Energy of Water Waves Induced by Submarine Landslides

Pure appl. geophys. 157 (2000) 281–3180033–4553/00/030281–38 $ 1.50+0.20/0

Energy of Water Waves Induced by Submarine Landslides

STEFANO TINTI1 and ELISABETTA BORTOLUCCI1

Abstract—Water waves generated by submarine landslides may constitute a serious hazard forcoastal population and environment. These masses may be giant, as documented by several examples inrecent history and by numerous geological traces of paleo-events. A theoretical investigation on wavegeneration and wave energy is performed here by using a model that is based on some simplifyingassumptions. The landslide is treated as a rigid body moving underwater according to a prescribedvelocity function. Water waves are governed by the shallow-water wave equations, where water velocityis constant through the water layer and vertical velocity is negligibly small. Geometrically simple basinsare considered with either constant depth or constant slope, since attention is focused on the fundamen-tal characteristics of the generation process. Analytical 1-D solutions as well as 1-D and 2-D numericalresults obtained by means of a finite-element model are used to gain understanding of the energy transferfrom a moving body to the water. From the 1-D examples, it is found that if slide duration is sufficientlylong, water usually gains energy in the form of waves until a saturation point is reached, when bodymotion is no longer capable of producing a net transfer of energy from the rigid body to water.Finite-duration motions of a body moving at constant speed along a flat ocean floor can be used ascanonical examples, since bottom slopes cannot significantly change the generated wave pattern.Typically, two trough-crest systems are developed that travel in opposite directions, with the leadingcrest in the direction of the slide and the leading trough toward the other direction. The amplitude ofthe former is generally higher, with amplitude controlled by the Froude number (ratio of body velocityto long waves phase celerity) and wavelength dictated by landslide length. Generation and propagationof 2-D cases show a more complicated pattern, since lateral radiation plays an important role. Some ofthe features present in the 1-D models are observed in 2-D wavefields, however substantial differencesarise. The most significant difference is that no energy saturation takes place in 2-D, since the bodytransfers energy to the water as long as it moves.

Key words: Energy propagation, energy transfer, shallow water approximation, submarine land-slide, tsunami.

1. Introduction

Water waves generated by submarine slumps and landslides may be verydangerous for coastal areas and are a natural hazard worthy of careful investiga-tion. In several places of the world’s oceans marine geologists have identified hugeunderwater landslide bodies that were set in motion because of an excess of gravity

1 Dipartimento di Fisica, Settore di Geofisica, University of Bologna, Viale Carlo Berti Pichat, 8,40127 Bologna, Italy. Fax: 051 2095058.

Stefano Tinti and Elisabetta Bortolucci282 Pure appl. geophys.,

load or due to some other triggering mechanism, and moved tens or hundreds ofkilometers from their original position, mostly in correspondence with steep shelfslopes, to their final position in bathyal zones. Remarkable examples are theAgulhas slide case at the continental margin off South Africa (DINGLE, 1977), theStoregga slides off northwestern Norway with volume exceeding 5000 km3 (BUGGE

et al., 1987, 1988), and the slides on the Hawaiian Ridge (MOORE et al., 1989,1994). These extraordinarily large slides were very likely accompanied bytransoceanic tsunamis involving their entire basin of pertinence and sweeping withtremendous energy the neighboring coasts. There is the hypothesis, though notuniversally accepted, that the Hawaiian Island of Lanai was hit by a tsunami waveexceeding 375 m as the consequence of a massive landslide occurring 105 Ka ago 50km southwest of the island (see MOORE and MOORE, 1988; the tsunami modelingby JOHNSON and MADER, 1994; and the alternative view by GRIGG and JONES,1997). Direct evidence of tsunami deposits that might be attributed to the secondStoregga slide about 6000–7000 years B.P. have been recently recognized on theeastern Scotland and Norway coasts (SMITH et al., 1985; DAWSON et al., 1988,1996; BONDEVIK et al., 1997), and numerical simulations have proven that theentire Norway Sea should have been affected by the ensuing tsunami with several-meter amplitude waves attacking the coasts (HARBITZ, 1992). Gravity is the chiefdriving force for submarine landslides, but probably earthquake shaking is the mostfrequent triggering mechanism for instability and for mass motion initiation in opencontinental margin slopes (LEE and EDWARDS, 1986), though slope failure on thesteep flanks of island- or submarine-volcanoes associated with volcanic activity is afurther well-known process of volcanic edifice transformation, and possible tsunamigeneration (see HOLCOMB and SEARLE, 1991; MOORE et al., 1994, for the HawaiiIslands; see also KOKELAAR and ROMAGNOLI, 1995, and TINTI et al., 1999, forStromboli in Italy).

The interest in underwater mass movements recently increased just after recog-nition that submarine earthquakes can generate tsunamis by direct sea-floor dis-placement and, in addition to this, can trigger landslides acting as sources ofsecondary tsunamis that might be locally much stronger and ravaging than theearthquake-induced waves. Mass motions in the form of turbidity currents arespeculated to have occurred in concomitance with the 1929 Grand Banks earth-quake off Newfoundland, Canada (HEEZEN and EWING, 1952), and on thecontinental slope of Africa in the Alboran Sea (eastern Mediterranean) as the resultof the 1954 earthquake destroying Orleansville, today named El Asnam (HEEZEN

and EWING, 1955; SOLOVIEV et al., 1992). The final effect was that several deep-seaseated communication cables were found to be seriously damaged after the earth-quakes. A very recent example of a seismically induced tsunami that was locallystrongly reinforced by water waves generated by slumps comes from Flores,Indonesia. In December 1992 this island was affected by an MS=7.5 earthquake,and the consequent toll of victims due to the earthquake and tsunami was about

Ocean Waves by Submarine Landslides 283Vol. 157, 2000

2000. An accurate post-event survey carried out soon after the disaster (YEH et al.,1993) could take measurements of runup and water penetration along most of thecoastline hit by the tsunami, and find that runup heights observed in some localitiesof eastern Flores (over 10 m with a maximum of 26.2 m in a place calledRiangkroko) were abnormally larger by a factor 2–4 than the mean value of thesurrounding area. These values could not be explained by model calculationsperformed by assuming the earthquake as the only tsunami source (IMAMURA andKIKUCHI, 1994, and HIDAYAT et al., 1995). In addition, direct evidence of coastalslumping and submarine landslides was found in proximity of the abnormal values,which was the proof that, besides seismic faulting, mass movements were signifi-cantly active in exciting water waves (IMAMURA et al., 1995).

