For Publication in: Encyclopedia of Physical Science and Technology - Academic Press 1 Tsunamis Steven N. Ward Institute of Tectonics, University of California at Santa Cruz I. Tsunami = Killer Wave? period, wavelength, and velocity. Unlike com- II. Characteristics of Tsunamis mon sea waves that evolve from persistent sur- III. Tsunami Generation face winds, most tsunamis spring from sudden IV. Tsunami Propagation shifts of the ocean floor. These sudden shifts can V. Tsunami Shoaling originate from undersea landslides and volca- VI. Tsunami Hazard Estimation noes, but mostly, submarine earthquakes parent VII. Tsunami Forecasting tsunamis. Reflecting this heritage, tsunamis often are called seismic sea waves. Compared with GLOSSARY wind-driven waves, seismic sea waves have pe- dispersive: Characteristic of waves whose velocity riods, wavelengths, and velocities ten or a hun- of propagation depends on wave frequency. The dred times larger. Tsunamis thus have pro- shape of a dispersive wave packet changes as it foundly different propagation characteristics and moves along. shoreline consequences than do their common eigenfunction: Functional shape of the horizontal cousins. and vertical components of wave motion versus depth in the ocean for a specific wave frequency. I. Tsunami = Killer Wave? geometrical spreading: Process of amplitude reduction resulting from the progressive expan- Perhaps influenced by Hollywood movies, some sion of a wave from its source. people equate tsunamis with killer waves. Cer- shoal: Process of waves coming ashore. Shoaling tainly a five-meter high tsunami born from a waves slow, shorten their wavelength, and grow great earthquake is impressive. So too is the fact in height. that a tsunami can propagate thousands of kilo- wavenumber: Wavenumber k equals 2p divided by meters and still pack a punch. Upon reaching wavelength l . Large wavenumbers associate shore, tsunami waves shoal and become more with short waves and small wavenumbers asso- menacing still. Understandably, the most dam- ciate with long waves. aging cases of natural hazards come to mind, but it is important to keep perspective. Tsunamis Tsunamis are gravity waves that propagate near over a meter or two in height are not common. A the ocean surface. Tsunamis belong to the same submarine earthquake greater than magnitude family as common sea waves that we enjoy at the M8 must happen to make a wave of this size. On beach; however, tsunamis are distinct in their a global average, about one M8+ earthquake oc- mode of generation and in their characteristic curs per year. Of these, maybe 1-in-10 strikes Ward: Tsunamis 2 under the ocean with a fault orientation favorable In classical theory, the phase c(w), and group for tsunami excitation. Thus, tsunamis that in- u(w) velocity of surface gravity waves on a flat duce widespread damage number about one or ocean of uniform depth h are two per decade. Although one’s concepts might be cast by rare “killer tsunamis”, many more be- gh tanh[k( )h] c( ) = (1) nign ones get lost in the shuffle. Today, ocean k( )h bottom pressure sensors can detect a tsunami of a and few centimeters height even in the open sea. Be- cause numerous, moderate (»M6.5) earthquakes é 1 k( )h ù u( ) = c( )ê + ú (2) can bear waves of this size, “baby” tsunamis oc- ë 2 sinh[2k( )h]û cur several times per year. They pass by gener- ally unnoticed, except by scientists. Perhaps Here, g is the acceleration of gravity (9.8 m/s2) while swimming in the surf, the reader has al- and k(w) is the wavenumber associated with a ready been in a tsunami! Whether killer waves or sea wave of frequency w. Wavenumber connects ripples, tsunamis span three phases: generation, to wavelength l (w) as l (w)=2p/k(w). Wave- propagation and shoaling. This article touches number also satisfies the relation gently on each. 2 = gk( )tanh[k( )h] (3) II. Characteristics of Tsunamis A. Tsunami Velocity, Wavelength, and Period This article reviews classical tsunami theory. Classical theory envisions a rigid seafloor over- lain by an incompressible, homogeneous, and non-viscous ocean subjected to a constant gravi- tational field. Classical tsunami theory has been investigated widely, and most of its predictions change only slightly under relaxation of these assumptions. This article draws upon linear the- ory that also presumes that the ratio of wave am- plitude to wavelength is much less than one. By and large, linearity is violated only during the final stage of wave breaking and perhaps, under Figure 1. (top panel) Phase velocity c(w) (solid lines) and extreme nucleation conditions. group velocity u(w) (dashed lines) of tsunami waves on a flat