Maximum run-up produced by tsunami wave trains entering bays of variable cross section

Investigation of the behavior of various types of tsunami wave trains entering bays is of practical importance for coastal hazard assessments. Furthermore, a mathematical algorithm for quick analysis of the run-up in bays for a large number of incident wave scenarios is also a practical need for tsunami hazard assessments. The linear shallow water equations admit two types of solutions inside an inclined bay with parabolic cross sections: energy-transmitting modes and modes with spatial decay towards the inland tip of the bay. In the low-frequency limit there is only one mode susceptible of transmitting energy to the inland tip. A full solution for the run-up requires taking into account these two types of modes and the scattered field outside, leading to mathematical complications. However, in the long wave limit, this complication can be avoided if one imposes the free surface at the bay mouth being equal to twice the disturbance associated with the incident wave in the open sea. The run-up produced by the solution obtained from this Dirichlet boundary condition can be easily calculated using a series of images. In this model no energy is allowed to escape from the bay; therefore the error arising from the simplification of the boundary condition at the bay mouth grows with time. Nevertheless the maximum run-up occurs before this error becomes significant. If the standard deviation of a Gaussian-shaped incident wave is 8 times the square root of the width of the bay, then this simple solution overestimates the first maximum of the run-up only by 12% compared to the exact solution calculated by means of an integral equation. This overestimation is partly due to the fact that the Dirichlet boundary conditions violate the continuity of the fluxes at the bay mouth. The solution associated with the Dirichlet boundary condition is perturbed in order to match fluxes inside and outside of the bay. The height of the first maximum of the run-up coming from the perturbation theory is in excellent agreement with the “exact” solution. This perturbation theory can also be applied to narrow bays with an arbitrary cross section as long as their depth does not change significantly in the longitudinal direction. The method developed here can also be used to calculate maximum run-up in noninclined bays of arbitrary cross section.

Figure 1

The bathymetry of the laterally infinite case featuring a coastal wall, a shallow shelf connecting to an infinite ocean with a flat bottom. Here $x=0$ is the shoreline, and $x=x_0$ is the position of the toe. The depth of the shallow shelf is $H_1$, and the depth of the open sea is $H_0$.

Figure 2

Incident wave approaching a shallow shelf of infinite width. (a) The continuous curve is the normalized free surface disturbance at $x=x_0$ (the toe of the slope) for an incident wave given by (2). The frequency, $\omega$, of the incident wave is $\sqrt{g{H}_1}\pi /\left(2x_0\right)$ and $x_0/H_0=20$. The ratio, $H_0/H_1$, is 200. The broken curve is twice the amplitude of the incident wave at the toe. (b) The continuous curve is the time evolution of the normalized run-up for an incident wave of Gaussian shape given by $I_{0}exp\left[-4\frac{g{H}_1}{x_0^2}{\left(t+\frac{x-x_0}{\sqrt{g{H}_0}}\right)}^2\right]$ for the same bathymety as (a). The dashed curve is the decay envelope given by $(r_{\max }/I_0)exp\left[-|\text{Im}{\omega }_k\left|\right(t-t_{\max }\right)]$, where ${\omega }_k$ is a natural frequency of the radiating mode given in Eq. (5). Note that the imaginary part of the frequencies of all modes is equal. For both panels the width is infinite.

Figure 3

Run-up generated by a Gaussian incident wave given by $exp\left\{-{\left[(x+\sqrt{g{H}_0})(t-T/2)/\left(0.4x_0\right)\right]}^2\right\}$ entering a rectangular bay of aspect ratio $x_0/\left(2Y_0\right)=10$. The continuous curve is the “exact” solution of integral equation, dashed curve is $r^{\left(0\right)}\left(t\right)$ (undisturbed solution of Dirichlet problem), and the stars are $r^{\left(0\right)}\left(t\right)+r^{\left(1\right)}\left(t\right)$. The timescale $t_0$ is $Y_0/\sqrt{gH}$, and $T$ is the duration of sampling. Maximum run-up associated with $r^{\left(0\right)}$ and $r^{\left(0\right)}+r^{\left(1\right)}$ is 4.0 and 3.6, respectively. The “exact” maximum is 3.52.

Figure 4

(a) The inclined bay with a parabolic cross section seen from above. The broken curves are contours of equal depth given by $z_k=-k{H}_{\max }/8$ with $k=1,2,\cdots ,7$. Any point $(x,y,z)$ at the bottom of the bay satisfies $z=-\alpha x+y^2/y_0$ with $H_{\max }=Y_0^2/y_0$. (b) The bay seen from the side depicting the plane, $y=0$. The slope, $\alpha$, is $H_{\max }/x_0$.