Generation of water waves by motion of underwater masses has been longinvestigated, but due to the phenomenon complexity, a comprehensive theoryexplaining all facets of landslide evolution and waves production is still missing.Numerous reports on field observations (see JøRSTAD, 1968; see also HAMPTON etal., 1996 for a review), on laboratory experiments (IWASAKI, 1982; WATTS, 1998) aswell as numerical simulations of landslide-induced tsunamis (HARBITZ, 1992;HARBITZ et al., 1993; ASSIER RZADKIEWICZ et al., 1997; MADER, 1997; KOWALIK,1997; TINTI et al., 1999) can be found in the literature. In this paper the attentionwill be focused on the topic of wave energy, in an attempt to explore two particularissues of special importance: 1) the rate by which energy is transferred from themoving body to water; 2) the energy propagation in the sea while the landslide ismoving and after that landslide has come to an end. To this purpose only idealizedtheoretical cases will be examined since they are amenable to simple treatment andcan lead to a simple interpretation of the fundamental physical processes. This is astrategy that has often been used, even in the context of landslide research, to focuson essential features while avoiding distracting complexities (see PELINOVSKY andPOPLAVSKY, 1996; JOHNSGARD and PEDERSEN, 1996). With this in mind, thepresent study will be restricted to the analysis of water waves generated by rigidbodies, though more complex models of landslide-generated waves have beenproposed in the literature in which the sliding mass is treated as a fluid interactingwith the surrounding fluid (JIANG and LEBLOND, 1994; ASSIER RZADKIEWICZ etal., 1997; HEINRICH et al., 1999) or as a solid structured in blocks interacting witheach other (TINTI et al., 1999). The rigid body approximation is appropriate for allcases of collapses of rocky material and for slumps as well as for the initial stagesof the evolution of many underwater landslides, before they evolve to debris andturbidity flows. Since rigid bodies are believed to be more effective in exciting waterwaves than deformable bodies where rheology plays an important role (JIANG andLEBLOND, 1994; LEBLOND and JONES, 1995), the simplification adopted hereimplies an overestimation of the body-to-water energy transfer, and it is significantsince it places a relevant upper bound. To the same effect also points the secondrestriction used in the paper, that is the adoption of a hydraulic model based on


shallow-water approximation. It is known that sea bottom changes transfer to thesea surface, and that the transfer function is a low-pass filter that leaves practicallyunaltered wavelengths that are substantially larger than water depth, while sensiblyreducing all smaller scales (KAJIURA, 1970; HAMMACK, 1973). Assuming long-waveapproximation entails that all wavelengths are taken to be as effective as tsunamigeneration as the largest scale perturbations, thus enhancing the excitation processand resulting in larger water waves. With these considerations in mind, the nextsections will be devoted to the basic formulation of the problem, and subsequentlyto the analysis of the underwater motion of rigid bodies, starting first with 1-Dbasins, namely channels, and moving then to 2-D domains, in order to stresssimilarities and differences.

2. Formulation of the Problem

The inhomogeneous form of the shallow water approximation to the Navier-Stokes hydrodynamic equations has been often used to study waves generated bydisplacements of the ocean bottom due either to rupture of a submarine fault(HWANG and DIVOKI, 1970; TUCK and HWANG, 1972) or to the underwatermotion of a body (HARBITZ, 1992; TINTI et al., 1999). Let us concentrate on thissecond case and then assume that the instantaneous ocean depth h at the point ofcoordinates x and y and at time t may be expressed as the algebraic sum of twocontinuous functions as

h(x, y, t)=h0(x, y)−hS(x, y, t) (1)

where, following the sketch given in Figure 1, hS is the height of the moving slideand h0 is the depth of the stable floor, not depending on time. Notice that a positive

Figure 1Schematic representation of the landslide. The total water depth is h, positive downward, the landslide

height is hS, positive upward, and the depth of the basin bottom is h0.


value of hS represents a moving mass, which is the case of most interest here, whilea negative value would represent a moving floor depression. Under the assumptionthat �hS � is considerably smaller than h0, if moreover nonlinear advective andviscous terms as well as bottom friction can be neglected, the governing equationsfor an inviscid nondispersive fluid take on the simple form

(tz+9h · (h0v)=(thS, (2a)

(tv+g9z=0. (2b)

Here z is the water elevation above still level, v is the horizontal water velocityvector, assumed to be constant over the water column, and g is the vertical gravityacceleration. Symbols 9 and 9h denote gradient and horizontal divergence opera-tors, respectively. The inhomogeneous term that is present in the continuity eq. (2a)is the only term in the set of eqs. (2) whereby the ocean water and the body movingon the ocean floor can exchange energy. Typically, it is expected that it is the bodythat transfers part of its energy to the water, thereby exciting waves, but there is notheoretical constraint for the transfer to take place the opposite way. Indeed it ispossible to envisage conditions where bottom forcing results in energy subtractionfrom the water. In order to gain a better insight to this issue, it is convenient tonotate the expressions for the water energy contained in the water column of depthh and horizontal infinitesimal surface dA, as follows

dEP

dA=

12

rgz2, (3a)

dEK

dA=

12

rhv2, (3b)

dEdA

=dEP

dA+

dEK

dA. (3c)

