Tunamis on a deep open sea and on a sloping beach -- a mathematical theory
Tadayoshi Kano
TL;DR
The paper asks how large, long-wavelength shallow-water waves (Airy-type) on a deep ocean accelerate into violent tunamis on sloping coasts. It builds a Friedrichs-expansion-based, dimensionless framework for two-dimensional water waves, introducing tunamis variables $P$ and $Q$ with $\gamma^2=\Gamma-b(x)$ to couple surface dynamics with seabed slope $b_x$. The Tunamis equations $P_t + (\gamma+u+\frac{b_x}{P_x})P_x = 0$ and $Q_t - (\gamma-u-\frac{b_x}{Q_x})Q_x = 0$ reveal that when $P_x$ or $Q_x$ approach zero near the shore, the $\frac{b_x}{P_x}$ term induces instantaneous, unbounded propagation speeds, explaining inland/outward blow-up. The work links the Tunamis system to Airy shallow-water theory, clarifies the mechanism of near-shore intensification, and discusses implications for coastal hazard assessment and early-warning strategies, grounded in rigorous hyperbolic-structure analysis of the depth-variant wave problem.
Abstract
Approaching a sloping beach, shallow water surface waves of Airy get suddenly $ +\infty$ or $ -\infty$ propagation speed at the point of surface $x = x_0$, say, where the tangent $\varGamma_x$ of the surface $y = \varGamma$ "coincide" with that $b_x$ of the water-bottom $y = b(x)$, losing the cruising sound speed of propagation so high on a deep open sea. That is, the tunamis gain instantaneously a $ +\infty$ propagation speed just before the crest as $(\varGamma_x - b_x)(x) \to +0$, $x \to x_0\!-\!0$ , and a $ -\infty$ propagation speed just after the trough as $(\varGamma_x - b_x)(x) \to -0$, $x \to x_0\!-\!0$. We would have thus a big crush between the crest rushing forward and the trough rushing backward. This is a mathematical structure of tunamis "on" a sloping beach, in particular.