Fully nonlinear Boussinesq equations for long wave propagation and run-up in sloping channels with parabolic cross sections

Abstract

A general framework for derivation of long wave equations in narrow channels, and their transformation to Lagrangian coordinates is briefly established. Then, fully nonlinear Boussinesq equations are derived for channels of parabolic cross sections. The simplified version with normal nonlinearity is compared with corresponding models from the literature, and propagation properties are discussed. A Lagrangian run-up model is adapted to the fully nonlinear set. This model is tested by means of controlled residues and by a well-controlled comparison to exact analytic solutions from the literature. Then, run-up of solitary waves in simple geometries is simulated and compared to a semi-analytic solution that is derived for propagation and run-up in a composite channel. The dispersive model retains the higher run-up height in a parabolic channels, as reported in the recent literature for NLSW solutions, as compared to a rectangular channel.

Appendices

Appendix 1: The standard Boussinesq and the KdV equation

After deletion of the nonlinear parts in E (4) reads

$$\begin{aligned} p=p_s+{\tilde{\eta }}+\mu ^2\left( \widetilde{\frac{\partial \phi }{\partial t} } -\frac{\partial \phi }{\partial t} \right) +O(\epsilon \mu ^2,\mu ^4). \end{aligned}$$

Insertion of this expression in the momentum equation followed by averaging then yields

$$\begin{aligned} {{\overline{D}}}{\overline{u}}=-\frac{\partial {\tilde{\eta }}}{\partial x}+\mu ^2\left( \overline{\left( \frac{\partial ^2\phi }{\partial t\partial x}\right) }-\frac{\partial \ }{\partial x}\left( \widetilde{\frac{\partial \phi }{\partial t} }\right) \right) +O(\epsilon \mu ^2,\mu ^4), \end{aligned}$$

(23)

and

$$\begin{aligned} \displaystyle {{\overline{D}}}{\overline{u}}=-\frac{\partial {\tilde{\eta }}}{\partial x} +\mu ^2\left( \left\{ \frac{4h_m^2}{21}+\frac{h_m}{45\beta }\right\} \frac{\partial ^3 {\overline{u}}}{\partial x^2\partial t} \displaystyle +h_m'\left\{ \frac{2h_m}{3}+\frac{1}{18\beta }\right\} \frac{\partial ^2 {\overline{u}}}{\partial x\partial t} +\frac{2h_mh_m''}{5}\frac{\partial {\overline{u}}}{\partial t} \right) +O(\epsilon \mu ^2,\mu ^4). \end{aligned}$$

(24)

When compared to equation 69 of Teng and Wu (1992), there are differences in the reciprocal $\beta$ terms. This is apparently due to a switching of the outer spatial derivative and the surface ($\,\widetilde{\ }$ ) average in the equation equivalent to (23) herein (see equation 38 in the reference).

To derive a KdV equation for unidirectional waves, it is most convenient to rescale according to ${\tilde{\eta }}=\epsilon {\hat{\eta }}$ and ${\overline{u}}=\epsilon {\hat{u}}$ and assume that $\epsilon =\mu ^2$ is appropriate. For $h_m=h_0=\hbox {const}$. the momentum and continuity equation become

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial {\hat{u}}}{\partial t}+\epsilon {\hat{u}} \frac{\partial {\hat{u}}}{\partial x}=-\frac{\partial {\hat{\eta }}}{\partial x}+\epsilon M \frac{\partial ^3 {\hat{u}}}{\partial x^2\partial t}+O(\epsilon ^2),\\ \displaystyle \frac{\partial {\hat{\eta }}}{\partial t}+\epsilon {\hat{u}} \frac{\partial {\hat{\eta }}}{\partial x}=-\nu (h_0+\epsilon {\hat{\eta }})\frac{\partial {\hat{u}}}{\partial x}+O(\epsilon ^2), \end{array} \end{aligned}$$

where M and $\nu$ are as in (18). Next, these equations are transformed by introducing a translating spatial variable, $\xi$, and a slow temporal variable $\tau$

$$\begin{aligned} \xi =x-c_0 t,\quad \tau =\epsilon t, \end{aligned}$$

where $c_0$ is as in (18). The equations become

$$\begin{aligned}&\,&\frac{\partial {\hat{u}}}{\partial \tau }+(\epsilon {\hat{u}} -c_0)\frac{\partial {\hat{u}}}{\partial \xi }=-\frac{\partial {\hat{\eta }}}{\partial \xi }+\epsilon M \frac{\partial ^3 {\hat{u}}}{\partial \xi ^3}+O(\epsilon ^2), \end{aligned}$$