Here r is the density of the water, while E, EP and EK denote total energy, potentialenergy and kinetic energy, respectively, and the symbol d/dA is used to designatesurface density. The energies contained within a finite volume of fluid delimited bythe sea surface on the top, by the ocean bottom with basal surface A, and by lateralvertical boundaries, may be computed by integrating the above expressions over thesurface A, that is

EP=&

A

dEP

dAdA, (4a)

EK=&

A

dEK

dAdA, (4b)

E=EP+EK. (4c)


After multiplying both sides of the continuity eq. (2a) by rgz, taking the scalarproduct of both members of the momentum eq. (2b) by the horizontal velocityvector rgh0v, and combining the results, it is straightforward to obtain

(t

dEdA

+r9h · (c2zv)=rgz (thS (5)

where

c=gh0 (6)

is the well-known expression of the phase velocity of linear long waves. Afterintegrating over the basal surface A and applying Gauss’ theorem, the aboveequation becomes

(tE+&

C

F · n dl=&

A

dQdA

dA (7)

where C is the close boundary of the surface A and the unit vector n is normal toC everywhere and pointing outward. The horizontal vector F defined by

F=rc2zv (8)

may be interpreted as the instantaneous flux of energy across the infinitesimal lineelement dl of the curve C, whereas the scalar quantity dQ/dA, defined as

dQdA

=rgz (thS (9)

is a surface density term representing a local source or sink of energy for the waves.Whenever this term vanishes, wave energy is conserved since any energy changeover the volume is exactly compensated by energy flows across the boundary C.However, if it differs from zero, energy is injected into, or extracted from, thesystem according to whether the water elevation z and the height rate of the slide(thS (or, equivalently, the vertical velocity of the sea bottom, positive upward) havethe same or opposite sign. Seen in another way, it can be stated that any floor upliftwhich is concomitant with a water trough has the effect of reducing the amplitudeand the energy of the wave, while it will increase amplitude and energy of a watercrest. But the opposite action will be exerted by a sea-floor subsidence on the samewave systems.

3. Analytical Solutions for a Flat Channel

If it is assumed that h0(x, y)=H and that hS=hS(x, t), then the solution of thesystem of eqs. (2) can be taken to be independent from y, with no component ofvelocity in the direction of axis y, and the problem simplifies to 1-D:


(tz+H (xu=(thS, (10a)

(tu+g (xz=0, (10b)

where u is the x component of the fluid particle velocity. This set describes thewater waves produced by transversally uniform floor displacements taking place ina channel with a flat bottom. This case represents a good example to treat, since ithappens to be sufficiently simple to admit analytical solutions, and at the same timeto be not at all trivial, but satisfactorily instructive to allow an understanding of thebasic features. Therefore it is worthy of a careful analysis, which will be carried outin the next three subsections, where the steady forced solution and a solutionconsisting of a combination of forced and free waves will be worked out.

3.1 Steady Forced Wa6es in a Flat Channel

If the perturbation hS(x, t) represents a rigid body moving with constantvelocity V towards positive (V\0), or negative (VB0) x, i.e., if

hS(x, t)=hS(a), a=x−Vt, hS]0, (11)

then it is easy to show that system (10) admits the following analytical steadysolution

z(x, t)=Fr2

Fr2−1hS(x−Vt), Fr"1 (12a)

u(x, t)=gV

z(x, t), Fr"1 (12b)

where Fr= �V �/c is the Froude number, c being equal to gH. The expressions (12)yield inasmuch as Fr"1. In the special case where the body speed equals the wavephase velocity, and consequently Fr=1, no steady state exists. The situation isquite similar to that of a mechanical system subject to external forcing. A singlefrictionless oscillator responds to periodic forcing by attaining a steady state ofoscillation if and only if forcing takes place at frequencies differing from theeigenfrequency of the system, otherwise critical resonance would occur, and thesystem would collapse unless energy is dissipated by internal damping.

The energy carried by the steady wave is constant in time. Computing theenergy by substituting solutions (12) into the energy integrals (4) results in thefollowing expressions

EP=12

rgWR(0)Fr4

(Fr2−1)2 , (13a)

EK=12

rgWR(0)Fr2

(Fr2−1)2 , (13b)


E=12

rgWR(0)Fr2 1+Fr2

(Fr2−1)2 , (13c)

where W is the channel width and R(0) is the value of the function R(j) computedat j=0, which is defined as

R(j)=& +�

−�

hS(x−Vt)hS(x−Vt+j) dx=& +�

−�

hS(a)hS(a+j) da. (14)

This quantity, possessing the same dimensions as a volume, may be interpreted asthe autocorrelation of the bottom profile of the moving body hS(a), and will be ofgreater use in examining the cases of the next subsections. Solution (12) is a wavetravelling together with the moving underwater body. If the body moves towardspositive x at a supercritical speed (Fr\1), then it excites a positive steady waveaccompanied by a positive movement of water particles (u\0). But in case ofsubcritical speed (FrB1), the excited steady wave is a propagating trough withparticle movement reversed (uB0). The wave energy is unequally distributedbetween potential and kinetic. As may be seen from eqs. (13), the ratio

EP

EK

=Fr2 (15)

indicates that potential energy predominates at supercritical body speeds, but it isless at subcritical speeds. Energy tends to equipartition and wave amplitude tendsto grow unlimited in proximity of the critical case V=c, where a steady solutiondoes not exist.

