Talk:Tidal bore dynamics

Semi-empirical model

Schematic of Tsunami uprush.

A semi-empirical approach for estimating the area that is flooded by a tsunami was developed by Kriebel et al. (2017). This model is based on the energy balance of a tsunami bore that is rushing inland along a cross-shore transect [math]x[/math]. The location [math]x=x_R[/math] where the initial energy of the tsunami bore eventually dissipates completely marks the greatest inland flooding distance. The following assumptions are made. The initial energy per unit shoreline length of the tsunami bore [math]E_0[/math] is the sum of the potential energy and the kinetic energy, [math]E_0 = \rho g H_0 + \frac{1}{2} \rho u_0^2 [/math], where [math]H_0[/math] is the height of the tsunami bore at the shoreline and [math]u_0[/math] the depth averaged initial current velocity. After travelling some distance inland, the bore energy is [math]E(x) = \rho g h(x) + \frac{1}{2} \rho u^2(x)[/math], where [math]h(x)[/math] is the water level relative to the shoreline level and [math]u(x)[/math] the local current velocity. The water level [math]h(x)[/math] is the sum of the land elevation [math]z(x)[/math] and the water depth [math]D(x)[/math]. Energy loss is due to friction, which depends linearly on the current velocity squared with a friction coefficient [math]C_D(x)[/math]. The energy balance equation reads

[math]\dfrac{1}{\rho}\dfrac{d \, E}{dx} = g \, m(x) + g \dfrac{d \, D(x)}{dx} + \frac{1}{2} \dfrac{d \, u^2(x)}{dx} = - C_D(x) \, \dfrac{u^2(x}{D(x)} \, , \qquad (1)[/math]

where [math]m(x) \equiv dz/dx[/math] is the local bed slope, which is assumed to be positive. For solving this equation, the current velocity [math]u(x)[/math] must be known. Therefore the assumption is made that the current velocity is a known fraction of the bore celerity. This fraction decreases to the maximum tsunami intrusion distance [math]x_R[/math] according to [math]\, u(x) = \alpha \, c \, \sqrt{1 – \frac{x}{x_R}} \, [/math]. The bore celerity is given by [math]c = \sqrt{g\, D(x)}[/math]. Typical values of the Froude parameter [math]\alpha[/math] are close to 1. The energy balance equation then becomes

[math]\dfrac{d}{dx} \Big( [1+\frac{1}{2} \alpha \, (1 – \frac{x}{x_R})] D(x) \Big) + \dfrac{m(x)}{g} + C_D (x) \, \alpha \, (1 – \frac{x}{x_R}) = 0 \, . \qquad (2)[/math]

This equation can be numerically integrated to find the tsunami intrusion limit [math]x_R[/math], at the condition that the initial bore height [math]H_0[/math], the Froude parameter [math]\alpha[/math] and the friction coefficient [math]C_D(x)[/math] are known. The friction coefficient varies from place to place depending on the presence of obstacles such as trees, buildings and other relief elements. Numerical integration of Eq. (2) is simple, but the results heavily depend on correctly estimating the friction coefficient [math]C_D[/math] and the Froude parameter [math]\alpha[/math]. .

The semi-empirical model based on the energy balance equation (1) is questionable. A more correct energy balance equation should consider the damping of the bore energy when travelling inland. The energy flux of a bore travelling with celerity [math]c[/math] is given by [math]F=c\, E[/math], where [math]\, E = \rho \Big[ \frac{1}{2} D u^2 + g \int_z^{z+D} z'dz' \Big] \,[/math] is the bore energy transported inland per unit time. The energy dissipated per unit time during uprush is given by [math]\dfrac{dF}{dx}= - \rho C_D u^3[/math]. The dimensionless coefficient [math]C_D[/math] represents the dissipation due to bore turbulence and due to friction along the ground and obstacles. The energy balance equation thus reads:

[math]\dfrac{d}{dx} \Bigg[ c\, D \, \Big( \frac{1}{2} u^2 + g (\frac{1}{2} D + z) \Big) \Bigg] + C_D \, u^3 = 0 \, . \qquad (3)[/math]

This equation can be integrated in the same way as Eq. (1) with the assumptions [math]c = \sqrt{g\, D(x)} \,[/math] and [math]\, u(x) = \alpha \, c \, \sqrt{1 – \frac{x}{x_R}} \, [/math].

References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2025): Talk:Tidal bore dynamics. Available from http://www.coastalwiki.org/wiki/Talk:Tidal_bore_dynamics [accessed on 5-12-2025] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J