2.1. Nonlinear shallow water equations

As waves approach the shoreline and travel into shallower water, the wavefront steepens and the water surface behind the front flattens as the wave amplitude grows. Consequently, the length scales associated with horizontal variations of the water surface and velocity become much greater than the water depth, leading to decreased importance of frequency dispersion. Meanwhile, the growth of wave amplitude increases the importance of wave nonlinearity. Therefore, the conservation of mass and momentum are governed by the NSWEs. The two-dimensional NSWEs are

(2.1a) \begin{align} \frac {\partial h}{\partial t} + \frac {\partial }{\partial x}(hu) + \frac {\partial }{\partial y}(hv) &=0, \end{align}

(2.1b) \begin{align} \frac {\partial u}{\partial t} + u \frac {\partial u}{\partial x}+ v \frac {\partial u}{\partial y}+ \frac {\partial \eta }{\partial x} &= 0, \end{align}

(2.1c) \begin{align} \frac {\partial v}{\partial t} + u \frac {\partial v}{\partial x}+ v \frac {\partial v}{\partial y}+ \frac {\partial \eta }{\partial y} &= 0, \\[9pt] \nonumber \end{align}

where $(x,y)$ are cross-shore and alongshore coordinates, respectively, $t$ is time, $h$ is total water depth, $\eta$ is free surface displacement measured from still water level and $(u,v)$ are depth-averaged velocities in the $(x,y)$ directions (see definition sketch in figure 1). The origin of the coordinate system is located at the still water line (SWL) and $h_1=h-\eta$ is the undisturbed water depth without the bore. For a constant-sloped, non-deformable beach, $h_1 = -x$ and the offshore boundary of the domain is taken to be $x=-1$ . All the quantities have been made dimensionless (Carrier & Greenspan Reference Carrier and Greenspan1958) as

(2.2) \begin{align} t&=\frac {t^*}{t_0^*},\quad (x,y) =\frac {(x^*, y^*)}{l_0^*},\quad (u, v)=\frac {(u^*, v^*)}{u_0^*},\quad h=\frac {h^*}{h_0^*},\quad \text{where} \,\, \nonumber\\ l_0^* &= \frac {h_0^*}{s}, \quad u_0^*=\sqrt {g^*h_0^*},\quad t_0^*=\frac {l_0^*}{u_0^*}, \end{align}

where the asterisk ( $^*$ ) indicates a dimensional variable, $s$ is the beach slope, $h_0^*$ is the undisturbed water depth at the offshore boundary of the domain and $g^*$ is the gravitational acceleration.

Figure 1. Definition sketch for an obliquely approaching bore: (a) side view; (b) top view.

To solve for the flow due to an obliquely approaching bore, we first consider the magnitude of the approach angle, $\theta$ (see figure 1 b), and make the assumption that it is small. This is not a particularly restrictive assumption since even a 30 $^\circ$ approach angle, which would be considered highly oblique, has a value of $\theta \approx 0.5$ rad for which $ \theta \approx \sin \theta \approx \tan \theta$ and $\cos \theta \approx 1$ are reasonable approximations. Using the assumption of small $\theta$ , we aim to simplify the NSWEs (2.1) to derive a weakly two-dimensional set of governing equations. We do this by first considering Snell’s law, which when applied to the surf and swash zones, implies that the parameter

(2.3) \begin{equation} \varepsilon = \frac {\sin {\theta _0}}{U_{b0}} \end{equation}

remains constant. Here, $\theta _0$ is the approach angle and $U_{b0}$ is the dimensionless bore speed, with the subscript $0$ denoting that both are specified at the offshore boundary of the domain. We refer to $\varepsilon$ as the obliqueness parameter since it quantifies the bore obliqueness and it is a key dimensionless parameter in the theory that follows. Since $\theta _0$ is assumed to be small and $U_{b0}$ is $O(1)$ to leading order, $\varepsilon$ is small.

Following Ryrie (Reference Ryrie1983), we define a pseudotime, $\tau$ , as

(2.4) \begin{equation} \tau (t,y) = t - \varepsilon y , \end{equation}

which captures how the flow solution translates along the $y$ direction. In other words, the wave passes the same cross-shore position, $x$ , at the same pseudotime, $\tau$ , regardless of the alongshore position, $y$ . Equation (2.4) implies that the solution’s alongshore dependence is only determined through the composite variable $\tau$ . This effectively makes the problem one-dimensional, since the full two-dimensional solution in $(x,y,t)$ can be obtained in the one-dimensional coordinates $(x,\tau )$ .

Inserting the pseudotime definition (2.4) into the governing (2.1) and using the assumption of a small obliqueness parameter (i.e. small $\theta$ ), we obtain

(2.5a) \begin{align} \frac {\partial h}{\partial \tau } + \frac {\partial }{\partial x}(hu) &=0, \end{align}

(2.5b) \begin{align} \frac {\partial u}{\partial \tau } + u \frac {\partial u}{\partial x}+ \frac {\partial h}{\partial x} &= -1, \end{align}

(2.5c) \begin{align} \frac {\partial v}{\partial \tau } + u \frac {\partial v}{\partial x} -\varepsilon \frac {\partial h}{\partial \tau } &= 0 , \end{align}

which we call the weakly two-dimensional NSWEs. Here, we have chosen the cross-shore velocity to be $u = O(1)$ and the alongshore velocity to be $v = O(\varepsilon )$ in accordance with the small- $\theta$ assumption. Then, (2.5a –2.5b ) are $O(1)$ and we have dropped terms of $O(\varepsilon ^2)$ , and (2.5c ) is $O(\varepsilon )$ and we have dropped terms of $O(\varepsilon ^3)$ . Physically, the assumption of small obliqueness parameter implies that spatial gradients in the alongshore direction are weaker than those in the cross-shore direction. As first noted by Ryrie (Reference Ryrie1983), the advantage of the weakly two-dimensional NSWEs is a one-way coupling from cross-shore dynamics (2.5a –2.5b ) to the alongshore dynamics (2.5c ) allowing solutions to the cross-shore problem to be used to find solutions to the alongshore flow. The weakly two-dimensional NSWEs (2.5) can be also written in characteristic form as

(2.6a) \begin{align} \frac {{\rm d}\alpha }{{\rm d}\tau } &=0 \; \text{ along the curve } \ \frac {{\rm d}x}{{\rm d}\tau }=u+c, \quad \text{where } \alpha =u+2c+\tau , \end{align}

(2.6b) \begin{align} \frac {{\rm d}\beta }{{\rm d}\tau } &=0 \; \text{ along the curve } \ \frac {{\rm d}x}{{\rm d}\tau }=u-c, \quad \text{where } \beta =u-2c+\tau , \end{align}

(2.6c) \begin{align} \frac {{\rm d}\gamma }{{\rm d}\tau } &=0 \; \text{ along the curve } \ \frac {{\rm d}x}{{\rm d}\tau }=u, \quad \text{where } \gamma =\frac {v}{\varepsilon }-h-x-\frac {1}{2}u^2, \end{align}

where $c=\sqrt {h}$ is the local wave speed. Note that these characteristics travel only along the $x$ -axis due to the weakly two-dimensional nature of the equations.

2.2. Review of one-dimensional flow solutions

In anticipation of the two-dimensional solution in § 2.3, we first review the cross-shore solutions of a normally incident bore (figure 1 a) due to Ho & Meyer (Reference Ho and Meyer1962), Shen & Meyer (Reference Shen and Meyer1963), Peregrine & Williams (Reference Peregrine and Williams2001) and Antuono (Reference Antuono2010).

In figure 1(a), the bore travels with a speed $U_b$ and the water velocity behind and in front of the bore are $u_2$ and $u_1$ , respectively. The mass and momentum conservation across the bore follows Rankine–Hugoniot conditions (Stoker Reference Stoker1957)

(2.7a) \begin{align} U_b[\![ h]\!] &=[\![ uh ]\!], \end{align}

(2.7b) \begin{align} U_b [\![ uh ]\!] &= \bigg [\!\!\bigg [ \frac {h^2}{2}+u^2h\bigg ]\!\! \bigg ], \end{align}

where $[\![ f ]\!]=f_2-f_1$ indicates the jump of the quantity $f$ across the bore.