3.2 Wa6es Due to a Finite-duration Slide in a Flat Channel

The foregoing case of a wavemaker body moving at a constant speed isextremely ideal and maybe too far from reality to be considered an acceptableapproximation to a landslide. More requirements seem to be essential. A plausiblelandslide could be a body i) with a finite lateral extension WS, ii) which is in motionduring a finite interval of time with duration T, iii) with a variable velocity V(t), iv)along a slope. As simple as they are, the above conditions are yet too demandingfor an analytical solution to exist, and this problem requires application ofnumerical techniques as will be seen. It is however convenient to go as far aspossible with analytical means since these can provide solutions that can bethoroughly examined to gain information on basic important features. Of the fourrequirements given in the prior list let us consider only the second, and search fora solution of the system (2) in the case in which the forcing function has the form

hS(x, t)=h*S(x−Vt)[H0(t)−H0(t−T)] (16)

where H0(j) is the discontinuous Heaviside unit-step function, namely


H0(j)=0 for j50, H0(j)=1 for j\0. (17)

In the following the asterisk will be dropped from h*S and hS(x−Vt) will be usedin place of h*S (x−Vt) for convenience. Since there is no loss of generality inpresuming that V\0, hereafter this assumption will be always made. The motionlaw (16) implies that the body starts its motion with infinite acceleration at theinitial time t=0, then it advances with constant speed in the time interval [0, T ],after which it suddenly stops as the effect of an infinite negative acceleration. Therunout distance, say LR, is simply given by the product LR=VT. The water issurmised to be at rest before landslide initiation, and accordingly the initialconditions to impose are

z(x, 0)=u(x, 0)=0 Öx. (18)

Due to the linearity of the set of eqs. (2), in general the solution can be built upby superposing the forced wave given by eqs. (12) to suitable free waves. These arewave satisfying the homogeneous system associated with eqs. (2), i.e., a systemanalogous to set (2), but with no forcing term. As will be seen, it is necessary todistinguish solutions valid within the time interval [0, T ], during which the slide isactive, from solutions holding for larger times, when the slide is already at rest andits forcing action finished. These solutions will be hereafter designated respectivelyby in-slide and post-slide solutions, and subscripts IS and PS will be used for thesolving functions. Continuity of the waves is required at the stopping time t=T,which implies that

zIS(x, T)=zPS(x, T) Öx (19a)

uIS(x, T)=uPS(x, T) Öx. (19b)

The solution to this problem may be found through the method of characteristicsas proposed by HARBITZ and PEDERSEN (1992) and by HARBITZ and ELVERHøI,(1999), and can be written as follows

zIS(x, t)=z ISF (x, t)+z IS

+(x, t)+z IS−(x, t) 05 t5T, (20a)

uIS(x, t)=uISF (x, t)+uIS

+(x, t)+uIS−(x, t) 05 t5T. (20b)

Superscript symbols F, + and − are used to denote waves that are respectively theforced wave, the free wave propagating toward positive x and the free wavetravelling backward. The elevation function z IS

F (x, t), z IS+(x, t) and z IS

−(x, t) andtheir corresponding velocities, have different expressions for noncritical and criticalflows:

z ISF (x, t)=

Fr2

Fr2−1hS(x−Vt) uIS

F (x, t)=gV

z ISF Fr"1, (21a)


z IS+(x, t)= −

12

FrFr−1

hS(x−ct) uIS+(x, t)=

gc

z IS+ Fr"1, (21b)

j IS−(x, t)= −

12

FrFr+1

hS(x+ct) uIS−(x, t)= −

gc

z IS− Fr"1, (21c)

z ISF (x, t)= −

c2

th %S(x−ct) uISF =

gc�

z ISF −

12

hS(x−ct)n

Fr=1, (22a)

z IS+(x, t)=

14

hS(x−ct) uIS+ =

gc

z IS+ Fr=1, (22b)

z IS−(x, t)= −

14

hS(x+ct) uIS− = −

gc

z IS− Fr=1. (22c)

Here prime in h %S denotes differentiation with respect to its argument, that ish %S(a)=dhS(a)/da. Analogously, the post-slide solutions have the form:

zPS(x, t)=zPS+ (x, t)+zPS

− (x, t) t]T, (23a)

uPS(x, t)=uPS+ (x, t)+uPS

− (x, t) t]T, (23b)

where, again, expressions for zPS+ (x, t) and zPS

− (x, t) and their correspondingvelocities differ in case of noncritical and critical body speed and are given by

zPS+ (x, t)=

12

FrFr−1

[hS(x−VT−c(t−T))−hS(x−ct)] uPS+ =

gc

zPS+ Fr"1,

(24a)

zPS− (x, t)=

12

FrFr+1

[hS(x−VT+c(t−T))−hS(x+ct)] uPS− = −

gc

zPS− Fr"1,

(24b)

zPS+ (x, t)= −

c2

Th %S(x−ct) uPS+ =

gc

zPS+ Fr=1, (25a)

zPS− (x, t)=

14

[hS(x−2cT+ct)−hS(x+ct)] uPS− = −

gc

zPS− Fr=1. (25b)

In order to understand the characteristics of the exact solution listed above, weconsider the example of a noncritical flow. The in-slide system of waves, describedby eqs. (20) and (21), consists of three waves: one, z IS

F , travelling together with thewavemaker, is exactly the steady solution encountered in the previous subsection,the others, z IS

+ and z IS−, are free waves moving in opposite directions with speed c.

At the initial stage (t=0) these waves are perfectly superposed and cancel out tomatch the initial conditions of null water elevation and velocity. At subsequenttimes, waves separate progressively since travel with different speeds. The first wavethat differentiates is the backgoing impulse z IS

−. Later, the advancing waves separatefrom each other. The separation time of two waves depends on their relative speed.


