maximum shoreward [16].

Another factor that may alter the longshore current profile is the instability of longshore currents, as this may result in velocity fluctuations [17] and an extra lateral mixing effect [7,18]. Some related results based on numerical simulations presented that the extra lateral mixing resulted in a smoothing of the initial longshore current velocity profile [9,19]. Reniers [20] found experimentally that the shear instabilities significantly contributed to the momentum flux at the downstream end of the basin, which was comparable to the mixing associated with breaking wave-induced turbulence. Compared with studies exploring the influence of instability on profile shape, few studies have investigated the influence of profile shape on instability characteristics.

The present study aimed to provide a comprehensive classification of possible types of longshore current profiles that may exist on plane beaches. Two mild slopes, 1:100 and 1:40, were adopted in this experimental study, which have rarely been adopted in previous studies and allowed the mild-slope effect to be examined. Three analytical formulae were proposed to describe the observed profiles that agreed well with the laboratory data. Fluctuations in velocity and meandering of the dye path were also observed in the experiments. They were attributed to the shear instabilities of the longshore currents by comparing the results of the linear instability analysis and the energy spectra of the observed velocity time series. In this analysis, the effects of different types of velocity profiles on longshore current instabilities were also examined to show their different instability characteristics.

2. Experimental Setup and Instrumentation

2.1. Wave Basin and Experimental Layout

The experiments were performed in a wave basin of the State Key Laboratory of Coastal and Offshore Engineering at Dalian University of Technology in China. The basin was 55 m long, 34 m wide, and 0.7 m deep. At one end of the basin was the piston-type wave maker consisting of 70 paddles. The concrete plane beach model was rotated at an angle of 30° to the wave maker and had a longshore dimension of 29 m. The shortest distance from the toe of the beach to the wave maker was 8 m. Two wave guides perpendicular to the wave maker, terminated at the toe of the beach, were built on both sides of the horizontal bottom to prevent the waves from diffracting. The detailed experimental layout is shown in Figure 1. Also shown is the coordinate system with the origin at the upstream end of the still water line. The positive x-axis was directed offshore, and the positive y-axis was directed downstream. Two beach models with slopes of 1:40 and 1:100 were constructed to consider the slope effect. The widths of the beaches of the two slopes were kept at the same length of 18 m; therefore, different water depths on the horizontal bottom $h_0$ were adopted at 45 and 18 cm, respectively.

A passive recirculation system was adopted around the beach model, which consisted of channels with the same depth as the horizontal bottom. The channel width was 4.4 m at the two sides of the beach and 4.0–8.0 m behind the beach.

2.2. Experimental Wave Conditions

The incident wave conditions used in this study included regular and irregular waves with periods of T = 1.0, 1.5, and 2.0 s. With the experimental model scale 1:20, these periods corresponded to wave periods of 4.5~8.9 s for the prototype wave condition. The regular waves were monochromatic waves and the JONSWAP spectrum was used to generate irregular waves. The angle of incidence relative to shore normal was 30° at the wave maker.

Table 1. Experimental testing conditions for beach slopes of 1:100 and 1:40 and longshore current profile types.