Following Antuono (Reference Antuono2010), we rewrite the jump conditions as

(2.8a) \begin{align} U_b = u_1 + \sqrt {\frac {1}{2}\left (\frac {h_2^2}{h_1}+h_2\right )}, \end{align}

(2.8b) \begin{align} (u_2 - u_1) = (h_2-h_1)\sqrt {\frac {1}{2}\left (\frac {1}{h_2}+\frac {1}{h_1}\right )}. \end{align}

By writing $u_1$ and $u_2$ in (2.8b ) in terms of their respective forward characteristic variables, $\alpha _i=u_i+2c_i+\tau$ (where $i=1,2$ denote the flow data either side of the bore), we can write the jump in depth across the bore as the polynomial equation

(2.9) \begin{equation} z^6-9c_1z^4+8\sqrt {c_1}(\alpha _2-\alpha _1+2c_1)z^3-\left [ 2(\alpha _2-\alpha _1+2c_1)^2+c_1^2\right ]z^2+c_1^3 = 0, \end{equation}

where $z=c_2/\sqrt {c_1}$ . Since the bore is assumed to propagate into quiescent water on a beach of constant slope, we have $u_1=0$ and $h_1=-x_b$ , where $x_b \lt 0$ is the bore position, and hence $\alpha _1$ is known. To close the problem, Antuono (Reference Antuono2010) assumed $\alpha _2$ to be a known constant (see also Antuono & Brocchini (Reference Antuono and Brocchini2007) for a discussion of the required offshore boundary conditions in terms of the characteristic variable $\alpha$ ).

With that, we have a closed system with three unknowns $h_2, u_2, U_b$ and three equations. We first solve (2.9) for $h_2$ , then obtain $u_2$ from the definition of the forward moving characteristic variable (2.6a ), and find the bore speed $U_b$ via (2.8a ). Then, the bore position $x_b$ can be obtained by numerically integrating the bore speed

(2.10) \begin{equation} \frac {{\rm d}x_b}{{\rm d}\tau } = U_b(h_1,u_1,\tau ), \quad x_b(0) = -1, \end{equation}

where $\tau =t$ for the one-dimensional (cross-shore) problem.

The solution shows the well-known phenomenon of bore collapse: as the bore approaches the shoreline ( $h_1 \xrightarrow {} 0$ ), the water depth jump across the bore vanishes and the flow velocities $u_2$ and $U_b$ approach the same limit $U_s$ , which can then be interpreted as the initial shoreline velocity of the swash flow (Whitham Reference Whitham1958; Keller, Levine & Whitham Reference Keller, Levine and Whitham1960). This shoreline is defined as the moving point where $h=0$ . After bore collapse (i.e. after the bore cross the SWL at $x=0$ ), the Rankine–Hugoniot conditions are no longer applicable and the bore transitions into a moving shoreline in the ensuing swash. The velocity of this shoreline motion is given by $u_s = \alpha _2-\tau$ , which is known thanks to the constant $\alpha _2$ assumption, and it can be integrated to find the shoreline position.

So far, we have calculated the bore solution: the bore trajectory before bore collapse and the shoreline motion after bore collapse. To calculate the full flow field, we note that the values of $u_2$ and $c_2$ are known from the bore solution at any bore position $x_b$ , and hence we also have the resulting $\beta _2$ that is carried by the backward moving characteristics in the offshore direction. Similarly, in the swash, we have the flow variables on the moving shoreline, $u_s = \alpha _2-\tau$ and $h_s = 0$ , and hence the resulting $\beta _s$ . On backward moving characteristics that emanate from the bore (or the shoreline) at a point $(x_b,\tau _b)$ (or $(x_s,\tau _s)$ ) with characteristic variable valued $\beta _2$ (or $\beta _s$ ), the following equation holds at a general point $(x, \tau )$ :

(2.11) \begin{equation} x = x_b + \left (\frac {\alpha _2+3\beta _2}{4}\right )(\tau -\tau _b)-\frac {\tau ^2}{2}+\frac {\tau _b^2}{2}, \end{equation}

which allows for calculation of the value of $\beta (x,\tau )$ throughout the domain. With both $\alpha$ and $\beta$ values known, we can calculate the full flow field as

(2.12) \begin{equation} u(x,\tau ) = \frac {\alpha _2+\beta (x,\tau )}{2}-\tau , \quad c(x,\tau ) = \frac {\alpha _2-\beta (x,\tau )}{4}, \end{equation}

using again the constant- $\alpha$ assumption to close the system so that $\alpha = \alpha _2$ throughout the flow behind the bore (or the shoreline).

Although the problem is now fully solved, it is useful to understand what a constant $\alpha (-1,\tau )=\alpha _2$ implies about the offshore boundary condition in physical terms. First, since the forward moving characteristics travel faster than the flow, applying constant- $\alpha$ at the offshore boundary is sufficient to guarantee the flow field maintains a constant $\alpha$ value throughout the domain behind the bore, as required. Second, from the definition of $\alpha$ , we see that quiescent conditions at the offshore boundary at $t=0$ give $\alpha (-1,0)= 2$ , and thus one measure of the strength of the incoming bore that enters the domain at $t=0$ is given by $(\alpha _2 - 2)$ . Therefore, we refer to $(\alpha _2-2)$ as the bore strength. Finally, the implications of the constant $\alpha _2$ on the flow variables at the offshore boundary can be understood by writing $u(-1,\tau )=u^I(-1,\tau )+u^R(-1,\tau )$ and $c(-1,\tau )=1+c^I(-1,\tau )+c^R(-1,\tau )$ , where the superscript $I$ refers to the incoming subcritical flow and the superscript $R$ refers to the reflected (or outgoing) flow (Antuono Reference Antuono2010). We see that these terms are given by

(2.13a) \begin{align} u^I(-1,\tau )=\frac {\alpha _2-2-\tau }{2}, &\quad c^I(-1,\tau )=\frac {\alpha _2-2-\tau }{4}\quad \text{ and } \end{align}

(2.13b) \begin{align} u^R(-1,\tau )=\frac {\beta (-1,\tau )+2-\tau }{2}, &\quad c^R(-1,\tau )=-\frac {\beta (-1,\tau )+2-\tau }{4}. \end{align}

Thus, at the offshore boundary, both the velocity and the square root of the depth decrease linearly in time for the incoming bore. As we show in § 3, this resembles the physical situation for a saw-tooth shaped bore approaching a coast.

Aside from the above constant- $\alpha$ solution due to Antuono (Reference Antuono2010), there are the well-known results by Ho & Meyer (Reference Ho and Meyer1962) and Shen & Meyer (Reference Shen and Meyer1963), who found that the shoreline motion in the swash is only dependent on the initial shoreline velocity and behaves like a particle moving freely under gravity. Through asymptotic analysis of the singularity of the characteristics as the bore approaches the shoreline, they found the shoreline motion to be described by

(2.14a) \begin{align} x_s &= U_s\tau _s-\frac {1}{2}\tau _s^2, \end{align}

(2.14b) \begin{align} u_s &= U_s-\tau _s, \end{align}

where $x_s$ is the shoreline position, $u_s$ is the shoreline velocity and $U_s$ is the initial shoreline velocity as noted earlier. Here, we have defined a new time coordinate for the swash zone $\tau _s=\tau -\tau _c$ for convenience, where $\tau _c$ indicates the time of bore collapse.

The analytical solutions for the flow velocity and the free surface displacement behind the moving shoreline are (Ho & Meyer Reference Ho and Meyer1962; Shen & Meyer Reference Shen and Meyer1963; Peregrine & Williams Reference Peregrine and Williams2001)

(2.15a) \begin{align} h &= \frac {1}{9} \left (U_s-\frac {1}{2}\tau _s-\frac {x}{\tau _s} \right )^2, \end{align}

(2.15b) \begin{align} u &= \frac {1}{3}\left ( U_s-2\tau _s+2\frac {x}{\tau _s}\right ), \end{align}

where $x\gt 0$ is the cross-shore position above the still-water line. Note that these solutions are closely linked to a dam-break flow and also have a constant $\alpha$ behind the moving shoreline. Even though (2.15) is an asymptotic solution near the shoreline ( $(x_s - x) \ll 1$ ), it has been shown to predict laboratory data of swash flow very well for a large portion of the swash cycle (Pujara et al. Reference Pujara, Liu and Yeh2015b ).