If the axial length of the landslide is LS, then progressing waves z ISF and z IS

+ willfully separate at time TS=LS/�c−V �, while the backward propagating waveseparates from them at T %S=max(LS/2c, LS/(c+V)). It is easy to see that T %SBTS.It is worth observing that all waves have the same shape, that is given by the axiallandslide profile hS, with however different amplitude, which depends on theFroude number. The resulting wave configuration is that a trough propagates in thedirection opposite the landslide, while a system composed of a crest and a followingtrough travels in association with the landslide. It is interesting to note that theleading front is always a crest. Indeed, if Fr\1, it follows that z IS

F is positive andz IS

+ is negative, with z ISF moving faster. In the case where FrB1, the z IS

F is negativeand z IS

+ is positive, with z ISF slower than z IS

+.After time T, the body is at rest and a forced wave can no longer exist. From

eqs. (23) and (24) it can be seen that the global configuration constitutes four freewaves, two travelling forward (zPS

+ ) and two backward (zPS− ). Of these four waves,

two are identical to those that were created at the slide initiation phase. They resultto be unaffected by slide stopping and continue travelling at their own speed c.These are represented by the second terms in eqs. (24a) and (24b). On the otherhand, at slide stopping, the forced wave transforms into two new free waves,represented by the first terms in eqs. (24a) and (24b). The resulting wave configura-tion may be described as consisting of two crest-trough systems. The one movingforward has a leading positive wave, the other has a leading negative wave. Thesesystems separate from each other, since they go in opposite directions, but afterseparation their shape does not change anymore, because crest and trough traveltogether at the same speed. The final crest-trough separation is therefore onlydetermined by the duration T of the slide motion. If T is longer enough, moreprecisely if T\TS, then no wave interference can take place and wave are fullydistinct.

Figure 2 illustrates the evolution of water waves along a channel computedthrough eqs. (20)–(21) and (23)–(24) for a case of subcritical body motion(Fr$0.64), with a slide duration time T=80 s which is longer than T %S=19.5 s,but smaller than TS=88 s. The wavemaker profile used for all calculations is

hS(a)=A2�

1−cos�2p

LS

(a−a0)�n

if a� [a0, a0+LS ]; hS(a)=0 elsewhere

(26)

representing a bump-like body of maximum height A and length LS. During thegeneration phase, highlighted in the upper panels, the three-wave system is built upprogressively. Later, the two double-front systems visible in the lower panelspropagate rightward an leftward. The advancing front has crest and trough nearly,although not yet completely, separated. These are, however, fully distinct in thebackgoing system. Figure 3 portrays wave propagation for three supercritical cases.They differ as regards the Froude number value, and since they supposedly have


Figure 2Wave profiles generated by a bump-like (see eq. (26)), 1-km long and 1-m high, landslide in a 1-D flatchannel. The basin depth is 100 m. The body moves with constant speed (V=20 m/s), corresponding toa subcritical motion with constant Froude number (Fr$0.64), during the time interval [0, 80 s].Backward moving waves are well separated, while waves going down the channel are not, since TBT %

(see text).

the same slide runout distance LR their slide duration is inversely proportional toFr. For all cases slide initiation is assumed to occur at the initial time t=0. Thisexample can be used to point out further interesting features of the generatedwaves. We first concentrate on the backward impulses. The leading trough isproduced exactly at the beginning of wavemaker motion and thus, for all threecases examined here, they appear perfectly superposed in the diagrams of Figure 3.On the other hand, the backgoing crest is generated at the slide stopping time and,


Figure 3Wave profiles produced by a bump-like (see eq. 26)), 1-km long and 1-m high, landslide moving in a 1-Dchannel with uniform depth (100 m). The mass displacement takes place at constant velocity. Three casesof supercritical speed are shown superposed. Runout distance LR is 1.2 km for all cases. Forward wavescorresponding to Froude number closer to 1 are higher amplitude. Notice that troughs, being free wavesgenerated at the slide initiation times, are well synchronized. But crests, generated at the stopping time

T, which is different from case to case, appear as a sequence of waves in the graphs.

owing to the assumption of equal runout, it is generated first by the fastest body.Therefore, on the graphs they appear ordered in a sequence, their separationremaining constant since they move with the same speed c. As expected from eqs.(21c) and (24b), amplitude of both in-slide and post-slide backgoing impulsesincreases with Fr. Let us then take into account the waves going ahead. Comparedto backward waves they have larger amplitudes which increase as Froude numberapproaches 1. In the plots, the most advanced crest corresponds to the highestFroude number.


The special case of Fr=1 has the solution specified by eqs. (20) and (22), andby (23) and (25). It is worthwhile noticing that the forced wave z IS

F created duringslide motion, has amplitude linearly growing with time and shape which isproportional to h %S in place of hS. Typically, for ordinary slide profiles, h %S isnegative at the slide front and positive at the rear. Thus z IS

F represents a crest-trough system with a leading crest. A second observation is that j IS

F and z IS+ both

advance with the same speed c and cannot separate at any time. At the slidestopping instant, the progressing forced wave j IS

F stops growing, and at later timesit advances unchanged, keeping its speed and its amplitude. Notice that theconstant factor −cT/2 in eq. (25a) replaces the linearly changing factor −ct/2 ineq. (22a). This discussion aids comprehension of why it was not possible to find asteady solution to the problem addressed in the previous subsection, where thewavemaker was assumed to move at speed c for an infinite time. Indeed, if the slideduration T were increased unlimited, the forced wave amplitude would growsteadily until the solution would become meaningless.

3.3 Energy Associated with a Finite-duration Slide in a Flat Channel

The knowledge of the exact solution to the problem analyzed above enables usto calculate the energy integrals (4). It is convenient to treat the noncritical casefirst. After some tedious computations, with the aid of eqs. (20) and (21) as well asof the definition (14), it is possible to find the expression of the total energy of thewater waves during the slide motion which may be written as

EIS(t)=12

rgWFr2�2Fr2+1

(Fr2−1)2 R(0)−1

(Fr−1)2 R(bt)−1

(Fr+1)2 R(gt)n

Fr"1.

