et al. Earth, Planets and Space (2017) 69:117
Should tsunami simulations include a
nonzero initial horizontal velocity?
Gabriel C. Lotto
, Gabriel Nava
and Eric M. Dunham
Tsunami propagation in the open ocean is most commonly modeled by solving the shallow water wave equations.
These equations require initial conditions on sea surface height and depth-averaged horizontal particle velocity or,
equivalently, horizontal momentum. While most modelers assume that initial velocity is zero, Y.T. Song and collabo-
rators have argued for nonzero initial velocity, claiming that horizontal displacement of a sloping seaﬂoor imparts
signiﬁcant horizontal momentum to the ocean. They show examples in which this eﬀect increases the resulting
tsunami height by a factor of two or more relative to models in which initial velocity is zero. We test this claim with
a “full-physics” integrated dynamic rupture and tsunami model that couples the elastic response of the Earth to the
linearized acoustic-gravitational response of a compressible ocean with gravity; the model self-consistently accounts
for seismic waves in the solid Earth, acoustic waves in the ocean, and tsunamis (with dispersion at short wavelengths).
Full-physics simulations of subduction zone megathrust ruptures and tsunamis in geometries with a sloping seaﬂoor
conﬁrm that substantial horizontal momentum is imparted to the ocean. However, almost all of that initial momen-
tum is carried away by ocean acoustic waves, with negligible momentum imparted to the tsunami. We also compare
tsunami propagation in each simulation to that predicted by an equivalent shallow water wave simulation with vary-
ing assumptions regarding initial velocity. We ﬁnd that the initial horizontal velocity conditions proposed by Song and
collaborators consistently overestimate the tsunami amplitude and predict an inconsistent wave proﬁle. Finally, we
determine tsunami initial conditions that are rigorously consistent with our full-physics simulations by isolating the
tsunami waves from ocean acoustic and seismic waves at some ﬁnal time, and backpropagating the tsunami waves
to their initial state by solving the adjoint problem. The resulting initial conditions have negligible horizontal velocity.
Keywords: Tsunami, Tsunami modeling, Tsunami generation, Oﬀshore earthquake, Subduction zone, Shallow water
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Tsunamis from earthquake sources pose a signiﬁcant
threat to coastlines around the world and have killed
hundreds of thousands of people in the last few decades.
e Earth science community often relies on numerical
models to better understand tsunami physics (e.g., Titov
and Synolakis 1998; Tanioka and Satake 1996; Kowalik
2003), perform hazard assessments (e.g., Geist and Par
sons 2006; Borrero et al. 2006; González et al. 2009), and
plan for early warning (e.g., Titov et al. 2005; Melgar and
Bock 2013; Behrens et al. 2010).
For numerical modelers, the tsunami life cycle is often
divided into three parts: generation, propagation, and
inundation (Bernard et al. 2006; Mori et al. 2011; Titov
and Gonzalez 1997; Watts et al. 2003; Song and Han 2011).
is paper is entirely focused on the ﬁrst two parts, though
modeling inundation poses its own unique set of chal
lenges and is far from being completely understood.
It is fairly straightforward to model tsunami propaga
tion in the open ocean, as reviewed by Satake (2002).
e simplest and most common way to do so is by using
the linearized shallow water wave equations, based on
assumptions of inviscid, incompressible water; small
amplitude waves; and horizontal wavelengths greater
than the ocean depth. In one horizontal dimension, the
Department of Geophysics, Stanford University, Stanford, CA, USA
Full list of author information is available at the end of the article