To see the link between the asymptotic solution due to Ho & Meyer (Reference Ho and Meyer1962) and Shen & Meyer (Reference Shen and Meyer1963), and the constant- $\alpha$ solution due to Antuono (Reference Antuono2010), note that the initial shoreline velocity in the constant- $\alpha$ solution is $U_s=\alpha _2-\tau _c$ , which is a known constant from the bore solution. Therefore, $U_s$ is directly linked to the constant $\alpha _2$ and $u+2c+\tau _s=U_s$ remains invariant in the swash zone. Physically, since $h=0$ at the shoreline, all the information about the incoming flow before bore collapse is collected into $U_s$ , which becomes the only parameter needed to describe the swash flow. Moreover, since $U_s$ is directly dependent on $\alpha _2$ in the solution of Antuono (Reference Antuono2010), the swash flow can also be fully described by the constant $\alpha _2$ value in that scenario. We discuss the relationship between $U_s$ and $\alpha _2$ in further detail in § 2.6.

2.3. Weakly two-dimensional flow solution

To extend the cross-shore solution in § 2.2 to obliquely approaching bores, we consider a bore approaching the coast at an angle $\theta$ and with speed $U_b$ (figure 1 b). The flow velocity magnitude behind the bore is denoted as $u_b$ with components $u_2$ in the cross-shore direction and $v_2$ in the alongshore direction. The Rankine–Hugoniot condition for alongshore momentum conservation (see also (13c) of Ryrie (Reference Ryrie1983)) derived from (2.5c ) is

(2.16) \begin{equation} U_b \bigg [\!\!\bigg [ hv - \frac {1}{2}\varepsilon h^2\bigg ]\!\!\bigg ] = \left [\![ huv \right ]\!]. \end{equation}

A straightforward algebraic manipulation, combined with the cross-shore momentum Rankine–Hugoniot condition (2.7b ) under the condition $u_1 = v_1 = 0$ , allows us to write the alongshore velocity behind the bore as

(2.17) \begin{equation} v_2 = \varepsilon u_2 U_b. \end{equation}

Combined with the Snell’s law (2.3), this implies that the flow velocity behind the bore is perpendicular to the bore, where $u_2=u_b \cos \theta$ and $v_2 = u_b \sin \theta$ , and the forward moving characteristics in the $x$ direction behind the bore have

(2.18) \begin{equation} \alpha _2 = u_b\cos \theta + 2c_2 + \tau . \end{equation}

By writing $\tau$ in terms of $\alpha _1$ through (2.6a ), we get

(2.19) \begin{equation} u_b = \sec \theta (\alpha _2-\alpha _1 + 2(c_1-c_2)). \end{equation}

Following a similar procedure to § 2.2, we can obtain a polynomial equation for the jump in depth across the bore using (2.8b ), but with $u_2$ replaced with $u_b$ from (2.19) to account for the oblique approach angle. This gives

(2.20) \begin{equation} z^6\!-\!c_1z^4\!-\!c_1^2z^2\!+\!c_1^3 = 2\sec ^2\theta \left [ 4c_1z^4-4\sqrt {c_1}(\alpha _2-\alpha _1 \!+\! 2c_1)z^3 \!+\! (\alpha _2-\alpha _1 \!+\! 2c_1)^2z^2\right ], \end{equation}

where $z = c_2/\sqrt {c_1}$ as before. Using Snell’s law ( $\sin \theta = \varepsilon U_b$ ) and (2.8a ) with $u_1=0$ to write

(2.21) \begin{equation} \sec ^2\theta = \frac {1}{1-\frac {\varepsilon ^2}{2}(z^4+c_1z^2)}, \end{equation}

and inserting into (2.20) gives the final polynomial equation for the jump in depth across the bore. This, combined with (2.8a ) and

(2.22) \begin{equation} \frac {{\rm d}x_b}{{\rm d}\tau } = U_b(h_1,u_1,\tau ) \sec \theta , \quad x_b(0) = -1, \end{equation}

gives a closed system of equations for the obliquely approaching bore.

Thus far, the obliquely approaching bore problem has been treated as a fully two-dimensional problem. We can simplify the problem by assuming a small obliqueness parameter (small $\varepsilon$ ; see § 2.1) to derive the equations for the weakly two-dimensional case. Using Taylor series expansions of the trigonometric functions and dropping terms of $O(\varepsilon ^2)$ or higher, we find that the cross-shore bore dynamics are identical to the one-dimensional case. Therefore, for the weakly two-dimensional case, there is a one-way coupling from the solution for the purely cross-shore bore dynamics (§ 2.2) to the alongshore flow.

To obtain the alongshore flow at the shoreline after bore collapse, we observe that the shoreline is a moving free surface subject to the kinematic free surface boundary condition, which requires that fluid particles on the free surface remain on it. Accordingly, the normal component of the fluid velocity at the shoreline must match the rate of change of the shoreline position (Dean & Dalrymple Reference Dean and Dalrymple1991). This gives

(2.23) \begin{equation} u_s = \frac {\partial x_s}{\partial t} + v_s\frac {\partial x_s}{\partial y} \approx \frac {\partial x_s}{\partial \tau }. \end{equation}

Here, $u_s$ and $v_s$ are the cross-shore and alongshore components of the shoreline velocity, respectively, for the swash and the last approximation comes from dropping an $O(\varepsilon ^2)$ term. Therefore, the alongshore velocity component is neglected when determining the shoreline motion, consistent with the approach used to calculate the bore position in the weakly two-dimensional flow.

The evolution of the shoreline angle in the swash follows

(2.24) \begin{equation} \sin \theta \approx \tan \theta = \varepsilon u_s. \end{equation}

In fact, (2.24) is simply the small- $\theta$ approximation of Snell’s law for the most forward $\alpha$ characteristics that describes the swash shoreline motion, and is analogous to the small- $\theta$ approximation of Snell’s law for bore motion before collapse (2.3).

To calculate the full flow field, we follow the procedure outlined in § 2.2 for the cross-shore flow ( $u(x,\tau )$ and $h(x,\tau )$ ). For the alongshore flow, we note that the characteristic variable related to the alongshore motion is $\gamma = (v/\varepsilon )-h-x-({1}/{2})u^2$ , which is constant on curves where ${\rm d}x/{\rm d}\tau =u$ . From the bore solution, we can calculate $\gamma _2$ and then numerically integrate the path of the $\gamma$ characteristics using knowledge of the cross-shore flow field $u(x,\tau )$ . This allows us to compute $\gamma (x,\tau )$ from which we can extract $v(x,\tau )$ since $u(x,\tau )$ and $h(x,\tau )$ are already known. At the moving shoreline in the swash, since $h=0$ , the characteristics for $\alpha$ , $\beta$ and $\gamma$ all travel together with the shoreline, which moves at the speed $u_s$ . Therefore, from (2.17), $v_s=\varepsilon U_s^2$ , which shows that the alongshore component of the shoreline velocity remains constant throughout the swash. From $v_s$ and (2.14), the corresponding constant $\gamma _s$ travelling with the shoreline is

(2.25) \begin{equation} \gamma _s = \frac {1}{2}U_s^2. \end{equation}

The problem is now fully solved, with both the bore (and shoreline) path and the flow behind the bore (and the shoreline) obtained. Since we referred to Antuono’s cross-shore solution as the constant- $\alpha$ solution, we refer to our solution to the weakly two-dimensional system as the small- $\theta$ , constant- $\alpha$ solution.