(27a)

Here the arguments bt and gt have been introduced with the following meaning

bt= (V−c)t, (27b)

gt= (V+c)t. (27c)

The total energy of the generated waves changes with time, the time dependencebeing included just in the above defined terms bt and gt. The autocorrelationfunction defined by eq. (14) possesses properties that are relevant for the presentdiscussion. Since according to eq. (26) hS is assumed to be different from zero onlywithin the interval of length LS where the slide exists, and within which it takespositive values, it follows that

R(j)]0 Öj, (28a)

R(j)=0 j]LS. (28b)

Furthermore, the autocorrelation function attains its maximum at the origin, i.e.,


R(j)5R(0) Öj (28c)

the inequality being strictly valid for nonperiodic functions. Since nobody wouldexpect a landslide to possess a periodic profile, the expression (28c) is assumed tohold here as an inequality. Bearing this in mind, it is trivial to check that EIS(0)=0,since terms within brackets in eq. (27a) cancel out exactly, and it is easy to showthat, as time passes and arguments (27b) and (27c) grow larger and larger, functionsR(bt) and R(gt) become vanishingly small with the consequence that the energyapproaches the value

Emax=rgWR(0)Fr2 Fr2+1(Fr2−1)2 Fr"1 (29)

which is a maximum for EIS(t). This fact is significant and deserves more consider-ation. The movement of the wavemaker generates water waves, thus introducingenergy in the hydraulic system. If the motion duration is sufficiently long, thenduring body motion an instant arrives when water energy attains the value Emax,after which the water system evolves with constant energy. This means that themoving body is no longer able to induce a net energy transfer to water, and that theprocess has come to a saturation point. Mathematically, this situation occurs whenboth R(bt) and R(gt) equal zero, and this in turn corresponds to the tree-wavesystem z IS


−(x, t) being fully separated, which, as already seen,can take place only if T is larger than the separation time TS. Therefore, what isexpected is that Emax is the value of energy at the time TS, i.e. Emax=EIS(TS), andalso that if TBTS, then EIS(T)BEmax, while if T\TS, then EIS(T)=Emax. It isalso quite interesting to consider the quantity of energy conveyed by single waveswhen they do not overlap. The following expressions can be easily obtained

EISF =

12

rgWR(0)Fr2 Fr2+1(Fr2−1)2 Fr"1, (30a)

EIS+ =

14

rgWR(0)Fr2

(Fr−1)2 Fr"1, (30b)

EIS− =

14

rgWR(0)Fr2

(Fr+1)2 Fr"1. (30c)

Likewise Emax, these can be interpreted as the maximum possible values of energythat z IS


−(x, t) can attain. It is worthwhile observing that thefollowing identities hold

EIS+ +EIS

− =EISF =

12

Emax (31)

irrespective of the value taken on by the Froude number. In virtue of (31) it can bestated that the water system has the tendency to distribute energy evenly betweenthe forced wave and the sum of the excited free modes of oscillation.


After the moving body stops, the energy must be calculated by using post-slidesolutions (23) and (24). Since wave systems zPS

+ (x, t) and zPS− (x, t) travel opposite

directions, no doubt they will sooner or later separate, and their energy can beadded to obtain the total energy of the system. We introduce symbols EPS

� and EPS�

to designate the energy carried by zPS+ (x, t) and zPS

− (x, t), respectively. After somecalculations, it is obtained

EPS� =

12

rgWFr2

(Fr−1)2 [R(0)−R(bT)] Fr"1, (32a)

EPS� =

12

rgWFr2

(Fr+1)2 [R(0)−R(gT)] Fr"1, (32b)

where, according to definitions (27b) and (27c), it is meant that

bT= (V−c)T, (33a)

gT= (V+c)T. (33b)

No surprise that expressions (32) do not depend on time, since the total energy ofthe water cannot change after the forcing action of the body has ceased, since nodissipation is included in the governing set (2). It is also trivial to see that theidentity

EPS� +EPS

� =EIS(T) (34)

holds true, which is a consequence of the matching conditions (19). The correlationterms R(bT) and R(gT) depend on the overlapping between crest and trough of thewave systems, respectively advancing and regressing. If crest and trough happen tobe separated, then EPS

� and EPS� reach their maximum values. If these are denoted

by EPS+ and EPS

− , the following expressions may be written

EPS+ =

12

rgWR(0)Fr2

(Fr−1)2 Fr"1, (35a)

EPS− =

12

rgWR(0)Fr2

(Fr+1)2 Fr"1. (35b)

It is interesting to stress that the equalities

EPS+ =2EIS

+, (36a)

EPS− =2EIS

−, (36b)

EPS+ +EPS

− =Emax (36c)

are identically valid as well as the following chain of equations

EPS−

EPS+ =

EIS−

EIS+ =

�Fr−1Fr+1

�2

B1 Fr"1. (37)


Figure 4Total energy of water waves vs. time in case of constant-depth channel and constant-speed body,computed by means of formula (27a). Energy is normalized to 1/2rgWR(0), which is the value of energythat the water basin would gain in case of sudden vertical displacement of the sea floor, as occurs forsubmarine earthquakes. Curves correspond to different Froude numbers. All curves reach an energysaturation plateau for large times, which is the higher, the closer to 1 is the Froude number. Saturation

is reached later for smaller Froude numbers.