Figure 5

The positive continuous curve is the initial profile of the incident wave given by $exp\left[-\frac{1}{2}{\left(\frac{x-x_1}{a}\right)}^2\right]$. The incident wave is progressing in an inclined bay with a parabolic cross section. The bottom of the bay is $z=-\alpha x+y^2/y_0$. The depression is the wave profile at instant $t=2{\int }_0^{x_1}\frac{dx}{\sqrt{2g\alpha x/3}}$ (the denominator in the integrand is the wave speed). The broken curve is the wave profile at instant $t$ for the same incident wave progressing over a sloping bay with a rectangular cross section whose slope is $\frac{2}{3}\alpha$. In both geometries the averaged depths for a given $x$ are equal. The values of parameters are $x_1=x_0/4,a=0.02x_0$ ($x_0$ is an arbitrary length scale). Note that for the case of a sloping bay with a rectangular cross section the incident wave starts to be reflected before reaching $x=0$.

Figure 6

The continuous and the broken curves are the dispersion relations for an infinite channel with a parabolic cross section. Broken curves indicate a purely imaginary wave vector. The waves associated with such wave vectors decay exponentially in the $-x$ direction because they are proportional to $exp\left(\right|k\left|x\right)$. Continuous and broken curves were obtained using the routine eigh (see the Appendix). The equation of the bottom of the channel is $z=-H_{\max }(1-y^2/Y_0^2)$ where $H$ is its maximum depth and $Y_0$ is its half width. The stars are the dispersion relation given by $\omega =\sqrt{g2H_{\max }/3}\left|k\right|$ where $2H/3$ is the average depth of the parabolic channel. The circles are the dispersion relation obtained using Eq. (2.51) from Ref. [17]. The dots (two branches) are the dispersion relation obtained from (40).

Figure 7

The continuous curve represents the eigenfunction, $f_0(y,\omega )$, for $\omega Y_0/\sqrt{g\alpha x_0}=1$. See Eq. (30) for the definition of $f_{{\kappa }_0}$. The dashed and the dash-dotted curves are, respectively, $f_{{\kappa }_1}$ and $f_{{\kappa }_2}$. The function $f_{{\kappa }_0}$ is associated with the oscillatory solution in the $x$ direction. Note that $f_{{\kappa }_0}\left(y\right)$ is almost constant.

Figure 8

Absolute value of Fourier transform of a run-up divided by that of the incident wave is displayed as a function of the nondimensional frequency, $\omega Y_0/\sqrt{gH}$. The length of the inclined parabolic channel is $x_0=10Y_0$ ($Y_0$ is half width at the mouth of the bay).

Figure 9

Continuous curves are an “exact” run-up for an incident Gaussian waves calculated using (43). In (a) the incident wave in the open sea is $exp\left\{-{[x+\sqrt{g{H}_0}(t-T/2)-x_0]}^2{x}_0^{-2}\right\}$, and in (b) the narrower Gaussian is given by $exp\left\{-{[x+\sqrt{g{H}_0}(t-T/2)-x_0]}^2{\left(0.4x_0\right)}^{-2}\right\}$. The ratio $x_0/\left(2Y_0\right)$ is 10 for both cases. Here $x_0$ is the length of the bay, and $2Y_0=2\sqrt{\alpha x_0{y}_0}$ is the width of the mouth of the bay [see (15) for the geometry of the bay]. The timescale $t_0$ is given by $Y_0/\sqrt{g{H}_0}$. The broken curves are the run-up associated with the Dirichlet problem $\eta (t,x_0^{-})=2{\eta }_{\mathrm{inc}}^{\mathrm{open}}(t,x_0^{+})$ [see (21)]. Parameter $T$ is the duration of the sampling of the incident wave. The solution associated with the Dirichlet boundary condition overestimates the maximum run-up by 1% in (a) and 12% in (b). The solutions associated with the Dirichlet boundary conditions are not damped. Note that two-way travel time along this bay are $97.98t_0$ =$2\sqrt{6}{x}_0/\sqrt{g{H}_0}$.

Figure 10

The continuous curves are the function $\mathrm{\Gamma }\left(\tau \right)$ displayed as a function of nondimensional time $\tau /t_0$ with $t_0=Y_0/\sqrt{g{H}_0}$. See (64) for the definition of $\mathrm{\Gamma }$. The horizontal dot-dashed lines are $\sqrt{\langle H\rangle /H_0}$ [see (67)]. The dashed curves are an asymptotic approximation given by (68). For a triangular bay the depth is $H\left(y\right)=H_0(1-|y|/Y_0)$ (depth independent of $x)$. For a rectangular bay the depth is uniform with $H=H_0$.

Figure 11

Run-up generated by a Gaussian incident wave given by $exp\{-{[(x+\sqrt{g{H}_{\max }})(t-T/2)/\left(0.4x_0\right)]}^2\}$ entering an inclined parabolic bay of aspect ratio $x_0/\left(2Y_0\right)=10$. The continuous curve is the “exact” solution found using (43), the broken curve is $r^{\left(0\right)}\left(t\right)$ (undisturbed solution of the Dirichlet problem), and the stars are $r^{\left(0\right)}\left(t\right)+r^{\left(1\right)}\left(t\right)$. The timescale $t_0$ is $Y_0/\sqrt{g{H}_{\max }}$, and $T$ is duration of the sampling.