There is one subtlety related to the offshore boundary condition left to examine. We discussed in § 2.2 the physical interpretation of the offshore boundary condition in terms of the incoming flow (2.13). Here, we note that there is no obvious equivalent specification for incoming alongshore flow $v^I(-1,\tau )$ , since we do not specify boundary conditions for $\gamma$ at the offshore boundary. It turns out that the $\gamma$ characteristics emanating from the bore at $x=-1$ in fact initially travel onshore, resulting in a gap in the alongshore flow solution for small, positive values of $(x+1)$ and $t$ , i.e. for small times near the offshore boundary (see Appendix A). At later times, the problem is alleviated by offshore moving $\gamma$ characteristics arriving at $x=-1$ . In lieu of a well-defined boundary condition for the incoming alongshore velocity, we postulate that the flow behind the bore maintains a constant incoming angle $\theta _0$ at the offshore boundary. From kinematics, the incoming alongshore velocity $v^I$ can then be specified as

(2.26) \begin{equation} v^I(-1,\tau ) = u^I(-1,\tau ) \tan \theta _0. \end{equation}

With this, we now have the information required to compute the solution throughout the domain for $x\geqslant -1,\tau \geqslant 0$ .

Finally, we note that there also exists a different analytical solution to the alongshore flow for the weakly two-dimensional system due to Ryrie (Reference Ryrie1983). Assuming a constant $\gamma (x,\tau _s)=\gamma _s$ throughout the swash, and using the cross-shore flow solution in (2.15) due to Peregrine & Williams (Reference Peregrine and Williams2001), Ryrie (Reference Ryrie1983) showed that the alongshore flow is given by

(2.27) \begin{equation} v = \frac {\varepsilon }{3}\left [ 2U_s^2 + \frac {x^2}{\tau _s^2} -U_s\tau _s + \frac {3}{4}\tau _s^2 + 2x \right ]. \end{equation}

We examine the differences between our small- $\theta$ , constant- $\alpha$ solution and Ryrie’s (Reference Ryrie1983) constant- $\gamma$ solution (2.27) in further detail in §§ 2.4 and 3.

In summary, our small- $\theta$ , constant- $\alpha$ solution for an oblique bore consists of the purely cross-shore flow ((2.8)–(2.12)) derived under the constant- $\alpha$ assumption (Antuono Reference Antuono2010), combined with the alongshore velocity behind the bore obtained from Snell’s law (2.17) and the small- $\theta$ assumption. In § 2.4, we numerically compute the path of $\gamma _2$ characteristics originating from the bore and calculate the alongshore velocity using its definition (2.6c ).

2.4. Numerically computed results

Following Antuono (Reference Antuono2010), we compute the bore position $x_b$ from numerical integration of the bore speed (2.10) using a fourth-order Runge–Kutta scheme with an adaptive time step to ensure accuracy during bore collapse given by

(2.28) \begin{equation} \Delta \tau _2 = A\left[1-(1-h_1)^n\right], \quad \text{with} \quad n=12,\quad A=10^{-4}. \end{equation}

The bore speed ( $U_b$ ), the flow variables immediately behind the bore ( $u_2$ , $v_2$ and $h_2$ ), and the resulting $\beta _2$ and $\gamma _2$ values, are calculated at each time step by solving the polynomial in (2.9) and using (2.17). The angle $\theta$ at each time step is calculated using (2.3) or (2.24). With this procedure, the absolute error between the prescribed value of $\alpha _2$ and the value computed from numerical calculation remained less than $10^{-10}$ .

To compute the full flow field ( $u(x,\tau )$ , $h(x,\tau )$ and $v(x,\tau )$ ), we use a grid with $\Delta x=5 \times 10^{-4}$ and $\Delta \tau =5 \times 10^{-4}$ . We use linear interpolation to compute the $\beta (x,\tau )$ field after using (2.11) at each time step of the bore solution. With this, both $\alpha$ and $\beta$ fields are known and we can compute the cross-shore flow ( $u(x,\tau )$ and $h(x,\tau )$ ) using (2.12). We compute the alongshore flow ( $v(x,\tau )$ ) from the path of each $\gamma _2$ characteristic by numerically integrating (2.6c ) using a fourth-order Runge–Kutta scheme, using linear interpolation to obtain $\gamma (x,\tau )$ , and then substituting the cross-shore flow solution into the definition of $\gamma$ .

For $x\lt 0$ , our methods yielded solutions for $h_1 \geqslant 10^{-12}$ , after which $\Delta \tau _2$ became too small to handle numerically. For $x\geqslant 0$ , we used the same methods with the known shoreline motion $u_s = \alpha _2 - \tau$ . The resulting small discontinuity in $u_2$ and $u_s$ near $x=0$ led to numerical errors of $O(10^{-2})$ in the $\beta$ and $\gamma$ characteristics travelling close to the shoreline. Moreover, the imposition of the approximate boundary conditions for the incoming alongshore flow velocity (2.26) also generated errors of a similar magnitude. However, these numerical errors are confined to be very close to the shoreline and the offshore boundary, and do not introduce any additional errors to the solution away from these regions (see Appendix A).

Figure 2. Flow properties along the bore for $\alpha _2=2.3$ and $\varepsilon = 0.24$ : (a) $U_b$ (black line), $u_2$ (thin blue line); (b) $h_2$ (black line), $h_1$ (thin blue line); (c) $\theta$ ; (d) $v_2$ (black line), $\gamma _2$ (red dotted line).

Figure 2 shows the flow properties along the bore for $\alpha _2=2.3$ and $\varepsilon =0.24$ . We chose $\alpha _2=2.3$ to reproduce the cross-shore solution in Antuono (Reference Antuono2010). In figure 2(a), we see that $u_2$ converges with $U_b$ during bore collapse and then decreases linearly with time during the swash. In figure 2(b), we see that there is a very rapid decrease in the bore height during bore collapse. The angle $\theta$ is illustrated in figure 2(c), whose evolution can be understood from the small- $\theta$ approximation to Snell’s law (2.24), which shows that $\tan \theta$ follows the evolution of the bore speed $U_b$ so that $\theta$ decreases during the bore’s approach and then rapidly increases during bore collapse. In the swash zone, we observe a linear decrease of the shoreline angle with time, as expected from (2.14b ) and (2.24). The alongshore velocity $v_2$ and gamma characteristics $\gamma _2$ , illustrated in figure 2(d), are relatively constant, but rapidly increase during bore collapse. After bore collapse, both $\gamma _2$ and $v_2$ are constant regardless of the angle $\theta$ .

Figure 3. Timeseries of (a–d) flow velocities and (e–h) water depths for $\alpha _2=2.3$ , $\varepsilon = 0.24$ : (a–d) $u$ (black solid line), $v' = v/\varepsilon$ (dashed blue line), $\gamma$ (dotted red line); (e–h) $h$ (black solid line); analytical solutions $u$ and $h$ ((2.15), thin magenta solid lines) and $v'$ ((2.27), thin orange dashed line).

Figure 3 shows timeseries of the flow field at various $x$ locations. We show the alongshore velocity as $v'=v/\varepsilon$ so that its magnitude is of the same order as the cross-shore velocity. For the swash zone ( $x \geqslant 0$ ), we observe that the analytical solutions for the cross-shore flow (2.15) agree very well with our solution for larger $x$ and $\tau$ . However, the analytical solution for alongshore velocity (2.27) is much larger than our solution. This difference stems from how bore collapse influences the alongshore dynamics, which is not taken into account in Ryrie’s analytical solution. We see from figure 2(d) that $\gamma _2$ and $v_2$ increase rapidly during bore collapse, which means that $\gamma$ characteristics close to the shoreline have larger values. This is reflected in figure 3(c,d), where we can see $\gamma$ is close to zero far from the shoreline, but rapidly increases near the shoreline. Since the analytical solution in (2.27) assumes that $\gamma = \gamma _s$ is constant, it predicts a larger $v$ than our small- $\theta$ , constant- $\alpha$ solution.

Figure 4 shows snapshots of the flow velocities and free surface displacement at various times. Similar to figure 3, the analytical solutions agree well with our solution close to the shoreline and for larger $\tau$ except for the alongshore velocity, which show a discrepancy. This discrepancy decreases for increasing $\tau$ , but is still prominent for the reasons mentioned above.