This can be given the simple interpretation that energy possessed in the in-slidestage by the forced wave is converted into post-slide free wave modes in such a waythat the ratio of the energies of the advancing and regressing free waves isconserved. Most importantly, from eq. (37) it is learned that the energy that tendsto be radiated in the same direction in which the body moves is always larger thanthe energy going backward, the ratio depending only on the Froude number. Forbody speeds close to the critical value c, practically the amount of backwardradiation is negligibly small, while at largely subcritical and supercritical flows, thetwo energies tend to a balance. Figure 4 displays diagrams of total energy vs. timefor various Froude numbers. For all cases the slide duration T is taken to begreater than the separation distance TS, and accordingly EIS(t) can attain themaximum allowed value Emax which depends remarkably upon Fr. These graphsalso can be used to study cases of shorter slide duration. The energy function willcoincide with the plotted curve until the time t=T, and then will remain constant,therefore having a value less than Emax.

The critical case corresponding to Fr=1 can be dealt with by making use ofin-slide solutions (20) and (22) and post-slide solutions (23) and (25). The finalresult is that expressions (27a) and (29) have to be replaced by


EIS(t)=18

rgW [R(0)+2c2t2R1(0)−R(2ct)] Fr=1 (38)

and by

Emax=18

rgW [R(0)+2c2T2R1(0)] Fr=1 (39)

where R1(j) is the autocorrelation function of h %S, i.e.,

R1(j)=& +�

−�

h %S(a)h %S(a+j) da. (40)

As expected from the linear time-dependence of eq. (22a), the energy growsquadratically during the generation phase, and no intrinsic energy saturation of thewater body can ever be reached. The wave system travelling forward is composedby two waves z IS

F and z IS+ advancing with the same speed c, and cannot separate.

Therefore there is no possibility to distinguish here between the maximum valuesEIS

F and EIS+ as was done for Fr"1 in expressions (30a) and (30b). It is only

possible to provide an expression for the maximum energy, say EIS+F, conveyed

jointly by the overlapping waves, that is

EIS+F=

18

rgW�1

2R(0)+2c2T2R1(0)

nFr=1. (41a)

As regards the backgoing wave z IS−, it is easy to see that expression (30c) can be

replaced by

EIS− =

116

rgWR(0) Fr=1 (41b)

which implies the identity

EIS+F+EIS

− =Emax. (42)

After the end of the wavemaker motion only one advancing wave survives, whiletwo free waves propagate backward. It is easy to see that expressions (32) are to besubstituted by

EPS� =

14

rgWc2T2R1(0), Fr=1 (43a)

EPS� =

18

rgW [R(0)−R(2cT)]. Fr=1 (43b)

The second term in (43b) represents the contribution due to the interferencebetween the two regressing waves, generated at slide initiation and at slide stopping.The expressions for EPS

+ and for EPS− corresponding to eqs. (35) are now

EPS+ =EPS

� Fr=1, (44a)


EPS− =

18

rgWR(0) Fr=1. (44b)

The ratios between the maximum energies carried by the regressing and theadvancing wave systems are

EIS−

EIS+"

EPS−

EPS+ =

R(0)c2T2R1(0)

Fr=1 (45)

that should be compared with formula (37). As the forcing duration grows, thepost-slide ratio tends to become continuously smaller and, in agreement with thebehavior exhibited by expression (37) as the Froude number approaches 1.

It is interesting to compare energy associated with tsunamis generated bylandslides, to that of tsunamis produced by earthquakes. In this latter case, thesource is impulsive and acts for a very short time (1–40 s), that is a timeconsiderably less than the typical periods of the excited waves (102–103 s). Usually,the action of the earthquake is incorporated in tsunami models as an initialcondition for the water waves: the initial sea level is displaced from the still waterlevel, and the displacement is taken to be equal to the vertical displacement of thesea floor produced by the earthquake, while the initial horizontal velocity of thewater is taken to be zero. Consequently, tsunamis are modelled as free waves,possessing initially only potential energy, and, if no dissipation is taken intoaccount, the total tsunami energy remains constant during its evolution. This is animportant feature for earthquake-induced tsunamis, and it is worthwhile stressingthat just in virtue of this property the initial tsunami energy has been proposed asthe basic parameter to measure the magnitude of such class of tsunamis (MURTY

and LOOMIS, 1980). More detailed considerations of a comparison between theenergy of tsunamis induced by earthquakes and landslides, are given in theAppendix.

4. Numerical Solutions for a Channel

The detailed treatment of the constant-velocity slide moving down a flat channelwas justified by the availability of closed analytical formulas and the consequentpossibility of readily exploring the basic characteristics of the generated waves. Inthis section the investigation will be carried out further to examine more realisticcases, where the body moves with variable velocity along a flat channel or down anincline. The analysis is executed by using numerical means to calculate the solution.A finite-element technique, that is suitable for shallow-water 2-D problems (TINTI

et al., 1994) and that was applied in previous papers to study tsunamis generated byearthquakes and by landslides (see e.g., TINTI and GAVAGNI, 1995; TINTI andPIATANESI, 1996; PIATANESI and TINTI, 1998; TINTI et al., 1999), is used here ona rectangular mesh covering a channel with triangular elements. The channel has


lateral vertical walls and open ends. All quantities are assumedly uniform across thechannel, and therefore this basin is appropriate to study 1-D cases with wavespropagating down the channel, with channel axis being coincident with the x axisof the reference frame used for the basic eqs. (2).