Case	Name	Wave Type	Slope	T (s)	${\mathit{H}}_{\mathit{i}}\left(\mathbf{cm}\right)$	${\mathit{\xi }}_0$	${\mathit{x}}_{\mathit{b}}$ (m)	Profile Type
1	RMT1H1	Regular	1:100	1.0	2.52	0.076	5.0	Quasi-Gaussian
2	RMT1H2	Regular	1:100	1.0	4.90	0.054	8.5	M-shape
3	RMT1H3	Regular	1:100	1.0	5.95	0.049	12.0	M-shape
4	RMT2H1	Regular	1:100	1.5	2.53	0.119	6.0	Quasi-Gaussian
5	RMT2H2	Regular	1:100	1.5	5.30	0.082	9.5	M-shape
6	RMT2H3	Regular	1:100	1.5	5.53	0.080	12.0	M-shape
7	RMT3H1	Regular	1:100	2.0	3.16	0.150	7.0	Quasi-Gaussian
8	RMT3H2	Regular	1:100	2.0	4.80	0.122	10.0	M-shape
9	RMT3H3	Regular	1:100	2.0	5.62	0.112	10.0	M-shape
10	IMT1H1	Irregular	1:100	1.0	2.56	0.075	8.5	Quasi-Gaussian
11	IMT1H2	Irregular	1:100	1.0	2.72	0.073	9.0	Quasi-Gaussian
12	IMT1H3	Irregular	1:100	1.0	3.71	0.062	9.5	M-shape
13	IMT2H1	Irregular	1:100	1.5	2.56	0.118	7.5	Quasi-Gaussian
14	IMT2H2	Irregular	1:100	1.5	2.85	0.112	8.5	M-shape
15	IMT2H3	Irregular	1:100	1.5	3.57	0.100	12.0	M-shape
16	IMT3H1	Irregular	1:100	2.0	2.44	0.170	8.0	Quasi-Gaussian
17	IMT3H2	Irregular	1:100	2.0	2.96	0.155	10.0	Quasi-Gaussian
18	IMT3H3	Irregular	1:100	2.0	3.63	0.140	12.0	M-shape
19	RST1H1	Regular	1:40	1.0	5.80	0.130	3.0	Quasi-Rayleigh
20	RST1H2	Regular	1:40	1.0	8.60	0.107	5.0	Quasi-Rayleigh
21	RST1H3	Regular	1:40	1.0	10.50	0.096	6.0	Quasi-Rayleigh
22	RST2H1	Regular	1:40	1.5	6.50	0.184	4.0	Quasi-Rayleigh
23	RST2H2	Regular	1:40	1.5	9.20	0.154	5.0	Quasi-Rayleigh
24	RST2H3	Regular	1:40	1.5	11.50	0.138	9.5	Quasi-Rayleigh
25	RST3H1	Regular	1:40	2.0	6.00	0.255	4.5	Quasi-Rayleigh
26	RST3H2	Regular	1:40	2.0	9.50	0.203	7.0	Quasi-Rayleigh
27	RST3H3	Regular	1:40	2.0	10.50	0.193	8.0	Quasi-Rayleigh
28	IST1H1	Irregular	1:40	1.0	4.05	0.155	4.0	Quasi-Rayleigh
29	IST1H2	Irregular	1:40	1.0	5.63	0.132	5.0	Quasi-Rayleigh
30	IST1H3	Irregular	1:40	1.0	6.76	0.120	9.5	Quasi-Rayleigh
31	IST2H1	Irregular	1:40	1.5	4.49	0.221	4.5	Quasi-Rayleigh
32	IST2H2	Irregular	1:40	1.5	6.94	0.178	8.0	Quasi-Rayleigh
33	IST2H3	Irregular	1:40	1.5	8.13	0.164	9.5	Quasi-Rayleigh
34	IST3H1	Irregular	1:40	2.0	3.38	0.340	5.0	Quasi-Rayleigh
35	IST3H2	Irregular	1:40	2.0	5.71	0.262	6.5	Quasi-Rayleigh
36	IST3H3	Irregular	1:40	2.0	7.20	0.233	9.0	Quasi-Rayleigh

shows the wave conditions used in this study, where T and $H_i$ are the wave period and incident wave height for regular waves, respectively, or the peak period and root-mean-square wave height for irregular waves, respectively. The values of Iribarren number ${\xi }_0=\tan \beta /\sqrt{H_0/L_0}$ ($\tan \beta$, $H_0$, and $L_0$ being beach slope, deep water wave height, and deep water wave length, respectively) are listed in Table 1 to indicate breaking wave type. All values were smaller than 0.5, which indicated that the wave breaking types observed on the two beaches were spilling breakers. The locations of the wave breaking point $x_b$ are listed to show the surf zone width. The types of measured longshore current profiles are also given in Table 1. For each wave condition, three runs were conducted to test the repeatability of the experimental results.

2.3. Instrumentation and Setup

The wave free surface elevations were measured with 60 capacitance-type wave gauges mounted on three carriages (Arrays 1, 2, 3) located at the three cross-shore sections (see Figure 2), where 14 gauges were placed at y = 7.0 m, 32 gauges at y = 12.0 m, and 14 gauges at y = 17.0 m. The gauge intervals included 0.5, 1.0, and 2.0 m. The first two were adopted for Array 2 to measure the breaking point and rapid variation of wave height in the surf zone, and the last two were adopted for Arrays 1 and 3 to investigate the alongshore uniformity of wave height. A sampling rate of 50 Hz was set for all 60 wave gauges.