Figure 5 shows timeseries of the flow velocity at $x=-1$ . In figure 5(a), the cross-shore velocity shows good agreement with its inferred value from the offshore boundary conditions ( $u^I(-1,\tau )$ , (2.13)), until the solution becomes supercritical at $\tau _{\textit{crit}} \approx 1.664$ (Antuono Reference Antuono2010). On the other hand, in figure 5(b), the expected $v'^I(-1,\tau )$ ((2.26) divided by $\varepsilon$ ) diverges from $v'(-1,\tau )$ earlier than $\tau _{\textit{crit}}$ . This shows that the incoming flow does not maintain a constant angle, even before the flow becomes supercritical, due to the reflected flows altering the direction of the alongshore flow.

Figure 4. Snapshots of (a–d) flow velocities and (e–h) free surface displacements for $\alpha _2=2.3$ , $\varepsilon = 0.24$ : (a–d) $u$ (black solid line), $v' = v/\varepsilon$ (dashed blue line), $\gamma$ (dotted red line); (e–h) $\eta$ (black solid line), beach surface (thin dashed line); analytical solutions $u$ and $h$ ((2.15), thin magenta solid lines) and $v'$ ((2.27), thin orange dashed line).

Figure 5. Timeseries of (a) cross-shore and (b) alongshore velocities at $x=-1$ for $\alpha _2=2.3$ , $\varepsilon = 0.24$ : (a) $u$ (black solid line), $u^I(x=-1)$ (thin green solid line); (b) $v' = v/\varepsilon$ (dashed blue line), $v'^I(x=-1)$ (thin green dashed line).

Figure 6. Minimum alongshore velocity and $\gamma$ as a function of cross-shore position in the swash zone for $\alpha _2=2.3$ and $\varepsilon = 0.24$ : $v_{\textit{min}}$ for our small- $\theta$ , constant- $\alpha$ solution (black solid line), $v_{\textit{min}}$ for Ryrie’s (Reference Ryrie1983) analytical solution (thin blue solid line), minimum $\gamma$ for our small- $\theta$ , constant- $\alpha$ solution (thin orange dotted line) and predictions of the minimum alongshore velocities $\tilde {v}_{\textit{linear}}$ (green dash–dotted line), $\tilde {v}_{\textit{nonlinear}}$ (red dashed line) and $\tilde {v}_{\textit{R}}$ (thin magenta dashed line).

A similar observation is made in figures 3 and 4, where $v$ changes sign from positive to negative for $x \lt 0$ , despite the fact that $v_2 \gt 0$ immediately behind the bore, as shown in figure 2. This reversal is primarily attributed to the reflection on the slope influencing the local flow direction, as anticipated from the behaviour of $v'^I$ at the offshore boundary. In contrast, for $x \gt 0$ , the alongshore velocity $v$ remains positive, owing to the steep increase in $\gamma$ near the bore collapse. Physically, this implies that while bore-induced momentum dominates in the swash zone ( $x \gt 0$ ), reflected flows play an increasingly important role in the surf zone ( $x \lt 0$ ), where they can overcome the initial momentum and reverse the alongshore flow direction.

2.5. Explicit expressions for alongshore velocity

For predicting alongshore fluxes of fluid, solute, sediment, etc., it would be useful to have explicit formulae of the alongshore velocity as a function of cross-shore position. Since the alongshore velocity timeseries in the swash zone show that $v$ is close to its minimum value for the majority of the swash cycle and given that the higher values of $v$ at the start and end of the swash cycle are likely to be disproportionately affected by bed friction (Pujara et al. Reference Pujara, Liu and Yeh2015b ), we develop predictions from the theory for the minimum alongshore velocity, $v_{\textit{min}}$ .

We first consider how the minimum alongshore velocity varies with cross-shore distance in the swash zone. Figure 6 shows $v_{\textit{min}}$ for Ryrie’s analytical solution (2.27) and for the small- $\theta$ , constant- $\alpha$ solution for $\alpha _2=2.3$ and $\varepsilon =0.24$ . We also include data of $\gamma _{\textit{min}}$ for the small- $\theta$ , constant- $\alpha$ solution. While $v_{\textit{min}}$ from the analytical solution shows an almost-linear increase with $x$ , we observe that $\gamma _{\textit{min}}$ in our solution increases with $x$ as characteristics carrying larger $\gamma$ values start travelling up the beach with higher initial velocity, resulting in a nonlinear increase in $v_{\textit{min}}$ with $x$ in our solution.

We present different explicit expressions for $v_{\textit{min}}$ : a linear expression for Ryrie’s analytical solution (denoted as $\tilde {v}_{\textit{R}}$ ), a linear expression for our small- $\theta$ , constant- $\alpha$ solution (denoted $\tilde {v}_{\textit{linear}}$ ), and an expression for a nonlinear approximation for our small- $\theta$ , constant- $\alpha$ solution (denoted $\tilde {v}_{\textit{nonlinear}}$ ). These are given by

(2.29a) \begin{align} \tilde {v}_{\textit{R}} &= \frac {\varepsilon }{18}\left ( 10 U_s^2+ 16 x\right ), \end{align}

(2.29b) \begin{align} \tilde {v}_{\textit{linear}} &= \frac {\varepsilon }{18}\left ( U_s^2+ 16 x\right ), \end{align}

(2.29c) \begin{align} \tilde {v}_{\textit{nonlinear}} &= \frac {\varepsilon }{18}\left (13 U_s^2 +10 x - 12 U_s^{1.5}(U_s^2-2x)^{0.25}\right ). \end{align}

To obtain $\tilde {v}_{\textit{R}}$ , we find the linear fit between the minimum velocity at the start of the swash zone (i.e. $v_{\textit{min}}(x=0) = ({5}/{9})\varepsilon U_s^2$ ) and the velocity at the maximum run-up location (i.e. $v(x=({1}/{2})U_s^2) = v_s$ ). For our small- $\theta$ , constant- $\alpha$ solution, the linear approximation $\tilde {v}_{\textit{linear}}$ accounts for the difference in $\gamma$ compared with Ryrie’s constant- $\gamma$ assumption. In particular, since Ryrie’s solution has constant $\gamma = \gamma _s$ whereas our solution yields $\gamma \approx 0$ near the lower swash region ( $x \approx 0$ , see figure 6), we subtract $\varepsilon \gamma _s = ({1}/{2})\varepsilon U_s^2$ from $\tilde {v}_{\textit{R}}$ to approximate $\tilde {v}_{\textit{linear}}$ . To more accurately capture the steep increase in $v_{\textit{min}}$ near the maximum run-up, we find the nonlinear approximation, $\tilde {v}_{\textit{nonlinear}}$ by starting with $\tilde {v}_{\textit{nonlinear}} = a_1 + a_2 x + a_3 (U_s^2 - 2x )^{a_4}$ and determining the coefficients $a_i$ ( $i = 1$ – $4$ ) from known constraints. As in the linear case, we impose the known conditions at $x=0$ : the value of $v_{\textit{min}}$ and its slope $\partial v_{\textit{min}}/\partial x$ . Additionally, we enforce $v(x = ({1}/{2})U_s^2) = v_s$ at the maximum run-up. Fixing $a_4 = 0.25$ , which yields the best agreement with the numerical results, leads to the expression presented in (2.29c ).

