Swash flow due to obliquely incident bores

TL;DR

This paper addresses swash flow resulting from obliquely incident bores by extending the constant-$\alpha$ cross-shore solution to a weakly two-dimensional regime under a small effective angle $\varepsilon$. The authors derive a one-way coupling from cross-shore bore dynamics to alongshore flow, using a small-$\theta$ approximation and Snell's law to relate bore obliquity to alongshore motion, and validate the theory with large-scale laboratory experiments that measure depths, velocities, and shoreline motion. They provide explicit expressions for the minimum alongshore velocity and show that the alongshore predictions align well with data, while highlighting the role of the non-constant alongshore characteristic gamma $\gamma$ in improving accuracy over prior constant-$\gamma$ theories. The work offers a practical framework for predicting alongshore transport in nearshore zones and suggests directions for incorporating friction, dispersion, and wave–swash interactions in future modeling efforts.

Abstract

We present a new solution to the nonlinear shallow water equations and show that it accurately predicts the swash flow due to obliquely approaching bores in large-scale wave basin experiments. The solution is based on an application of Snell's law of refraction in settings where the bore approach angle $θ$ is small. We use the weakly two-dimensional nonlinear shallow water equations [Ryrie (J. Fluid Mech., vol. 129, 1983, p. 193)], where the cross-shore dynamics are independent of, and act as a forcing to, the alongshore dynamics. Using a known solution to the cross-shore dynamics [Antuono (J. Fluid Mech., vol. 658, 2010, p. 166)], we solve for the alongshore flow using the method of characteristics and show that it differs from previous solutions. Since the cross-shore solution assumes a constant forward-moving characteristic variable, $α$, we term our solution the `small-$θ$, constant-$α$' solution. We test our solution in large-scale experiments with data from fifteen wave cases, including normally incident waves and obliquely incident waves generated using the wall reflection method. We measure water depths and fluid velocities using in situ sensors within the surf and swash zones and track shoreline motion using quantitative imaging. The data show that the basic assumptions of the theory (Snell's law of refraction and constant-$α$) are satisfied and that our solution accurately predicts the swash flow. In particular, the data agrees well with our expression for the time-averaged alongshore velocity, which is expected to improve predictions of alongshore transport at coastlines.

Figures (17)

• Figure 1: Definition sketch for an obliquely approaching bore: (a) side view; (b) top view.
• Figure 2: Flow properties immediately behind 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 3: Timeseries of flow velocities (a-d) and water depths (e-h) for $\alpha_2=2.3$, $\varepsilon = 0.24$. (a-d): $u$ (black solid line), $v' = v/\varepsilon$ (dashed blue line), $\gamma$ (dotted red line), $u^I(x=-1)$ (thin purple solid line), $v^I(x=-1)$ (thin green dashed line); (e-h): $h$ (black solid line); analytic solutions: $u$ and $h$ (Eqs. \ref{['eq:Peregrine Williams']}, thin magenta solid lines), $v'$ (Eq. \ref{['eq:Ryrie']}, thin orange dashed line).
• Figure 4: Snapshots of flow velocities (a-d) and water depths (e-h) 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); analytic solutions: $u$ and $h$ (Eqs. \ref{['eq:Peregrine Williams']}, thin magenta solid lines), $v'$ (Eq. \ref{['eq:Ryrie']}, thin orange dashed line).
• Figure 5: Minimum alongshore velocity and $\gamma$ as a function of cross-shore position in the swash zone for $\alpha_2=2.3$ and $\varepsilon = 0.24$: $v_{\text{min}}$ for our small-$\theta$, constant-$\alpha$ solution (black solid line), $v_{\text{min}}$ for ryrie1983longshore's analytic solution (thin blue line), minimum $\gamma$ for our small-$\theta$, constant-$\alpha$ solution (red dotted line), and predictions of the minimum alongshore velocities $\Tilde{v}_C$ (dashed orange line) and $\Tilde{v}_R$ (thin dashed magenta line).