Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays

Abstract

We present an analytical study of the propagation and run-up of long waves in piecewise sloping, U-shaped bays using the cross-sectionally averaged shallow water equations. The nonlinear equations are transformed into a linear equation by utilizing the generalized Carrier–Greenspan transform (Rybkin et al. J Fluid Mech 748:416–432, 2014). The solution of the linear wave propagation is taken as the boundary condition at the toe of the last sloping segment, as in Synolakis (J Fluid Mech 185:523–545, 1987). We then consider a piecewise sloping bathymetry, and as in Kanoglu and Synolakis (J Fluid Mech 374:1–28, 1998), find the linear solution in the near shore region, which can be used as the boundary condition for the nonlinear problem. Our primary results are an analytical run-up law for narrow channels and breaking criteria for both monochromatic waves and solitary waves. The derived analytical solutions reduce to well-known solutions for parabolic bays and plane beaches. Our analytical predictions are verified in narrow bays via a comparison to direct numerical simulation of the 2-D shallow water equations.

Appendix: Nonlinear Run-Up Alternate Solution

In this Appendix, we solve the complete nonlinear run-up of the wave in Region 2 by finding the potential function for the nonlinear run-up from the linear solution. This is an alternate method to the one used in Sect. 4 that results in the same solution. Using the CG transform, we show that both linear and nonlinear run-ups have the same wave potential. Then, we find this potential from the linear solution, thus finding the complete nonlinear run-up of the wave in Region 2. This method has the advantage of not having singularities in the solution to the general boundary value problem for the potential function, as shown in Eq. (19).

To solve the cross-sectionally averaged fully nonlinear SWEs in Region 2, Eq. (2), we use the same generalized CG transform Rybkin et al. (2014) as used in Sect. 4. The transform is defined by Eq. (13) Garashin et al. (2016) . The nonlinear SWEs for U-shaped bays, Eq. (2), reduce to the single second-order, linear PDE, Eq. (14), under this transform.

Recall that the initial conditions in Region 2 are $u^{(2)}(x,t=0)=0$ and $\eta ^{(2)}(x,t=0)=0$. In addition, recall that the boundary conditions of region 2 are that the wave is bounded at the shoreline, and that at $x=x_0$, we have $\eta ^{(2)}_\mathrm{nonlinear}(x=x_0,t)=\eta ^{(2)}_\mathrm{linear}(x=x_0,t)$. We also assume that the wave height and velocity are small at the boundary. Under the CG transform, Eq. (13), our initial and boundary conditions on the potential become

$$\begin{aligned} \Phi (\lambda =0,\sigma )=0~,~~~~ \Phi _{\lambda }(\lambda =0,\sigma )&=0~,~~~~ \Phi _{\sigma }(\lambda ,\sigma =0)=0~,\nonumber \\ \frac{\Phi _\lambda (\lambda =g\alpha t,\sigma =\sigma _0)}{2g}&=\eta ^{(2)}_\mathrm{linear}(x=x_0,t)~, \end{aligned}$$

(37)

where $\sigma _0 = \sqrt{(4g\alpha x_0(m+1))m^{-1}}$, and $\eta ^{(2)}_\mathrm{linear}(x=x_0,t)$ is given by Eq. (11).

Instead of directly solving Eq. (14) with the given initial and boundary conditions, as we did in Sect. 4, we instead use the linear solution of the run-up in Region 2 to find the nonlinear wave potential. To this effect, we use the linear version of the CG transform to solve for the linear propagation of the wave in Region 2. We thus first linearize the 1D SWEs for U-shaped bays by assuming $\eta (x,t) \ll h(x)$ and $u(x,t) \ll \sqrt{g(m+1)m^{-1}h(x)}$. We thus neglect the nonlinear terms $u\eta _x$, $(m+1)m^{-1}\eta u_x$ and $uu_x$ in Eq. (4).

We also linearize the CG transform by assuming that u and $\eta$ are small, such that $u \ll g \alpha t$ and $\eta \ll \frac{\sigma ^2m}{4g(m+1)}\approx h(x)$, and that terms of $O(u^2)$ can be neglected. Using these assumptions on the nonlinear CG transform, Eq. (13), we get the linear CG transform, Eq. (16).

Note that the same assumptions were made in linearizing the SWEs and the CG transform. The linear CG transform can be used to solve the linear SWE. This transform applied to Eq. (4) gives us

$$\begin{aligned} \Phi _{\lambda \lambda }-\Phi _{\sigma \sigma }-\frac{m+2}{m\sigma }\Phi _\sigma = 0~. \end{aligned}$$

(38)

Note that Eqs. (38) and (14) are identical. In addition, note that the shoreline corresponds to $\sigma =0$ for the linear CG transform as well as the nonlinear transform. Our initial conditions in Region 2 and our boundary conditions at $x=x_0$, under the linear CG transform, will be identical to Eq. (37).

As the governing equations for both the linear and nonlinear potentials are identical, and all initial and boundary conditions for both potentials are identical, the potentials will be identical. We thus find the linear potential from the solution to the linear shallow water wave equations, and the linear CG transform. The solution to the linear wave propagation in Region 2 is now given by Eq. (11). We use the linear CG transform, Eq. (16), to transform this solution into $(\sigma ,\lambda )$ coordinates to see that

$$\begin{aligned} \eta ^{(2)}(\sigma ,\lambda ) = \frac{2}{\gamma } \int \limits _{-\infty }^{\infty } A_{i}(\kappa /\gamma )\left( \frac{\sigma _0}{\sigma }\right) ^\frac{1}{m}\frac{J_{\frac{1}{m}}\left( \kappa \sigma \right) e^{-i\kappa (\sigma _0/2- \lambda )}}{J_{\frac{1}{m}} \left( \kappa \sigma _0 \right) - i J_{\frac{1}{m} + 1} \left( \kappa \sigma _0 \right) } d\kappa , \end{aligned}$$

where $\gamma =c/g\alpha$ and $\kappa \sigma _0=2kx_0$. Now, using Eq. (16), we find the linear potential to be

$$\begin{aligned} \Phi (\sigma ,\lambda ) = - \frac{i4g}{\gamma } \int \limits _{-\infty }^{\infty } \frac{A_{i}(\kappa /\gamma )}{\kappa }\left( \frac{\sigma _0}{\sigma }\right) ^\frac{1}{m}\frac{J_{\frac{1}{m}}\left( \kappa \sigma \right) e^{-i\kappa (\sigma _0/2- \lambda )}}{J_{\frac{1}{m}} \left( \kappa \sigma _0 \right) - i J_{\frac{1}{m} + 1} \left( \kappa \sigma _0 \right) } d\kappa . \end{aligned}$$

(39)

As we have proven, this will also be the nonlinear wave potential in Region 2. Note that this potential, Eq. (39), is identical to the potential found in Sect. 4, Eq. (21).