Figure 6 shows that $\tilde {v}_{\textit{R}}$ provides an excellent fit to the $v_{\textit{min}}$ found from Ryrie’s analytical solution (2.27) whereas $\tilde {v}_{\textit{linear}}$ only captures the $v_{\textit{min}}$ data for our small- $\theta$ , constant- $\alpha$ solution accurately for small $x$ since it was derived using such an approximation. The nonlinear fit $\tilde {v}_{\textit{nonlinear}}$ performs better than $\tilde {v}_{\textit{linear}}$ for our small- $\theta$ , constant- $\alpha$ solution. All three expressions offer practical, explicit estimates of the alongshore velocity and confirm that $v_{\textit{min}}$ increases monotonically with cross-shore distance up to the maximum run-up location. While $\tilde {v}_{\textit{nonlinear}}$ exhibits improved accuracy across the domain, frictional effects are expected to dominate near the maximum run-up (Hogg & Pritchard Reference Hogg and Pritchard2004; Chanson Reference Chanson2009; Pujara, Liu & Yeh Reference Pujara, Liu and Yeh2016), where deviations from the predicted $v_{\textit{min}}$ are more likely to occur. Moreover, $\tilde {v}_{\textit{linear}}$ is more useful for isolating the dependence on cross-shore position. For these reasons, we focus our comparison with experimental measurements on $\tilde {v}_{\textit{R}}$ and $\tilde {v}_{\textit{linear}}$ in § 4.

2.6. Application to laboratory or field data

Applying the solutions mentioned above to laboratory or field data requires several considerations. First, we must consider the real-world domain in which the NSWEs are valid and the location where we can apply the constant- $\alpha$ boundary condition. Since dispersion is neglected and bores are treated as discontinuities, the NSWEs are valid in shallow water depths where waves have already broken and the velocity field is (nearly) depth uniform. Then to apply the constant- $\alpha$ boundary condition, it becomes necessary to specify the value of $\alpha _2$ , which is the only free parameter in the cross-shore solution.

To understand the relationship between $\alpha _2$ and the properties of the incoming bore, we consider two common definitions of bore strength: Whitham’s (Reference Whitham1958) definition, which quantifies the degree to which the incoming bore speed is supercritical, is given in dimensionless form by $(U_{b0} - 1)$ , where $U_{b0}$ is the dimensionless bore speed at the offshore boundary of the domain where the dimensionless water depth is unity. And Antuono’s definition, which quantifies the magnitude of the fluid velocity immediately behind the bore, is given in dimensionless form by $(\alpha _2 - 2)$ at the offshore boundary where the dimensionless water depth is unity and at $t=0$ . In both cases, it is clear that the bore strength increases with the value of $\alpha _2$ .

However, as noted in § 2.2, all information about the incoming bore is transferred to the initial shoreline velocity $U_s$ after bore collapse, which governs shoreline movement and is the swash zone proxy for the $\alpha _2$ value at the offshore boundary. Figure 7 shows the relationship between these two variables, which is monotonic as expected. Since $U_s$ is more easily observed and measured compared with $\alpha _2$ , which requires measurements of both flow velocity and water depth in the surf zone, it is easier to use data of $U_s$ and infer the implied $\alpha _2$ using figure 7.

Figure 7. The initial shoreline velocity $U_s$ with respect to $\alpha _2$ .

For the alongshore component of the flow solution, the only free parameter is $\varepsilon$ since there is a one-way coupling from the cross-shore solution to the alongshore flow in the weakly two-dimensional NSWEs. Again, it is simpler to infer $\varepsilon$ from data of the shoreline motion compared with measurements of the bore during its approach to the beach. We can obtain $\varepsilon$ from measurements of the shoreline motion using the small-angle approximation of Snell’s law in (2.24).

Apart from specifying the parameters to drive the solution, there is an additional consideration when applying the constant- $\alpha$ solution to data. In figures 3 and 4, we observe that the water depth approaches zero at all $x$ locations for large times. This occurs because the $\beta _2$ characteristics that emanate from the shoreline travel offshore without any hindrance. However, in a real-world swash flow, a hydraulic jump or backwash bore would likely form during the downwash and prevent the $\beta$ characteristics from propagating farther offshore. Thus, it is important to bear in mind that the long-time behaviour of the solution is non-physical due to the fixed nature of the offshore boundary condition.

3.1. Experimental set-up

Large-scale experiments were conducted in the Directional Wave Basin at O.H. Hinsdale Wave Research Laboratory at Oregon State University (Corvallis, OR, USA). Figure 8 shows the basin set-up where one end of the basin is equipped with a vertically hinged (so called ‘snake-type’) wave maker with 29 boards and 30 actuator points. The boards reorient relative to one another according to actuator motion, producing smooth multidirectional waves. The opposite end of the basin has a smooth metal beach with a uniform slope of $s=1/10$ . The laboratory coordinate system used in this study has its origin at the centre of the SWL, with $X$ being the cross-shore coordinate positive onshore and $Y$ being the alongshore coordinate positive in the direction incoming oblique bores are expected to drive an alongshore flow.

Figure 8. Experimental set-up (not to scale).

The basin centreline ( $Y=0$ ) was instrumented with one wire wave gauge (WG2, ImTech RWG) in the constant-depth region and two pairs of ultrasonic wave gauges (USWG) (Senix TS-30S1) and acoustic Doppler velocimeters (ADV) (Nortek Vectrino side-looking probes) at $X = -1.2$ m (USWG1, ADV1) and $X=0.8$ m (USWG3, ADV3), respectively. To quantify alongshore variability of the obliquely incident waves, two additional wire wave gauges (WG1 and WG3) were deployed at the same cross-shore location as WG2 in the constant depth region. Similarly, two additional pairs of USWG and ADV (USWG2 and ADV2, USWG4 and ADV4) were deployed at the same cross-shore location as USWG3 and ADV3. These sets of sensors were separated by $2.0$ m in the alongshore direction. Our analysis will focus on data from station 1 (S1, where USWG1 and ADV1 are located) and station 3 (S3, where USWG3 and ADV3 are located). All wave gauges and ADVs were programmed to collect data at a sampling frequency of 100 Hz and were synchronized through the data acquisition system (DAQ) (National Instruments PXI-6259). To account for drift in the calibration of the wire wave gauges, we collected calibration data at the start and end of the whole experiment by slowly filling or draining the basin and recording the change in voltage. The calibration coefficients used for each experiment are the values from linear interpolation in time.

Apart from the in situ sensors, an overhead camera with a sensor size of $1920 \times 1080$ pixels was also installed to capture images of the swash zone at a frame rate of $29.97$ Hz. These images allowed for quantifying the shoreline motion. To synchronize the camera images with the DAQ, we installed a red light-emitting diode (LED) within the camera field of view, which turned on and off in an unambiguously random sequence. We recorded the voltage signal from the LED, which indicated whether it was on or off, using the DAQ and matched the LED signal from the camera with the DAQ signal to synchronize the image data with the data from the in situ sensors.

The still-water depth, which was constantly measured and monitored from a pressure sensor in the basin, was kept constant during all experiments at $h_0^*=0.80$ m. Note, as before the $*$ superscript denotes dimensional variables, but the $0$ subscript denote variables in the constant depth region for the laboratory experiments, which is different to the theory section where the $0$ subscript denoted the offshore boundary of the theoretical domain.

To generate swash events at normal and oblique angles, we used two different methods. We generated solitary waves to obtain normally incident bores, which is a well-established method for generating isolated, but energetic swash events (Pujara et al. Reference Pujara, Liu and Yeh2015b ). However, we found in preliminary experiments that using obliquely generated solitary waves did not produce alongshore-uniform oblique bores due to their interaction with the basin sidewalls, in spite of the fact that the measurement section was outside of the most obvious reflection and diffraction zones near the walls. We found that obliquely propagating solitary waves undergo sufficient reflection and diffraction at the sidewalls that alongshore variability inevitably develops during wave travel. Under such alongshore variability, the alongshore component of the hydrostatic pressure gradient generated an unwanted alongshore flow, causing the situation to deviate from an ideal obliquely incident bore. To minimize this alongshore variability, we instead used the sidewall reflection method (Dalrymple Reference Dalrymple1989; Mansard & Miles Reference Mansard and Miles1993) to generate obliquely incident waves.

We used the implementation of the Dalrymple’s (Reference Dalrymple1989) method in the AwaSys wave generation software (v7, Meinert, Andersen & Frigaard Reference Meinert, Andersen and Frigaard2017). This implementation is based on an inverse diffraction calculation to obtain the target wave height and phase outside of the interference area, assuming linear superposition of the generated waves towards and away from the reflecting wall. We generated very long, regular oblique waves which were used as a proxy for solitary (or single) waves. The wave maker trajectories from AwaSys were then used to drive the motion of the different wave paddles. By also recording the trigger signal from the wave paddle, we were able to clearly define the start time for all sensors and cameras for each wave case.