The longshore currents were measured with 29 Acoustic-Doppler Velocimeters (ADVs) mounted on cross-shore and alongshore carriages (see Figure 2). The cross-shore carriage was for the measurement of the longshore current profiles, whereas the alongshore carriage was for the measurement of the alongshore variations in longshore currents. The cross-shore carriage was located at y = 14.5 m and had 18 ADVs with different intervals of 0.5, 1, and 2 m for the beach slope of 1:100 (distributed from x = 1.0 m to x = 12.0 m) and 0.3, 0.4, 0.5, and 1 m for the beach slope of 1:40 (distributed from x = 0.2 m to x = 9.0 m). The smaller intervals (0.3, 0.4, and 0.5 m) were for recording the rapid variation in longshore current profiles in the surf zone. The alongshore carriage was located at x = 4.0 m for slope 1:100 or x = 2.5 m for slope 1:40 and had 12 ADVs placed with a uniform interval of 2 m (from y = 2.5 m to y = 24.5 m). All the ADV sensors were set at 1/3 of the local water depth from the bottom, which was near the location of the depth-averaged velocity. The side-looking sensors were adopted for the shallow water region to measure the velocities near the shoreline. The minimum water depth that could be measured was 1.0 cm. All the ADVs were wirelessly connected to the laboratory computers. The ADV sampling rate was 20 Hz and the sampling duration was 450 s for regular waves and 700 s for irregular waves. Unusual velocity spikes in the recorded time series of velocities were expected to occur when air bubbles from breaking waves penetrated to the depth of the ADV sensors in very shallow water, and they could be removed during post-processing using the despiking routine developed by Islam and Zhu [21].

In order to visualize the motion of the longshore currents, especially the meandering motion, blue dye was continuously released in the surf zone through a long, thin tube 8 mm inner diameter (see Figure 2). The releasing point was at x = 4.0 m and y = 7.0 m for the beach slope of 1:100 and at x = 2.5 m and y = 7.0 m for the beach slope of 1:40. The dye path was recorded at an interval of 1 Hz using a charge coupled device (CCD) system mounted 11.6 m high above the water surface.

3. Results and Analysis

3.1. Longshore Velocity Profile Types and Analysis

Based on the data collected in this study, the measured mean velocity profiles of the longshore currents on beach slopes of 1:40 and 1:100 were found to have different geometrical features. Figure 3 shows that the quasi-Rayleigh profile occurred on the beach slope of 1:40 for all considered regular and irregular waves. The quasi-Rayleigh profile was convex in the shoreward side between the shoreline and maximum velocity of the profile and had one inflection point in the seaward side.

Figure 4 shows that the quasi-Gaussian profile occurred on the beach slope of 1:100 for tests with smaller incident wave heights. For regular waves, they occurred for the smallest wave height among the three wave heights considered. For irregular waves, they occurred for two lower wave heights among the three wave heights for T = 1.0 s and T = 2.0 s and occurred for the smallest wave height for T = 1.5 s. The quasi-Gaussian profile had inflection points in both the shoreward and seaward sides of the profile. The part between the shoreline and shoreward inflection point was concave, and this was the major difference from the quasi-Rayleigh profile.

Figure 5 shows that the M-shape profile occurred on the beach slope of 1:100 for tests in which the quasi-Gaussian profile did not occur, so they occurred for testing conditions of larger waves ($H_i$ ≥ 0.04 m for regular waves, $H_i$ ≥ 0.028 m for irregular waves). Different from the above two types of profiles, the M-shape profile had two peaks. M-shape profiles are usually observed on barred beaches in laboratory experiments [5,11,22], and the two peaks correspond to wave breaking on the sand bar and near the shoreline, respectively. The M-shape profiles on plane beaches are due to the ceasing of wave breaking in the middle of the surf zone [12].

The following analytical formulae were proposed to describe the three types of longshore current profiles:

$$V(x)=C_{1}x\exp [-{(\frac{x}{{\alpha }_1})}^n] \mathrm{for} \mathrm{Quasi}-\mathrm{Rayleigh} \mathrm{profile}$$

(1)

$$V(x)=C_1{x}^3\exp [-{(\frac{x}{{\alpha }_1})}^n] \mathrm{for} \mathrm{Quasi}-\mathrm{Gaussian} \mathrm{profile}$$

(2)

$$V(x)=C_1{x}^3\exp [-{(\frac{x}{{\alpha }_1})}^n]+C_2\exp [-{(\frac{x-x_r}{{\alpha }_2})}^2] \mathrm{for} \mathrm{M}-\mathrm{shape} \mathrm{profile}$$

(3)

where x is the cross-shore coordinate pointing seaward with the origin at the shoreline, and $x_r$ is the cross-shore position of the offshore peak for the M-shape profile. The exponent n is limited to being an integer, and coefficients $C_1$, $C_2$, ${\alpha }_1$ and ${\alpha }_2$ can be determined by fitting the formula to the measured profile using the nonlinear least squares method. Figure 3, Figure 4 and Figure 5 also present the fitted curves using Equations (1)–(3) and list the proper exponent n together with the coefficients.

