## 1. Introduction

[2] Frequency dispersion may affect tsunamis propagating over long distances in the ocean [*Kajiura and Shuto*, 1990]; this has been observed in the 2004 Indian Ocean [*Kulikov*, 2006; *Horrillo et al.*, 2006; *Grilli et al.*, 2007] and the 2011 Tohoku [*Løvholt et al.*, 2012] tsunami events. Significance of dispersive effects depends on wavelength, water depth and travel distance. In an ideal ocean of constant water depth, *Kajiura* [1963] defined a parameter

and suggested that dispersive effects must be considered in a tsunami if *P* < 4 , where *h* is still water depth, *d* is travel distance, and *λ* is the length of tsunami source along major propagation axis. In practice, *λ* may also be taken as the characteristic wavelength of the tsunami source [*Løvholt et al.*, 2012]. In a real ocean, uneven bathymetry may also cause dispersive transformation of long waves [*Madsen and Mei*, 1969]. Frequency dispersion is comparatively stronger along tsunami propagation directions and in deep water [*Dao and Tkalich*, 2007]. Neglecting dispersive effects may result in overestimated wave heights [*Horrillo et al.*, 2006; *Grilli et al.*, 2007] and coastal runups [*Ioualalen et al.*, 2007].

[3] The dispersive propagation of tsunamis can be simulated with the Boussinesq-type models [e.g.,*Horrillo et al.*, 2006; *Løvholt et al.*, 2008, 2012]. The Boussinesq-type models involve the solution of complicated mathematical equations, and require a lot of computational resources. In recent years, parallel computing technology has been implemented into the Boussinesq-type models, making them applicable to real-time operational tsunami simulations [e.g.,*Kirby et al.*, 2009]. Dispersive waves may also be simulated with the long wave models that include non-hydrostatic pressure [e.g.,*Stelling and Zijlema*, 2003; *Yamazaki et al.*, 2009], or match the physical dispersion with numerical dispersion [e.g., *Imamura and Shuto*, 1989; *Cho et al.*, 2007].

[4] At 17:40:10 UTC on 29 September 2009, a moment magnitude (Mw) 8.1 earthquake occurred in the outer trench slope near the northern end of Tonga Trench. The epicenter was at 15.51°S, 172.03°W, approximately 200 km south of the Samoan Islands, and 350 km northeast of Tonga. This event was an outer-rise earthquake caused by the bending of the Pacific Plate before it enters the Kermadec-Tonga Subduction Zone, and was characterized by a prevailing normal fault mechanism. Within two minutes, the first earthquake was followed by two major interplate thrust-faulting slips in the Kermadec-Tonga Subduction Zone with a combined Mw equal to 8.0 [*Beavan et al.*, 2010; *Lay et al.*, 2010]. A tsunami generated by this earthquake doublet caused severe damage to the Samoan Islands and Tonga. A maximum runup of 17.6 m was reported on Tutuila Island of American Samoa; on Tafahi Island of Tonga, a 22.4 m runup was observed [*Okal et al.*, 2010]. At DART (“Deep-ocean Assessment and Reporting of Tsunamis”) station 46407, nearly 7800 km northeast of the epicenter, a leading wave of 4.2 cm wave height was observed to be followed by a train of shorter trailing waves. Some trailing waves have periods shorter than 400 s. By employing the average water depth of 4100 m in the Pacific Ocean, the wavelength-to-depth ratio of these shorter waves is estimated to be below 20, suggesting that frequency dispersion may be important for this event.

[5] In this study, we investigate the dispersive effects of the 2009 Samoa tsunami through numerical simulations. Tsunami propagation in the Pacific Ocean is first simulated with a weakly nonlinear and dispersive Boussinesq model. The model equations are equivalent to those derived by *Nwogu* [1993], but described in geographic coordinates to consider the earth surface curvature [*Zhou et al.*, 2011]. A high-order finite-difference scheme is applied to solve these equations with a fourth-order accuracy in both time and space. The model has a linear dispersion relation accurate to*O*(*k*^{2}*h*^{2}), where *k*is wave number. This event is also simulated with a non-dispersive shallow-water-equations (SWE) model. In order to minimize numerical errors, the same high-order numerical scheme is also applied in the SWE model. The dispersive effects are evaluated by comparing the computational results between the two models.

[6] We further investigate the application of MOST (“Method of Splitting Tsunamis”) to simulate dispersive tsunami propagations. In MOST, the two-dimensional SWE are split into a pair of one-dimensional systems and solved independently through a finite-difference scheme that is accurate to the second order in space and first order in time [*Titov and Synolakis*, 1998]. *Burwell et al.* [2007] indicated that the numerical dispersion of MOST can be applied to match the physical frequency dispersion at operational resolutions. In their study, *Burwell et al.* [2007] demonstrated the effects of numerical dispersion in a series of numerical experiments with regular waves in constant water depth in one dimension. The application of MOST to dispersive waves in a complex water environment is yet to be investigated.