Table 1 summarizes the properties of the 16 different wave cases used in the experiments. There were three cases of solitary waves for the zero-incident-angle tests (W14-16) and 13 cases for regular waves (W1–13) that include zero-incident-angle waves and obliquely incident waves generated using the sidewall reflection method. We generated the regular waves at two different periods, $T_0^* = 10.4$ s and $T_0^* = 8.4$ s, while also varying the wave height and incident angle. Analysis of WG2 data confirmed that the generated waves were of the desired periods.

Table 1. Wave properties. The subscript 0 denotes data related to the constant depth region and the subscript C denotes data related to the bore collapse location. CL are collapsing breakers and PL are plunging breakers. $T_0^*=\infty$ indicates solitary wave. $\theta _c$ and $U_{s,m}$ are omitted for the $T_0^* = 8.4$ s wave cases due to wave breaking occurring outside the camera ROI, and for the solitary wave cases due to camera vibrations. $\theta _0$ and $\varepsilon$ are excluded for the purely cross-shore wave cases. $U_{s,\alpha }(S1)$ for W13 is omitted due to the sensor noise.

For all cases, the incident wave height in the constant depth region, $H_0^*$ , was obtained by averaging the wave heights measured at WG1–3, which is reported in table 1. We also report the wave breaker type according to visual observations. Specifically, $T_0^* = 8.4$ s waves displayed collapsing breakers, which exhibited weak bore collapse with a substantial amount of white foam on the bore front. We noticed that the swash period of these waves was close to the wave period, so that there may have been a very weak interaction between the swash events of successive waves that could influence their swash flows (Power, Holman & Baldock Reference Power, Holman and Baldock2011; Pujara et al. Reference Pujara, Liu and Yeh2015a ; Meza-Valle & Pujara Reference Meza-Valle and Pujara2024). On the other hand, the $T_0^* = 10.4$ s waves exhibited plunging breakers and it was clear that each swash event was independent so that the entire swash cycle from bore collapse to backwash bore generation were unaffected by the preceding or following wave.

Finally, we also report the nominal wave incidence angle in the constant depth region, denoted as $\theta _0$ . We estimated $\theta _0$ using the measured time delay, $\Delta t^*$ , between the arrival of the wave crests at WG1–3. By using $c^* = \sqrt {g^*(h_0^* + H_0^*)}$ for the phase speed, and the alongshore separation distance between WG1 and WG3, $\Delta y^* = 4\text{ m}$ , we found the angle $\theta _0$ from the relation $\Delta y^* \sin \theta _0 = c^* \Delta t^*$ . While this method is not exact, it provides a useful estimate of the nominal wave incidence angle.

3.2. Data processing

Measurements in the surf and swash zones are known to suffer from data quality issues related to bubbles, shallow water depths, and wave-to-wave variability. To improve data quality, we rely on various data processing methods for quality control. We use slightly different methods for the regular waves (W1–13) and the normal solitary waves (W14–16), but in all cases, the end product is reliable and accurate data of the mean (ensemble averaged) flow.

For W1–13, we remove obviously non-physical spikes in the USWG data by removing data points where the magnitude of the derivative of the surface elevation timeseries exceeds 3 m s⁻¹. For the ADV data, we remove data points where the correlation coefficient (COR) is lower than 80 % and the signal-to-noise ratio (SNR) is lower than 14 dB. After that, we ensemble average the data to obtain the mean flow timeseries. In this averaging, we only use the data from the fifth to 14th waves generated by the wave paddle. The first four waves are excluded because wave gauge data clearly show signs of wave paddle start up, whereas waves after the 15th wave are excluded to avoid effects due to recirculation flows within the basin. We then identify individual swash cycles at each measurement station by detecting a sudden increase in water depth from the USWG data. By resampling each swash cycle onto a dimensionless time vector normalized by its swash duration with a resolution of 0.01 (100 data points), we obtain a total of 30 waves (10 waves times three repeated runs of each wave case) that are used in ensemble averaging to obtain the mean flow.

To check whether data for obliquely incident waves is contaminated by significant alongshore variability in water depths and flow velocities (W1–13), we compare data across ADV2–4 and USWG2–4. Across all wave cases that are included in the analysis, we found a maximum alongshore difference in measured velocity at any time to be smaller than 0.2 m s⁻¹ (which is less than $5\,\%$ of the velocity scale of swash flow). Similarly, we found a maximum alongshore difference in measured water depth to be smaller than 30 $\%$ of the maximum water depth at any time, which was less than 3.2 cm for plunging breakers and 2.1 cm for collapsing breakers. With the minimum alongshore distance between the sensors being $2.0$ m, this corresponds to an alongshore slope in water depth of less than $1.6 \times 10^{-2}$ . These values being small, we believe the influence of the alongshore variability is minimal, and the ensemble-averaged flow field closely resembles an alongshore-uniform bore approaching a beach.

For W14–16, we conducted 10 repetitions for each case. Unlike the regular waves, these experiments are highly repeatable since there is no wave-to-wave variability. After matching the time vectors using the wave paddle start trigger signal, we compute ensemble averages across the 10 waves (repeated experiments) and compute the mean flow. Before calculating the ensemble averages, we removed spikes in the USWG data as before. However, for the ADV data, we used a slightly different procedure to remove spurious data points. The SNR and COR for the solitary waves were lower than for the regular waves, likely because the solitary waves did not uniformly mix the seeding tracer particles in the basin. Therefore, we first removed data points with a COR value lower than 40 $\%$ and after that, we applied a modified version of the phase-space thresholding method (Goring & Nikora Reference Goring and Nikora2002). This iterative method, based on normal probability distribution theory, was designed for steady flow, zero-mean data (e.g. channel flow after subtracting the mean velocity). Since the solitary wave experiments were found to be highly repetitive (see also Pujara et al. (Reference Pujara, Liu and Yeh2015b )), we removed the mean flow computed from the ensemble averaging. The algorithm was then applied iteratively, as designed. We required a minimum of five data points when calculating the ensemble-averaged velocity. If fewer than five data points were available (due to having been removed for poor quality, for example), the ensemble-averaged velocity was not calculated. Additionally, before the first iteration, obvious outliers (velocity differences larger than $0.7$ m s⁻¹ and $0.2$ m s⁻¹ for the cross-shore and alongshore directions, respectively) were removed while obtaining the ensemble-averaged velocity. During the iteration loops, more noisy data points were removed and the ensemble averaged velocity was updated at each iteration. The iteration process ended when no further data points were flagged for removal. The remaining velocity data were then used to compute the final ensemble average and used as the mean flow. More details of this data processing pipeline are provided in Appendix B.

Figure 9. Example of image rectification: (a) raw image; (b) rectified image. Red dots indicate the points used to calculate homograph transformation matrix and the green box denotes the region of interest (ROI) for calculating the shoreline movement.

Figure 10. Example of image processing to identify shoreline position: (a) raw image converted to greyscale; (b) after background removal; (c) after binarization with partially detected shoreline coloured in red; (d) fitted shoreline in blue on raw greyscale image.

We calculated the shoreline velocity and shoreline angle using image data from the overhead camera. A single video was recorded for each wave case. Since the camera was installed obliquely to the beach, we first rectified the raw images following methods previously outlined in Sung et al. (Reference Sung, Choi, Ha, Lee and Park2022). Figure 9 shows an example of a raw image and its rectified version. Briefly, using points with known laboratory coordinates (red dots in figure 9), we calculated the homography matrix which transforms the raw image into a rectified image with a resolution of 1 cm per pixel. The pixel intensities of the rectified image were calculated using bilinear interpolation. In the rectified image, we calculated the shoreline movement within the ROI (dashed green box in figure 9 b). The cross-shore position of the ROI was informed by the change in colour of the beach surface offshore from the selected ROI, which makes it challenging to distinguish the shoreline. The alongshore extent of the ROI was chosen to be within the centre of the image where effects of lens distortion were small and between the regions where overhead lighting caused significant glare on the beach surface.

