1 Landslide Tsunami Steven N. Ward Institute of Geophysics and Planetary Physics University of California at Santa Cruz Santa Cruz, CA 95064 USA Abstract In the creation of “surprise tsunami”, submarine landslides head the suspect list. Moreover, improving technologies for seafloor mapping continue to sway perceptions on the number and size of surprises that may lay in wait offshore. At best, an entirely new distribution and mag- nitude of tsunami hazards has yet to be fully appreciated. At worst, landslides may pose seri- ous tsunami hazard to coastlines worldwide, including those regarded as immune. To raise the proper degree of awareness, without needless alarm, the potential and frequency of landslide tsunami have to be assessed quantitatively. This assessment requires gaining a solid under- standing of tsunami generation by landslides, and undertaking a census of the locations and extent of historical and potential submarine slides. This paper begins the process by offering models of landslide tsunami production, propagation and shoaling; and by exercising the the- ory on several real and hypothetical landslides offshore Hawaii, Norway and the United States eastern seaboard. I finish by broaching a line of attack for the hazard assessment by building on previous work that computed probabilistic tsunami hazard from asteroid impacts. 1. Introduction far larger than expected given the earthquake size. In contemplating the great number and im- Earthquakes generate most tsunami. Rightly so, mense extent of seabed failures known to have tsunami research has concentrated on the hazards frequented recent geological history, the biggest posed by seismic sources. The past decade how- surprise just may arrive without any precursory ever, has witnessed mounting evidence of tsu- seismic warning at all -- a tsunami sprung from a nami parented by submarine landslides. In fact, spontaneous submarine landslide. submarine landslides have become prime sus- pects in the creation of “surprise tsunami” from The geography of earthquakes only casually small or distant earthquakes. As exemplified by resembles the geography of submarine landslides. the wave that devastated New Guinea's north Tsunami excitation mechanisms between earth- coast in 1998 (Tappin, et. al., 1999; Geist, 2000), quakes and landslides differ substantially too. surprise tsunami can initiate far outside of the Accordingly, an entirely new distribution and epicentral area of an associated earthquake, or be magnitude of tsunami hazards has yet to be fully V1.8 11/20/00 Accepted for Publication, Journal of Geophysical Research. Ward: Landslide Tsunami 2 appreciated. Landslides may pose perceptible amplitude to wavelength is much less than one, tsunami hazards to areas regarded as immune – but it does not otherwise restrict tsunami wave- the Gulf of Mexico, the North Sea, or the eastern length. In particular, I do not make a long seaboard of the United States. Many of these ar- wave/shallow water assumption. eas, unaccustomed to earthquakes and earthquake tsunami, lie on flat and passive continental mar- Linear wave theory may not hold in every gins. A 5-meter sea wave striking there would situation, but in the application here, the assump- over-run far more territory than a similar wave tion of linearity presents far more advantages than hitting a rugged shore, such as California's north draw-backs. Foremost among the advantages is coast. superposition. Superposition means that complex tsunami waveforms can be spectrally decom- To raise a proper degree of awareness, without posed, independently propagated, and then recon- needless alarm, the potential and frequency of structed. Superposition also means that complex landslide tsunami must be assessed quantitatively. landslide sources can be synthesized from simple To make this assessment, geophysicists must first ones. For all but monster landslides, linear theory garner an understanding of tsunami generation by breaks down only during the final stage of tsu- landslides equal to what they know about tsunami nami shoaling and inundation when the wave generation by earthquakes. Second, geologists height exceeds the water depth (Synolakis, 1987). and oceanographers must undertake a census of Taking waves the last mile to the beach does re- the locations and sizes of historical and potential quire full hydrodynamic calculation; however, if submarine landslides, and estimate their occur- one is willing to stand off a bit in water no shal- rence rates. Setting the stage for a quantitative lower than the height of the incoming waves, then hazard analysis, this paper begins to tackle