Modified Kadomtsev–Petviashvili equation for tsunami over irregular seabed

Abstract

We derive an asymptotic equation governing the trans-ocean propagation of tsunami from source to the continental shelf. Focus is on disturbances originated from a slender fault of finite length. The variable sea depth is assumed to consist of a slowly varying mean and random fluctuations. The method of multiple scales is used to derive a Kadomtsev–Petviashvili equation with variable coefficients. Modifications by one- and two-dimensional random irregularities are shown to affect the wave speed, dissipation and additional dispersion. The result can be used to facilitate physical insight with modest numerical efforts.

Appendix: Equivalence of random forcing term in 1D

For constant mean depth ($h=1$), Mei and Li (2004) found the random forcing term to be,

$$\begin{aligned} \left\langle b(x) \frac{\partial \zeta ^{(1)}}{\partial x} \right\rangle= & {}\,\varGamma (0) \frac{\partial \zeta ^{(0)}}{\partial \sigma } +\frac{1}{4}\frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2} \int _{-\infty }^\infty \varGamma (\xi ) {\text{d}}\xi \\ &+ \frac{1}{8} \int _{-\infty }^\infty \varGamma \left( \frac{\sigma -\sigma ^{\prime}}{2}\right) \frac{\partial ^2 \eta _0}{\partial \sigma ^{{\prime}2}} \, {\text{d}}\sigma ^{\prime} +\frac{1}{4}\int _{-\infty }^\infty P\left( \frac{\sigma -\sigma ^{\prime}}{2} \right) \frac{\partial ^3 \eta _0}{\partial \sigma ^{{\prime}3}} \, {\text{d}} \sigma ^{\prime} \end{aligned}$$

(7.1)

where $P(\xi )$ is

$$\begin{aligned} P(\xi ) = \int _{|\xi |}^\infty \varGamma (u) \, {\text{d}}u, \end{aligned}$$

(7.2)

In this form the physical influence of random scattering is clear. The first term represents reduction in phase speed. The second and the third terms give rise to dissipation, while the last term augments dispersion. Let us prove that it is equivalent to (4.16).

Note first that

$$\begin{aligned} \frac{{\text{d}} P(\xi )}{{\text{d}} \xi } = -\frac{d |\xi |}{{\text{d}} \xi } \varGamma (\xi ) = -\text {sgn}(\xi ) \varGamma (\xi ). \end{aligned}$$

(7.3)

Let us start from (4.16)

$$\begin{aligned}&\int _0^\infty \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi \nonumber \\&\quad =\frac{1}{2} \int _0^\infty \bigl [ 1 + \text {sgn}(\xi ) \bigr ] \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi \nonumber \\&\quad = \frac{1}{2} \int _{-\infty }^\infty \bigl [ 1 + \text {sgn}(\xi ) \bigr ] \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi \nonumber \\&\quad = \frac{1}{2} \int _{-\infty }^\infty \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi - \frac{1}{2}\int _{-\infty }^\infty \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}} P(\xi ) \nonumber \\&\quad = \frac{1}{2} \int _{-\infty }^\infty \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi +\frac{1}{2}\int _{-\infty }^\infty P(\xi ) \frac{\partial ^3 \zeta ^{(0)}}{\partial \sigma ^2 \partial \xi }(\sigma +2\xi ) \, {\text{d}} \xi \nonumber \\&\quad = \frac{1}{2} \int _{-\infty }^\infty \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi +\int _{-\infty }^\infty P(\xi ) \frac{\partial ^3 \zeta ^{(0)}}{\partial \sigma ^3}(\sigma +2\xi ) \, {\text{d}} \xi . \end{aligned}$$

(7.4)

Equation (4.16) becomes

$$\begin{aligned} \left\langle b(x) \frac{\partial \zeta ^{(1)}}{\partial x} \right\rangle= &{}\,\varGamma (0) \frac{\partial \zeta ^{(0)}}{\partial \sigma } +\frac{1}{2}\frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2} \int _0^\infty \varGamma (\xi ) {\text{d}}\xi +\frac{1}{4} \int _{-\infty }^\infty \varGamma (\xi ) \frac{\partial ^2 \zeta ^{(0)}}{\partial \sigma ^2}(\sigma +2\xi ) \, {\text{d}}\xi \nonumber \\&+\frac{1}{2}\int _{-\infty }^\infty P(\xi ) \frac{\partial ^3 \zeta ^{(0)}}{\partial \sigma ^3}(\sigma +2\xi ) \, {\text{d}} \xi . \end{aligned}$$

(7.5)

By letting $\sigma ^{\prime} = \sigma +2\xi$, (7.1) is recovered.