For the quasi-Rayleigh profiles, the fitting results using Equation (1) showed that n = 4 was suitable for regular wave conditions and n = 2 for irregular wave conditions. The profiles of irregular wave conditions became broader than those of regular wave conditions due to the smoothing effects of irregular waves, so a smaller n was chosen for irregular wave conditions. Equation (1) has been used by Bowen and Holman [6] and Falques and Iranzo [21] for analyzing the linear instability of longshore currents.

For the quasi-Gaussian profiles, the fitting results using Equation (2) showed that n = 5 was suitable for regular wave conditions and n = 3 for irregular wave conditions. Equation (2) is similar to the formula suggested by Allen et al. [9] and Feddersen [10], with the difference of the first power being changed from 2 to 3.

For the M-type profiles, the fitting results using Equation (3) showed that n = 5 was suitable for regular wave conditions and irregular wave conditions. Equation (3) is the new formula introduced in the present study, which has a major difference from the former two types since the profile has two peaks. For the M-shape profiles, the difference between the regular and irregular wave conditions lay in the trough and distance between the two peaks: the two quantities were smaller for irregular waves than for regular waves. The reason was explained by Yan et al. [12] in which the formation mechanism of this profile was discussed.

3.2. Instability Characteristics of Longshore Currents

Figure 6 shows the measured time series of cross-shore and longshore velocities and the photos of dye traces for tests IST1H2, RMT3H1, and RMT3H3. The tree tests corresponded to the quasi-Rayleigh, quasi-Gaussian, and M-shape profiles, respectively. The measured velocities were observed to fluctuate temporally and spatially around the mean longshore current with long periods. The linear instability theory of shear flows [23] was applied to demonstrate that the low-frequency fluctuations of velocities shown in Figure 6 were indeed the shear instabilities of longshore currents and corresponded to the most unstable linear modes of these instabilities. In the analysis, the effects of different types of velocity profiles on the longshore current instabilities were also examined to show their different instability characteristics.

The shallow water equations with the assumptions of a rigid lid, alongshore uniformity, linearized bottom stress, and neglecting turbulent eddy mixing given by ref. [10] are:

$$\frac{\partial }{\partial x}\left[uh\right]+\frac{\partial }{\partial y}\left[vh\right]=0,$$

(4)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-g\frac{\partial \eta }{\partial x}-\frac{\mu u}{h},$$

(5)

$$\frac{\partial u}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-g\frac{\partial \eta }{\partial x}-\mu \frac{v-V}{h},$$

(6)

where x and y are the cross- and alongshore coordinates, respectively, u and v are the cross- and alongshore velocities, respectively, η is the short wave-averaged water surface elevation above the still water level, h is the water depth with respect to the still water level, $\mu$ is the friction coefficient, and g represents gravity. Considering a perturbation velocity ($\tilde{u}$, $\tilde{v}$) about the mean alongshore current V(x), the total velocities are given by $u=\tilde{u}, v=V+\tilde{v}$. Eliminating η between Equations (5) and (6) by cross differentiation and linearizing in the perturbation variables ($\tilde{u}$, $\tilde{v}$) result in a perturbation equation in terms of stream function $\psi$:

$$(\frac{\partial }{\partial t}+V\frac{\partial }{\partial y})(\frac{1}{h}{\nabla }^2\psi -\frac{{\psi }_x{h}_x}{h^2})-{\psi }_y{(\frac{V_x}{h})}_x=-{(\frac{\mu }{h})}_x\frac{{\psi }_x}{h}-\frac{\mu }{h^2}[{\nabla }^2\psi -\frac{h_x{\psi }_x}{h}],$$