earth covered by oceans of 1, 2, 4 and 6 km depth. (bottom panel) Wavelength associated with each wave period. The ’tsunami window’ is marked. Ward: Tsunamis 3 short wave approximation (l <<h, 1/k<<h). Un- For surface gravity waves spanning 1 to 50,000s der a long wave approximation, kh® 0 and period, Fig. 1 plots c(w), u(w), and l (w). These equations (1) to (3) predict non-dispersive wave quantities vary widely, both as a function of propagation with c(w)= u(w)= gh . Long wave ocean depth and wave period. Waves whose ve- theory holds for the flat part of the curves in Fig locity or wavelength varies with frequency are 1. Under a short wave approximation, kh®¥ and called dispersive. During propagation, dispersion the equations predict dispersive propagation with “pulls apart” originally pulse-like waves into c(w)= 2u(w)= g ( ) 2 . Short wave theory their component frequencies. I emphasize below holds to the right in Fig. 1 where all the curves dispersion’s strong influence on attenuating tsu- lie atop each other. Waves in the tsunami win- namis. dow have intermediate character, behaving like shallow water waves at their longest periods and Common waves at the beach have periods near like deep-water waves at their shortest periods. 10s and wavelengths around 100m (see Fig. 1). Neither the long or short wave simplification Tsunamis on the other hand, because they are serves adequately in tsunami studies. A rigorous generated by seafloor shifts, must have wave- treatment requires an approach that works for lengths greater than three times the ocean depth waves of all lengths. at the point of their origination. (I explain later why this is so.) This fact fixes a short wave- B. Tsunami Eigenfunctions length bound on tsunamis near 10km. The di- Many properties of tsunamis can be understood mension of the sea floor disturbance itself fixes by examining their eigenfunctions. An eigen- the upper wavelength bound. The greatest earth- function describes the distribution of motion in a quakes might deform a region 500km across. tsunami mode of a particular frequency. Con- The left gray band of Fig. 1 colors the “tsunami sider coordinate system (x, y, z) where xˆ points window” (l =10 to 500km) that spans 100 to north, yˆ east, and zˆ down. Vertical (uz) and 2000s period. Waves in the tsunami window horizontal (ux) components of tsunami eigen- travel rapidly, reaching speeds of 160 to 250m/s functions normalized to unit vertical displace- (600-900km/hr) in the open ocean – about the ment at the sea surface are speed of a commercial jet airliner. Waves at the beach travel at 10m/s (40km/hr) – about the k( )g sinh[k( )(h - z)] u (w,z)= e i[k( )x - t] speed of a moped. The long period, great wave- z 2 cosh[k( )h] length, and high velocity of tsunamis help ac- - ik( )g cosh[k( )(h - z)] count for their destructive power. u (w,z)= ei[k( )x- t] (4) x 2 cosh[k( )h] Discussions of waves of length l in oceans of depth h sometimes include two simplifications: a Fig. 2 plots tsunami eigenfunctions versus depth long wave approximation (l >>h, 1/k>>h) and a in a 4km deep ocean at long (1500s), intermedi- ate (150s) and short (50s) periods. The little el- Ward: Tsunamis 4 lipses can be thought of as tracing the path of a functions of short waves do not reach to the sea- water particle as a wave of frequency w passes. floor, so the velocities of short waves are inde- At 1500s period (left, Fig. 2), the tsunami has a pendent of ocean depth (see right hand side of wavelength of l =297km and it acts like a long Fig. 1, top). The failure of short waves (l <<h) to wave. The vertical displacement peaks at the “feel” the seafloor also means that they can not ocean surface and drops to zero at the seafloor. be excited by deformations of it. This is the The horizontal displacement is constant through physical basis for the short wavelength bound on the ocean column and exceeds the vertical com- the tsunami window that I mentioned above. ponent by more than a factor of ten. Every meter of visible vertical motion in a tsunami of this III. Excitation of Tsunamis Suppose that the seafloor at points r0 uplifts in- bot stantaneously by an amount u z (r0) at time t (r0). Under classical tsunami theory in a uniform ocean of depth h, this sea bottom disturbance produces surface tsunami waveforms (vertical component) at observation point r=xxˆ +yyˆ and time t of ei [k ·r - (k)t] u surf (r,t) = Re dk F(k) z ò 2 k 4 cosh(kh) Figure 2. Tsunami eigenfunctions in a 4 km deep ocean with (5a,b) at periods 1500, 150 and 50s. Vertical displacements at - i[ k ·r0 - (k) ( r0 )] the ocean surface has been normalized to 1 m in each F(k) = dr ubot (r )e ò 0 z 0 case.