(25)

$$\begin{aligned}&\,&\frac{\partial {\hat{\eta }}}{\partial \tau }+(\epsilon {\hat{u}} -c_0)\frac{\partial {\hat{\eta }}}{\partial \xi }=-\nu (h_0+\epsilon {\hat{\eta }})\frac{\partial {\hat{u}}}{\partial \xi }+O(\epsilon ^2). \end{aligned}$$

(26)

To leading order, these imply $c_0{\hat{u}}={\hat{\eta }}+O(\epsilon )$ which is used to eliminate ${\hat{u}}$ from all $O(\epsilon )$ terms in (25) and (26). From the resulting two equations $\frac{\partial {\hat{u}}}{\partial \xi }$ is then eliminated and a single equation for ${\hat{\eta }}$ is obtained. The KdV equation on the form (18) then follows when ${\tilde{\eta }}, x$ and t are restored.

Appendix 2: A linear solution for the wave tank geometry

The incident wave in a geometry like the one in Fig. 4 will be modified when entering the slope. To assess this effect, we employ the availability of explicit local solutions to find a LSW solution with a sharp apex between the beach and the flat bottom at $x=\ell =\alpha ^{-1}$.

In the linear approximation, the incident wave of (11) and (12) may be expressed, for $x\le \ell$, according to

$$\begin{aligned} \eta =\nu x^{-\frac{1}{2}}I'(\tau _s),\quad u=\nu \left( \frac{1}{2}x^{-\frac{3}{2}}I(\tau _s)-\sqrt{\frac{3}{2\alpha }}x^{-1}I'(\tau _s)\right) , \quad \tau _s=2\left( \sqrt{\frac{3x}{2\alpha }}-\sqrt{\frac{3\ell }{2\alpha }}\right) +t+\frac{L}{c_0}-t_a \end{aligned}$$

(27)

where I is a potential related to ${\varPhi }, \nu$ is a constant and the last part of $\tau _s$ is included to make the expressions compatible with the incident wave as specified below. We skip the details, but note that (27) may also be readily verified by direct substitution into the LSW equations. If we assume that $I'$ defines a pulse of finite extent and omit the first term in the expression for u we obtain the optical approximation [see Didenkulova and Pelinovsky (2011b)] for which the energy is conserved and for which kinetic and potential energies are equal. On the other hand, this solution does not conserve volume. Instead the volume is decreasing as the pulse approaches the shore. Hence, in the full u in (27), we have the first term which gives a reduced kinetic energy, thereby also a deviation from energy equipartition, and a constant offshore cross-integrated flux in the wake of the incident pulse to account for the volume loss. Since this residual current is stationary, it is consistent with a surface elevation of finite extent. However, u does decrease toward deeper water since the cross-sectional area increases. Thus, there is a convective retardation and this is consistent with the weak trailing surface depression inherent in the nonlinear solution, as given by the last term of expression for $\eta$ in (11). While we could reproduce the trailing current by the wave paddle in the test case of Sect. 2.7, it is difficult to conceive how a surface pulse with a wake current could constitute the whole solution on the slope in a composite geometry as shown in Fig. 4. Even if the current could extend beyond the apex it would have to be finite in extent, since the incident wave from the constant depth region is confined. A finite current implies a surface gradient and a propagating wave. Hence, we must expect both modifications in the shape of the wave transmitted to the slope and a reflected wave.