(7)

where the stream function $\psi$ is defined by $h\tilde{u}=-{\psi }_y$, $h\tilde{v}={\psi }_x$, h is the still water depth, and $\mu$ is the coefficient of the linearized bottom stress. Assuming a solution to Equation (7) in the form of $\psi =\varphi (x)exp[ik(y-ct)]$ with cyclic real wave number k, celerity c and amplitude $\varphi (x)$ yields the governing equation for $\varphi (x)$:

$$(V-c-i\frac{\mu }{kh})({\varphi }_{xx}-\frac{h_x}{h}{\varphi }_x-k^2\varphi )-h{(\frac{V_x}{h})}_x\varphi -\frac{i}{k}{(\frac{\mu }{h})}_x{\varphi }_x=0,$$

(8)

where c and $\varphi$ may be complex, the subscript stands for the partial differentiation. The last term considers the spatial variation of $\mu$, as the depth dependent $\mu =(2/\pi )c_d{U}_0$ is adopted in this study. $c_d$ is bottom friction, and $U_0$ is the amplitude of wave horizontal orbital velocity ($U_0=H\sqrt{g/h}/2$ for regular waves and $U_0=H_{rms}\sqrt{g\pi /h}/4$ for irregular waves with $H_{rms}$ being the root-mean-square wave height). This equation is numerically solved by a second-order finite difference scheme and the following eigenvalue problem at each alongshore wave number k is reached:

$$[A]\left\{\varphi \right\}=c[B]\left\{\varphi \right\},$$

(9)

where A and B are N by N matrices and $\left\{\varphi \right\}$ is a vector of size N (N is the number of grid points). The expressions for [A] and [B] in Equation (9) are given in Appendix A. The corresponding boundary conditions are $\varphi (k,x)=0$ at the shoreline boundary x = 0 and offshore boundary x = x_sea. N = 221 grid points are adopted when x_sea = 22 m is used for the wave conditions considered. At each k (k is discretized between 0.01~2.0 m⁻¹ with an interval 0.01 m⁻¹), eigenvalue c and eigenvector $\varphi$ of the most unstable mode corresponding to the largest Im(kc) (>0) are selected.

Figure 7 presents the non-dimensional velocity gradients of the three profiles corresponding to the tests given in Figure 6 and the corresponding growth rate ${\omega }_i=k{c}_i$ versus alongshore wave number obtained by the linear instability analysis. The longshore current velocity was non-dimensionalized with the maximum velocity V_m and the x location was non-dimensionalized with the wave breaking location x_b. The velocity gradients in Figure 7 showed that there were different numbers of inflection points (corresponding to local extrema of V_x) for different profiles. Because the existence of inflection points is a necessary condition (Rayleigh condition) for the current to be unstable, the instability feature may differ for different types of velocity profiles [24]. Based on the results of the linear instability analysis, the effects of the three profile types on the characteristics of the longshore current instabilities could be examined. For the quasi-Rayleigh profile, there was only one inflection point (the minimum of V_x), which appeared in the backshear region of the profile. Correspondingly, there was only one fastest growing mode with a maximum growth rate at $k{x}_b$ = 1.45. For the quasi-Gaussian profile, there were two inflection points (the maximum and minimum of V_x), which appeared in the backshear and frontshear regions of the profile. Correspondingly, the growth rate curve had two peaks at $k{x}_b$ = 4.8 (Mode I) and 2.6 (Mode II). Without the damping effects of bottom friction, the growth rate of Mode I was larger than that of Mode II. However, with this damping effect, Mode II disappeared and Mode I became dominant. For the M-shape velocity profile, there were four inflection points (two maxima and two minima of V_x). Two appeared in the frontshear and backshear regions of the profile, and the other two appeared in the trough region between the two peaks. The growth rate curve also had two peaks, which were at $k{x}_b$ = 6.3 (Mode I) and 2.9 (Mode II), respectively. The growth rates of the two modes had similar values without the damping effects of bottom friction. With this effect considered, the growth rate of Mode II was a little larger than that of Mode I.

In order to compare the instability analysis results with those from the measurement data, the latter were obtained from the energy spectra of the observed velocity time series in Figure 6, calculated using the maximum entropy method [25]. Figure 8 shows the results for three runs of each wave condition. Although the three spectra were not identical, the major peaks were close. The representative frequencies of the velocity fluctuations could be determined, which corresponded to the peaks of the curve. The frequencies of the most unstable linear modes, which are indicated by dashed vertical lines, were also drawn in the figure for comparison. The results indicate that the major spectral peaks are close to the dashed lines, confirming that the long-period fluctuations in the velocity time series had frequencies in accord with those given by the linear instability analysis.

