A large eruption of the Hunga Tonga-Hunga Haʻapai volcano in Tonga on January 15, 2022, caused devastating disasters in nearby areas. The eruption generated air waves coupled with seawater waves propagating through the Pacific. Tsunamis, long-waves in the ocean, generated by the air wave from the eruption were observed in tide gauges and ocean-bottom pressure sensor network along the Japan Trench (S-net) in Japan. The network, which includes 150 pressure sensors connected by a cable with 30-km intervals, is operated by the National Research Institute for Earth Science and Disaster Resilience (NEID) in Japan (Uehira et al. 2012; Kanazawa 2013). A maximum tsunami amplitude of 1.2 m was observed at Amami in Japan, and thus, a tsunami warning was issued by the Japan Meteorological Agency.

Similar air–sea waves had previously been generated by the Krakatoa volcano eruption in 1883 (Harkrider and Press 1967), and the authors indicated that the air wave propagated at a speed of 312 m/s as the fundamental gravity mode in the atmosphere–ocean system. Nishida et al. (2014) indicated that the atmospheric pressure waves propagating at the acoustic speed were Lamb waves. This kind of air–sea coupled waves, which are often called to meteotsunamis, has already been studied without volcanic eruptions (Hibiya and Kajiura 1982; Fukuzawa and Hibiya 2020; Kubota et al. 2021). Saito et al. (2021) suggested that meteotsunamis are amplified when the velocity of the air waves is similar to that of the tsunami waves. This is called the Proudman resonance effect (Williams et al. 2021). In the case of the tsunami caused by the air wave from the volcanic eruption, the tsunami propagating near a deep trench should be amplified. Sekizawa and Kohyama (2022) investigated the sea-surface fluctuations by the 2022 Tonga eruption based on a one-dimensional computation.

In this study, the tsunami waves observed at ocean-bottom sensors of S-net in Japan were modeled by the air waves generated by the 2022 Tonga eruption. We discuss the characteristics of tsunami propagation, including the tsunami amplification near the Japan Trench, which has a depth of approximately 8 km.

### Data and method

Ocean bottom pressure data observed at 106 out of 150 sensors of S-net (National Research Institute for Earth Science and Disaster Resilience, 2019) were downloaded (Fig. 1). The observed data are filtered in the periods between 100 and 3600 s using the bandpass filter in Seismic Analysis Cord (SAC) developed by Goldstein et al. (2003) (Additional file 1: S1).

The governing equations to be solved by the numerical simulation are briefly explained. With linear approximation, Euler`s momentum equation (Dean and Dalrymple 1991) can be simplified as:

$$\frac{\partial u}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial x},$$

(1)

$$\frac{\partial v}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial y},$$

(2)

$$\frac{\partial w}{\partial t}=-g-\frac{1}{\rho }\frac{\partial p}{\partial z},$$

(3)

where *u*, *v* are the water velocities in the horizontal x, and y-directions, and *w* is the water velocity in the vertical z-direction (upward positive); *p* is the pressure, *g* is gravity acceleration, and \(\rho\) is the water density. In longwave theory, the vertical velocity is negligibly small compared with the horizontal velocities: the horizontal velocities in *x*- and *y*-directions (*u* and *v*) can be approximated to depth-averaged velocities (*U* and *V*). Using the former condition, Eq. (3) becomes

$$\frac{\partial w}{\partial t}=-g-\frac{1}{\rho }\frac{\partial p}{\partial z}=0.$$

(4)

Then, pressure *p* is \(p=-\rho gz+c\) based on a vertical integration of Eq. (4), where *c* is a function of *x*, *y* and *t*. As a boundary condition, pressure (*p*) is equal to be an atmospheric pressure (*p*_{0}) at the ocean surface (*z* = *h*). Pressure, *p*, then becomes \(p=\rho g\left(h-z\right)+{p}_{0}\). A value of *p*_{0} is an input to compute air–sea coupled waves. Furthermore, integrating Eqs. (1) and (2) along the depth and using the depth-averaged velocities and pressure, depth-integrated momentum equations in a two-dimensional horizontal plane are obtained as:

$$\frac{\partial U}{\partial t}=-g\frac{\partial h}{\partial x}-\frac{1}{\rho }\frac{\partial {p}_{0}}{\partial x},$$

(5)

$$\frac{\partial V}{\partial t}=-g\frac{\partial h}{\partial y}-\frac{1}{\rho }\frac{\partial {p}_{0}}{\partial y}.$$

(6)

Using a depth integration of the mass conservation equation, the continuity equation becomes

$$\frac{\partial h}{\partial t}=-\frac{\partial \left(d*U\right)}{\partial x}-\frac{\partial \left(d*V\right)}{\partial y},$$

(7)

where *h* is the wave height and *d* is the still water depth. The momentum Eqs. (5) and (6) for air–sea coupled waves and continuity Eq. (7) were numerically solved using a staggered grid system with an input of the atmospheric pressure gradient at each time step. The grid sizes of the numerical computation were set at 1.5 km in both *x* and *y-*directions.

The atmospheric pressures observed at approximately 1300 stations in Japan (Weathernews Inc., https://global.weathernews.com/news/16551/, 2022) showed that the pressure pulse (a peak amplitude of approximately 2 hPa, and the duration of 20–15 min) passed through Japan from southeast to northwest with a strike of − 44°. Figure 2 shows three snapshots of the amplitude distribution of pressure changes observed at those stations at 20:35, 20:45, 20:55 (Japan Standard Time) on the 15th of January. The shape of the pressure pulse was assumed to be half the wavelength of a sine wave. The half-wavelength was set to 300 km, which corresponded to a duration of 16 min using the speed of 312 m/s for the pressure pulse. The peak amplitude was set to 2 hPa. First, numerical computation in one dimension with a constant ocean depth of 5500 m was carried out to obtain the steady state of the air–sea coupled initial wave. The pressure pulse and the steady-state initial wave are entered into the two-dimensional computational domain from the low boundary along the *x*-axis (Fig. 3). The pressure pulse is continuously propagated in the *y*-direction with a constant velocity of 312 m/s. The bathymetry was rotated 44° clockwise (Figs. 1, 3) to match a strike of the pressure wave from Tonga. Finally, the observed pressure changes at ocean bottom (*p*_{b}) are calculated from the pressure changes due to ocean waves (*p*_{w}) and the atmospheric pressure (*p*_{0}) pulse as described by Kubota et al. (2021) and Saito et al. (2021):

$${p}_{\mathrm{b}}={p}_{\mathrm{w}}+{p}_{0 }\,\mathrm{where}\, {p}_{\mathrm{w}}=\rho gh.$$

(5)