Analysing tsunami generation through rapid accelerated sea-floor rise: a theoretical and numerical investigation based on Boussinesq-type model

Abstract

This paper describes tsunami generation through topographic rise with rapid acceleration. The phenomenon and the impact of the rapid bottom acceleration on tsunami generation were investigated both theoretically and numerically, using a Boussinesq-type model that accounts for vertical acceleration on the bottom. Analytical solutions for water depth and velocity were derived for scenarios where the bottom elevation increases according to a power law over time. The derived solutions showed good agreement with numerical simulation results. In the hydrostatic model, the solutions for both variation in water depth and velocity approached zero as the acceleration of the bottom elevation increases. In contrast, the Boussinesq-type model demonstrated that the solutions for velocity distribution changed with upward acceleration, while the solutions for water depth converged to a certain finite waveform. These findings reveal that the vertical acceleration on the bottom significantly influences the magnitude of the generated tsunami, especially affecting the velocity distribution.

Keywords:

• analytical solution
• Boussinesq-type model
• numerical simulation
• open channel flow: tsunami

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notation

$a\left(t\right)$	=	function of non-dimensional time $t$ (–)
$A_j\left(s\right)$	=	coefficients for each term in the polynomial expansion in Equation (12) $(j\ge 0)$
$C_f$	=	dimensionless constant measuring roughness of channel walls (–)
$\tilde{g}$	=	gravity acceleration (m s–2)
$\tilde{h}$	=	water depth (m)
$\widetilde{h_0}$	=	initial water depth (m)
$H_j\left(x\right)$	=	the Hermite polynomial
$\tilde{l}$	=	length scale in $\tilde{x}$-direction, which depends on the property of the bottom topography (m)
$L[z_b]$	=	the Laplace transform of $z_b$
$L[{\delta }_h]$	=	the Laplace transform of ${\delta }_h$
$n$	=	natural number power on time for the function $a\left(t\right)$ ($n\ge 3$)
$\tilde{t}$	=	time (s)
$t$	=	non-dimensional time (–)
$t_p$	=	non-dimensional computation period (–)
$\tilde{u}$	=	depth-averaged velocity (m s–1)
$\tilde{x}$	=	spatial coordinate (m)
$x$	=	non-dimensional spatial coordinate (–)
$\widetilde{z_b}$	=	bottom elevation (m)
$z_b$	=	non-dimensional bottom elevation (–)
$z_b(0,x)$	=	non-dimensional initial bottom elevation (–)
$z_b(t_p,x)$	=	non-dimensional bottom elevation at $t$ = $t_p$ (–)
$\widetilde{z_s}$	=	water surface elevation (m)
$z_s$	=	non-dimensional water surface elevation (–)
$\alpha$	=	coefficient for function $a\left(t\right)$ (–)
$\widetilde{{\delta }_h}$	=	perturbation of water depth (m)
${\delta }_h$	=	non-dimensional analytical solution of water depth (–)
${\delta }_h(0,x)$	=	initial perturbation of the water depth (–)
${\delta }_{h0}$	=	non-dimensional hydrostatic solution of water depth (–)
${\delta }_{\mathrm{hj}}$	=	additional parts of non-dimensional analytical solution of water depth due to vertical acceleration across depth ($j\ge 1$) (–)
${\delta }_{h_{b}j}$	=	additional parts of non-dimensional analytical solution of water depth due to vertical acceleration on bottom ($j\ge 0$) (–)
$\widetilde{{\delta }_u}$	=	perturbation of velocity (m s–1)
${\delta }_{u0}$	=	non-dimensional hydrostatic solution of velocity (–)
${\delta }_{\mathrm{uj}}$	=	additional parts of non-dimensional analytical solution of velocity due to vertical acceleration across depth ($j\ge 1$) (–)
${\delta }_{u_{b}j}$	=	additional parts of non-dimensional analytical solution of velocity due to vertical acceleration on bottom ($j\ge 0$) (–)
$\lambda$	=	non-dimensional length scale (–)