step linear theory should work fine. If nothing else, #1 above by offering elementary models of land- linear theory fixes a reference to compare predic- slide tsunami production, propagation and shoal- tions from various non-linear approaches. ing; and by exercising the theory on several real and hypothetical cases. In classical theory, the phase c(w), and group u(w) velocity of surface gravity waves on a flat 2. Tsunami characteristics under linear ocean of uniform depth h are theory. gh tanh[k( )h] c( ) = (1) The tsunami computed in this article derive k( )h from classical theory. Classical tsunami theory and envisions a rigid seafloor overlain by an incom- é 1 k( )h ù u( ) = c( ) + (2) pressible, homogeneous, and non-viscous ocean ëê 2 sinh[2k( )h]úû subjected to a constant gravitational field. Fur- thermore, I confine attention to linear theory. Here, g is the acceleration of gravity (9.8 m/s2) Linear theory presumes that the ratio of wave and k(w) is the wavenumber associated with a sea Ward: Landslide Tsunami 3 wave of frequency w. Wavenumber connects to the uplift. Tsunami trains (4a) are dominated by wavelength l (w) as l (w)=2p/k(w). Wavenumber wavenumbers in the span where F(k) is greatest. also satisfies the dispersion relation The peak of F(k) corresponds to the characteristic dimension of the uplift. Usually, large- 2 = gk( )tanh[k( )h] (3) dimensioned landslides produce longer wave- length, hence lower frequency tsunami than For surface gravity waves spanning 1 to 50,000 s small-dimensioned sources. period, Figure 1 plots c(w), u(w), and l (w). These quantities vary widely, both as a function of b) The 1/cosh(kh) term low-pass filters the ocean depth and wave period. source spectrum F(k). (1/cosh(kh)® 1 when kh® 0, and 1/cosh(kh)® 0 when kh®¥ , so the 3. Excitation of Tsunami by Sea Floor up- filter favors long waves.) Because of the low-pass lift filter effect of the ocean layer, only wavelengths of the uplift source that exceed three times the Suppose that the seafloor at points r0 uplift in- ocean depth (i.e. kh=2ph/l <»2) contribute much bot stantaneously at time t (r0) by amount u z (r0). to a tsunami. Under classical tsunami theory in a uniform ocean of depth h, this bottom disturbance stimu- c) The exponential term in (4a) contains all of lates surface tsunami waveforms (vertical com- the propagation information including travel time, ponent) at observation point r=xxˆ +yyˆ and time t geometrical spreading, and frequency dispersion. of (Ward, 2000) By rearranging equations (4a,b) as a Fourier- ei[ k· r- (k)t] u surf (r,t) = Re dk F(k) Bessel pair, vertical tsunami motions at r can also z ò 2 k 4 cosh(kh) be written as with (4a,b) - iw(k)t ¥ - i[ k· r0 - (k) ( r0 )] ¥ k dk e F(k) = dr ubot (r , (r ))e usurf (r,t)=Re J (kr)einq F (k) ò 0 z 0 0 z ò0 2pcosh(kh) å n n r0 n=-¥ with (5) 2 In (4), k=|k|, w (k) = gktanh(kh), dk=dk dk , bot i(w(k)t (r0 )- nq0 ) x y Fn (k) = òdr0 uz (r0) Jn (kr0 ) e dr0=dx0dy0, and the integrals cover all wavenum- r 0 ber space and locations r0 where the seafloor bot disturbance u z (r0)¹ 0. Here, q and q0 mark azimuth of the points r and Equation (4a) has three identifiable pieces: r0 from the xˆ axis, and the Jn(x) are cylindrical Bessel functions. Although (4) and (5) return a) The F(k) term is the wavenumber spectrum identical wavefields, for certain simply distrib- of the seafloor uplift. This number relates to the uted uplift sources (e.g. point dislocation models amplitude, spatial, and temporal distribution of of earthquakes), (5) might be easier to evaluate Ward: Landslide Tsunami 4 than (4). For instance, for radially symmetric up- W/2 L(t) lifts with t (r )=0, all of the terms in the sum save - iky y0 - i[k - (k)/v ]x 0 F(k) =u dy e dx e x r 0 (7) 0 ò 0 ò 0 F0(k) vanish, and (5) becomes -W/2 0 ¥ surf k dk cos[ (k)t] u (r,t) = J (kr)F (k) with kx = k · xˆ and ky = k · yˆ . The tsunami z ò0 2 cosh(kh) 0 0 waves from a simple slide at observation point r and time t are Fundamental formulas (4) and (5) presume in- stantaneous seafloor uplift at each point. This u L(t)W usurf (r,t)= 0 does not mean that landslides modeled by these z 4p2 i(k·r-w (k)t)- iX(k) formulas happen all at once. The function t (r0) e e sinX(k) sinY(k) ´ Re dk (8) that maps the spatial evolution of the slide is un- ò cosh(kh) X(k) Y(k) k restricted. Actually, the distinction between in- where stantaneous uplift and uplift over several, or sev- eral tens of seconds is not a big issue. Dominant kL(t) kW X(k) = kˆ · xˆ - c(k)/v ; Y(k) = kˆ · yˆ tsunami waves produced from landslides even in 2 ( r ) 2 ( ) fairly shallow water have periods exceeding 100 seconds.