Table 2. The most unstable mode parameters predicted by linear instability theory and those estimated from the measurements.

Test	Theory								Experiment		Matched Mode
	Mode I				Mode II
	k1 (rad/m)	$$\begin{gathered} {\mathit{\omega }}_{\mathit{i}1} \\ (\mathbf{rad}/\mathbf{s}) \end{gathered}$$	$$\begin{gathered} {\mathit{\lambda }}_1 \\ \left(\mathbf{m}\right) \end{gathered}$$	T1 (s)	k1 (rad/m)	$$\begin{gathered} {\mathit{\omega }}_{\mathit{i}2} \\ (\mathbf{rad}/\mathbf{s}) \end{gathered}$$	$$\begin{gathered} {\mathit{\lambda }}_2 \\ \left(\mathbf{m}\right) \end{gathered}$$	T2 (s)	T1 (s)	T2 (s)
IST1H2	0.29	0.0044	21.7	164.8					163.9		I
RMT3H1	0.68	0.0089	9.2	121.5	0.37	0.0059	17.0	191.2	126.1		I
RMT3H3	0.63	0.0048	10.0	99.4	0.29	0.0052	21.7	273.1	102.4	234.2	I, II

Note: k = cyclic real wave number; ${\omega }_i$ = the growth rate; $\lambda$ = the length of shear waves; T = the period of shear waves.

lists the measured periods determined from the peaks in Figure 8 and those given by the most unstable modes in the linear instability analysis. The results showed that for tests IST1H2 and RMT3H1, there was only one theoretical mode (Mode I) that matched the peak appearing in the energy spectra of Figure 8a,b. For test RMT3H3, two theoretical modes both matched the two spectral peaks in Figure 8c. These results showed that the long-period fluctuations in measured velocity were caused by shear instabilities and different types of velocity profiles led to different features of the shear instabilities. These three tests reflected the representative instability characteristics of different types of profiles, although there may be some minor differences between regular and irregular wave conditions.

4. Discussion

4.1. Experimental Results on Beaches with Steeper Slopes

The other two previous experiments of longshore current profiles on plane beaches with steep slopes were carried on a beach slope of 1:20 by Visser [4] and on a beach slope of 1:30 by Hamilton and Ebersole [6]. The wave conditions are briefly given in

Table 3. Wave conditions for Visser [4] and Hamilton and Ebersole [6].

Experiments	Wave Types	Slope	T (s)	d (cm)	${\mathit{\theta }}_{\mathit{i}} (°)$	Hi (cm)
Visser Case 4	Regular	1:20	1.02	35.0	15.4	7.8
Visser Case 5	Regular	1:20	1.85	34.8	15.4	7.1
Visser Case 6	Regular	1:20	0.7	35.0	15.4	5.9
H&E Test 6N	Regular	1:30	2.5	66.7	10.0	0.182
H&E Test 8E	Irregular	1:30	2.5	66.7	10.0	0.225

. Their results are also presented here to show the longshore current profiles on steeper beach slopes compared to those of this study and to check the applicability of the analytical formulae for the velocity profiles. Visser’s results were all for regular waves with the same incident angle and different wave heights and periods, while Hamilton and Ebersole’s results included one regular wave case and one irregular wave case.

Figure 9 and Figure 10 show the mean longshore current velocities measured on beach slopes of 1:20 and 1:30, respectively, and their profiles were all Quasi-Rayleigh profiles. Fitting the curves using Equation (1) with n = 2 was suitable for the regular wave cases on the beach slope of 1:20. For the slope of 1:30, n = 3 and n = 2 were suitable for regular and irregular wave cases, respectively.