If we combine (27) with the odd shoreline reflection, we have the full LSW solution on the slope. For $x\ge \ell$, the solution is

$$\begin{aligned} \eta =A_0 (Y(\tau _i)+P(\tau _r)),\quad u=\frac{A_0}{c_0} \left( -Y(\tau _i)+P(\tau _r)\right) ,\quad \tau _i=\frac{x}{c_0}+t-t_a,\quad \tau _r=-\frac{\ell -x}{c_0}+t-t_a, \end{aligned}$$

(28)

where $c_0$ is the shallow water speed over the flat bottom and Y and P represents the incident wave and the one reflected from the apex, respectively. If we restrict ourselves to the time period before the reflected wave from the beach reaches $x=\ell$, we may patch (27) and (28) by requiring continuous $\eta$ and u. Choosing $\nu =\sqrt{\ell }A_0$ and employing some elementary manipulation we obtain

$$\begin{aligned} I'-\frac{c_0}{4\ell }I=Y,\quad P=\frac{c_0}{4\ell }I. \end{aligned}$$

(29)

When the incident wave, $A_0 Y$, is specified this is a first order, linear ODE for I. Some Y shapes yield I as simple elementary function. However, from (19), we obtain (ignoring $c_1$) $Y(\tau )=\mathrm {sech}^2(kc_0\tau )$ and an I that may expressed by hypergeometric functions. It is then simpler to solve (29) with high accuracy by a Runge–Kutta method. Derivatives of any order may then be obtained, without quality loss, by repeated differentiation of (29). Combining the incident and reflected waves on the beach and invoking the limit $x\rightarrow 0^+$ we find the shoreline elevation

$$\begin{aligned} \eta _s=\frac{4A_0\ell }{c_0}I'', \end{aligned}$$

(30)

from which the maximum value, R, can be readily and accurately obtained by representing $I''$ by splines and differentiate. When the wavelength of the incident wave is small compared to $\ell$, the leading order approximation in (29) is $I'=Y$ and corrections may then be obtained by perturbation, or by manipulating the hypergeometric integral in the full solution. This yields

$$\begin{aligned} I'(\tau )\approx \mathrm {sech}^2(kc_0\tau )+\frac{1}{4k\ell }\left( 1+ \mathrm {tanh}(kc_0\tau )\right) ,\quad \frac{A_\ell }{A_0}\approx 1+\frac{1}{4k\ell },\quad \frac{R}{A_0}\approx \frac{8k}{3\sqrt{3}\alpha }\left( 1+\frac{\sqrt{3}}{8k\ell }\right) . \end{aligned}$$

(31)

where relative errors of order $(k\ell )^{-2}$ are implicit in all three relations and $A_\ell$ is the maximum surface elevation at $x=\ell$. We observe (see Fig. 8) that the bell shape of the solitary wave is transformed into a bell shape plus a long shelf when it passes the vertex. The height of the surface elevation is then also increased, from $A_0$ to $A_\ell$. Correspondingly, the maximum run-up height is increased as compared to the formula (21). Since linear and nonlinear theory yield the same R from initial conditions on the slope, the expression in (31) should apply when $A_0$ small, while the wave may be strongly nonlinear near the shore. The reflection from the apex is a long elevation. In principle, the incident wave as obtained above could be inserted into the theory outlined in Sect. 2.5.2. However, this is rather cumbersome and hardly worthwhile when the nonlinearities will not alter the maximum run-up height. If other properties are wanted, we would anyhow be better served by the numerical solutions of the Lagrangian NLSW.

An asymptotic solution is also readily obtained for long waves (solitary waves of low amplitudes) incident on steep beaches. The details are omitted, but the maximum run-up height becomes

$$\begin{aligned} \frac{R}{A_0}\approx 2+\frac{16}{9}(k\ell )^2, \end{aligned}$$

(32)

demonstrating that the run-up height approaches twice the amplitude in this limit, just as for the case with a plane beach.

If we instead of adding a horizontal channel section specified a paddle velocity, $A_0U/c_0$, on the slope, the ODE in (29) would be replaced by $I'-\frac{c_0}{2\ell }I=U$. If U was a pulse of finite duration, we would still have tails in the waves on the slope, as in the solution from (29).

Fig. 8

The linear surface elevation for the transmitted and reflected waves for $A_0=0.1$ and $\theta =3^\circ$. The curve marked “full” is computed with the Runge--Kutta solution for I, the one marked “pert.” corresponds to equation (31), while the curve market “opt.” displays the optical approximations in the sense that only the first term in the expression for $\eta$ from (31) is retained. The gray, vertical bar shows the position of the bottom apex