Our image processing procedure to identify the shoreline position involved standard methods including greyscale conversion, background removal, binarization using a threshold, and edge detection. Figure 10 shows examples of each step. We obtained the background image as the mean across all images of a given movie. By subtracting this background, we were able to enhance the contrast between the beach and the approaching shoreline (figure 10 b). We used a normalized binarization threshold of $0.09$ , and then removed regions affected by glare, by structures holding the sensors, and regions smaller than 300 pixels in area to give the final binarized image to be used for shoreline detection (figure 10 c). We detected the shoreline edge by searching for the lowest (most onshore) remaining pixel in each column (shown in red in figure 10 c). Finally, we fit a straight line to the edge detected points to find the shoreline position and angle (shown in blue in figure 10 d).

3.3. Properties of swash flow

3.3.1. Bore collapse and shoreline motion

Bore collapse is an important feature of the experimental data since it defines the start of the swash. Figure 11 illustrates the different laboratory coordinate systems and variables related to the bore collapse process in the experiments. We define the coordinate system relative to the SWL as $(X,Y)$ (see figure 8) and the coordinate system relative to the bore collapse location as $(x,y)$ . The distance between the origins of these two locations, denoted $X_c$ , quantifies where bore collapse occurs relative to the SWL, and $\theta _c$ quantifies the bore angle at collapse. Table 1 lists these values, which were measured by manually capturing the bore collapse at the basin centreline ( $Y=0$ ) in the camera images and reported as the ensemble mean values over 10 waves. The time of bore collapse is also used to define the swash zone pseudotime origin, $\tau _s = 0$ .

Figure 11. Diagram to illustrate shoreline properties.

It is interesting to note that the solitary waves collapse onshore of the SWL ( $X_c\gt 0$ ) since the fluid velocity is always pointed onshore, whereas the regular waves collapse offshore of the SWL ( $X_c\lt 0$ ) due to the presence of a wave trough and the associated offshore directed fluid velocity. Note, the collapse location was too far offshore of the camera ROI to be able to obtain a reliable measurement of $\theta _c$ and $X_c$ for the $T_0^* = 8.4$ s waves.

Figure 12. Shoreline motion data for W5 and W8: (a) shoreline position $x_s^*$ ; (b) angle $\theta$ ; (c) obliqueness parameter $\varepsilon$ . In (a) and (b), dots show raw data from 10 individual waves and solid lines show ensemble average. In (c), solid line shows the time evolution and red dashed line shows the time mean.

We track the shoreline motion after bore collapse, and extract the timeseries of shoreline position $x_s$ and angle $\theta$ . Figure 12(a,b) show example timeseries of this data for W5 and W8, where raw data from individual waves are shown as dots and the ensemble averaged values as solid lines. To reduce noise and allow for calculating the shoreline velocity in a single step, we compute the ensemble averages using linear fits over a moving window where the window size (typically $0.13$ – $0.32$ s) was selected to minimize the mean variance of the difference between the raw and ensemble average data.

3.3.3. Effective initial shoreline velocity, $U_s$

Figure 13. Effective initial shoreline velocity from in situ sensor data at different measurement stations: $U_{s,\alpha }(S1)$ (thin line), $U_{s,\alpha }(S3)$ (thick line) for W5 (red solid line), W8 (blue dashed line) and W15 (green dotted line). Vertical dashed lines indicate a window for calculating time mean.

As noted in § 2.6, we can use the initial shoreline velocity $U_s$ to infer the $\alpha _2$ of the incoming bore. Following Pujara et al. (Reference Pujara, Liu and Yeh2015b ), we estimate the effective $U_s$ for the swash flow in multiple ways.

The first is the measured initial shoreline velocity from camera data, $U_{s,m}$ . We extract this from the slope of the shoreline position (figure 12 a) at $\tau _s = 0$ and report it in table 1. However, we were only able to extract this for the $T_0^* = 10.4$ s waves. For solitary waves, it was not possible to obtain $U_{s,m}$ due to insufficient time resolution in the camera data and the extra noise due to vibrations from the plunging breaker that shook the camera. For the $T_0^* = 8.4$ s waves, the bore collapse location was offshore of the camera ROI as explained above.

The second is to use flow data from the in situ sensors combined with the definition of the forward moving characteristic, which gives $U_{s,\alpha } = u+2c+\tau _s$ . We calculate $U_{s,\alpha }(S1)$ and $U_{s,\alpha }(S3)$ , using the two different cross-shore measurement stations S1 and S3, respectively.

Figure 13 shows example timeseries of $U_{s,\alpha }(S1)$ and $U_{s,\alpha }(S3)$ for W5, W8 and W15. Other wave cases show behaviour that is very similar to the corresponding breaker type in figure 13. We observe that, overall, the $U_{s,\alpha }$ values remain constant. This confirms the applicability of the theory since it implies constant- $\alpha$ in the swash flow. The notable exception is $U_{s,\alpha }(S1)$ for W8, which we discuss further below. For all wave cases, we calculate the time mean $U_{s,\alpha }(S1)$ and $U_{s,\alpha }(S3)$ over the range $0.3 \leqslant \tau _s \leqslant 1.7$ (dashed vertical lines in figure 13) and use it to calculate the theoretical solution. These values are reported in table 1. The RSD values for $U_{s,\alpha }$ are $\lt 2.5\,\%$ across all wave cases. This excludes $U_{s,\alpha }(S1)$ for collapsing breakers, such as $U_{s,\alpha }(S1)$ for W8 in figure 13, for which the deviation is higher.

Here $U_{s,\alpha }(S1)$ is not constant for collapsing breakers, but rather increases during the swash cycle. This behaviour suggests that the NSWEs and constant- $\alpha$ solution may not accurately represent the flow at $x\lt 0$ for collapsing breakers. One reason for the increasing $U_{s,\alpha }$ at locations offshore of bore collapse could be due to incomplete bore collapse. Jensen, Pedersen & Wood (Reference Jensen, Pedersen and Wood2003) showed with particle image velocimetry data that the velocity field is not depth-uniform for surging breakers even when the bore is in very shallow depths. A depth-dependent velocity field indicates that dispersion effects are non-negligible, which in turn means the NSWEs are not applicable. Another possible reason could be the weak wave–swash interactions observed for collapsing breakers, which may have caused the incoming waves to break earlier than expected due to offshore directed momentum from the backwash. Additionally, the mass flux from the backwash of the previous wave might contribute to an increased water depth, leading to the observed increase in $U_{s,\alpha }(S1)$ over time. Lastly, our analysis of the NSWEs ignores the effects of bed friction, which is expected to decrease the rate of water drainage during backwash (Pedersen et al. Reference Pedersen, Lindstrøm, Bertelsen, Jensen, Laskovski and Sælevik2013) leading to larger water depths and therefore increase $U_{s,\alpha }(S1)$ .

However, we note that while $U_{s,\alpha }(S1)$ is not constant for collapsing breakers, $U_{s,\alpha }(S3)$ remains relatively constant, indicating that the constant- $\alpha$ solution is likely applicable to the swash flow in these cases. It may be informative to use a numerical solution with a time-varying $\alpha$ at the offshore boundary (Guard & Baldock Reference Guard and Baldock2007), but without further data it remains unclear whether the observed trend in $U_{s,\alpha }(S1)$ is due to a time-varying boundary condition, a breakdown of the shallow water approximation, or additional influences such as wave–swash interactions.

Briefly, we also mention that $U_{s,\alpha }(S1) \geqslant U_{s,\alpha }(S3)$ for all cases where the data is available. Since $U_{s,\alpha }$ quantifies the net energy in the flow, we attribute the decrease to losses due to friction and turbulence. The rate at which this loss occurs, and how it depends on breaker type, is left for future study.

To calculate the theoretical solution, we use the mean of $U_{s,\alpha }$ across time and across stations $S1$ and $S3$ for plunging breakers, and the mean $U_{s,\alpha }$ across time at station $S3$ for collapsing breakers.