4.2. Nondimensional Forms of Longshore Current Profiles

It can be seen from Figure 3 that the values of fitting coefficients $C_1$ and ${\alpha }_1$ differed for different wave conditions. In fact, they are related to the maximum of profile V_m and its corresponding location x_m, ${\alpha }_1=x_m{n}^{1/n}$ and $C_1=V_m{e}^{1/n}/x_m$, as Equation (1) has its maximum $V_m=C_1{\alpha }_1/{(ne)}^{1/n}$ at location $x_m={\alpha }_1/n^{1/n}$. Therefore, the nondimensional form of Equation (1) can be expressed as:

$$\frac{V}{V_m}=\frac{x}{x_m}\exp [\frac{1}{n}(1-{(\frac{x}{x_m})}^n)].$$

(10)

Figure 11 shows the nondimensionalized mean longshore current profiles on the beach slope of 1:40 given in Figure 3. Figure 11a shows the regular wave cases, and Figure 11b shows the irregular wave cases. Equation (10) was shown to have good agreement with the experimental data. From these nondimensional experimental results, it was found that the shape (indicated by n) of the Quasi-Rayleigh profile on plane beaches with slopes larger than 1:40 was almost independent of wave height and period, but dependent on the wave type. The present experiment only considered a smooth concrete beach and fixed incident wave angle; however, the n value may be affected by bottom roughness and the wave incident angle, and their effects need to be further investigated.

Similarly, for the quasi-Gaussian profile, the nondimensional form of Equation (2) can be expressed as:

$$\frac{V}{V_m}={(\frac{x}{x_m})}^3\exp [\frac{3}{n}(1-{(\frac{x}{x_m})}^n)].$$

(11)

Figure 12 shows the nondimensionalized mean longshore current profiles on the beach slope of 1:100 given in Figure 4. Figure 12a shows the regular wave cases, and Figure 12b shows the irregular wave cases. Equation (11) was also shown to have good agreement with the experimental data. The nondimensional expression given by Allen et al. [9] is also given in the figure:

$$\frac{V}{V_m}={(\frac{x}{x_m})}^2\exp [\frac{2}{n}(1-{(\frac{x}{x_m})}^n)], n=3, 6,$$

(12)

which cannot describe the shoreward side of velocity profile very well.

Figure 13 shows that as the incident wave height became larger on the beach slope of 1:100, local cessation of wave breaking occurred in the middle of the surf zone. This led to the formation of the M-shape profile. Because of the complexity of wave breaking and wave recovery processes, the coefficients in Equation (3) were not easily expressed by the values and locations of the M-shape profile’s two peaks.

For beaches with arbitrary bathymetry, there are usually sand bars and multiple wave breakings will occur on sand bars and near the shoreline [26]. Therefore, the longshore current profiles on barred beaches are similar to the M-type profile, but they may have several peaks.

5. Conclusions

This comprehensive laboratory study was undertaken to directly measure longshore current profiles on two plane beaches with mild slopes of 1:40 and 1:100 in a modern wave basin equipped with advanced piston-type wave makers consisting of 70 individual paddles. Both regular and irregular waves were measured using 60 capacitance-type wave gauges, and 29 ADVs were used to measure the longshore currents. Based on the collected laboratory data, three different types of cross-shore profiles of mean longshore current velocities were found on the two plane beaches, namely quasi-Rayleigh, quasi-Gaussian, and M-shape profiles. The quasi-Rayleigh profile occurred on the beach slope of 1:40 for all considered regular and irregular wave cases. The quasi-Gaussian profile occurred on the beach slope of 1:100 for tests with smaller incident wave heights. The M-shape profile occurred on the beach slope of 1:100 for tests with larger wave heights ($H_i$ ≥ 0.04 m for regular waves, $H_i$ ≥ 0.028 m for irregular waves) in which local cessation of wave breaking occurred in the middle surf zone. Three analytical formulae were also proposed to describe the three types of longshore profiles, which can be used for calculating longshore sediment transport and analyzing the shear instabilities of longshore currents.

Longshore current fluctuations observed in the experiment were found to be due to the shear instabilities of longshore currents. Linear instability analysis of longshore currents of three tests was conducted, showing that the three types of velocity profiles can lead to different features of longshore current instabilities. The quasi-Rayleigh profile had only one dominant mode due to the backshear of its profile. The growth rate curves of the quasi-Gaussian profile had two extrema due to the frontshear and backshear of its profile. With the damping effect of bottom friction, only the fastest growing mode became dominant for test RMT3H1. The M-type profile of test RMT3H3 also had two extrema in the growth rate curves and similar growth rates. With the damping effect of bottom friction, both of these two dominant modes can still exist.

In future research, the formation mechanisms of different longshore current profile types on plane beaches will be studied using wave-averaged and wave-resolving numerical models. The applicability of the existing expressions of bottom friction, surface rollers, shear dispersion, and wave breaking index will also be verified using the present experimental results on beaches with mild slopes.