4.1 Variable-6elocity Body in a Flat Channel

The first series of numerical experiments comprises computing waves generatedon a basin with constant depth by a body, which is supposedly a mass with shapegiven by eq. (26), that moves a distance LR in a given time T with a time-dependentvelocity V(t). Several velocity functions have been assumed, subject to the con-straints that V(0)=V(T)=0 and that the average value be equal to LR/T for all ofthem. The numerical values used here are values that can be expected in naturalexamples (see HARBITZ et al., 1991; HEINRICH et al., 1999; TINTI et al., 1999). Thegoal is to understand the effect of body velocity changes on wave generation.Figure 5 shows some snapshots of the wave profiles taken at different times forvarious trials corresponding to average Froude numbers larger than 1 (Fig. 5a) andsmaller than 1 (Fig. 5b). The last panel in both cases displays the velocity curvesused, together with the special case of a constant-velocity or box function. Thecurves are rather simple. The wavemaker is supposedly subject to acceleration inthe first phase of motion until it reaches its peak velocity, and then slows until itstops. For the variable-velocity cases illustrated in Figure 5a, the body speed passesfrom an initial subcritical regime (VBc) to a supercritical flow (V\c) duringwhich it achieves its maximum value, and then it reverts finally to subcritical. Onthe other hand, all cases exhibited in Figure 5b are pure subcritical flows, since thebody speed is always less than the wave celerity c. Typically, for real cases withmasses detaching on steep slopes the acceleration phase is shorter than thefollowing deceleration stage, but, for the sake of completeness, time histories takenhere for the body speed contemplate even cases where relative duration of acceler-ation and deceleration intervals are equal or reversed. Graphs show the axial profileof the excited waves, with all cases plotted together to favor intercomparison. Thefirst snapshots are taken during the generation phase, while the body is still moving.The most relevant remark that can be made on examining these results shown inFigures 5a and 5b is that all profiles are quite similar, differing only for minor,though relevant, details, especially after body motion has ceased (t\T). It isconfirmed that the body generates two crest-trough systems, one moving forwardand the other backward, the higher amplitude waves travelling in the same directionas the slide. In cases with the average Froude number larger than 1 (Fig. 5a), theforced wave is the leading positive wave that, for t\T, splits in two separate freewaves moving in opposite directions. When the average Froude number is smallerthan 1 (Fig. 5b), the forced wave is negative and follows the free leading wave.Despite the closeness of the wave profiles, there is a difference between the case of


constant-velocity and changing-velocity bodies that deserves attention. In theformer case, free wave production is practically concentrated at the initial and finalinstants of the motion when the body is subject to velocity discontinuities, or toinfinitely large accelerations. When speed varies continuously, then free wavegeneration occurs on a continuous interval of time and over a finite region of space.

Fig. 5.


Figure 5Numerically computed wave profiles in a 1-D, 6.5-km long, 100-m deep flat channel. Wavemaker bodymoves with time-dependent velocity in the interval [0, 30 s] over a distance of 1.5 km (a) and 0.3 km (b).The body is 1-km long and 1-m high. It initially covers the region between x=2 km and x=3 km. Asmany as six velocity functions, all which have the same average value (50 m/s in (a) and 10 m/s in (b)),have been considered. They are shown in the lower right-corner panel. Each wave-profile graph is asnapshot taken at a given time. Wave profiles corresponding to all six velocity time-histories are plotted

superposed to expedite comparison.


Table 1

Wa6e energy calculated through the definition integrals (4) for trials illustratedin Figure 5a. Velocity function Vi (i=0, 1, . . . , 5) are plotted in the last panelof Figure 5a. All 6alues are normalized to the total energy computed through eq.

(27a) for the constant-6elocity body (6elocity function V0)

EPS�EPS

�+EPS� EPS

�Velocity function

0.945V0=50 m/s 0.05510.0560.631V1 0.687

0.547V2 0.0550.6020.590 0.535 0.055V3

0.602 0.547 0.055V4

0.0580.616V5 0.674

This implies that waves are broader and that wave interference is partially destruc-tive, with the further consequence that the quantity of energy carried by the freewave systems is expected to be less compared to the constant-speed case. Table 1reports the wave energies calculated numerically for all trials plotted in Figure 5aby means of integrals (4) later than separation of the wave systems going left andright has taken place. Values are normalized over the total energy EIS(T) of thecorresponding constant-speed body that is computed through the formula (27a). Itis seen that the wave-field total energy reduces to 60–70 percent when body speedchanges smoothly, and that it is higher for those cases in which the body is subjectto larger accelerations. Therefore, it seems that body acceleration is a key-factor forthe efficacy of the wave production process. A further important observation is thatthe wave system most affected by the energy reduction is the one going forwardcarrying the quantity of energy EPS

� . The wave front moving backward is largelyminor, but its energy EPS

� is less drastically dependent upon the body velocity timehistory.

4.2 Variable-6elocity Body Sliding Down a Slope

The second series of experiments concerns a body sliding down an incline withconstant slope. The velocity functions used for body movement are identical tothose used in the previous subsection where channel depth was taken to be uniform.The basin is a rectangular channel whose depth passes gently from 100 m down to400 m over a distance of 5.5 km. The problem treated here is again 1-D, since theslide is supposedly uniform across the channel. The numerical results are shown inFigure 6 where wave profiles computed with the numerical code at different instantsare plotted together for all the considered velocity functions. The waves generatedresemble the wave systems computed for the cases studied above. Obvious wavemodifications are introduced by propagation over the bottom slope: amplitudesdecrease downslope according to Green’s law, and correspondingly wavelengths


Figure 6Wave profiles computed numerically in 1-D incline. It is 5.5-km long over the x axis, with depth rangingfrom 100 m to 400 m. The initial horizontal position of the landslide is between x=1.5 km and x=2.5km. Assumed body velocity functions are plotted in the last panel and are the same used for flat channelexperiments. Wave profiles are superposed in the various graphs, each corresponding to a given

evolution time.


increase, the reverse being true for waves climbing the slope. However, if we focuson the specific aspects concerning wave generation, changes are not dramatic. Thedouble crest-trough system is clearly visible in Figure 6 and is radiated down thechannel with a leading crest, while it propagates backward with a leading trough.Since local wave celerity is now a function of the position, the Froude numberassociated with the body motion changes not only because the body speed changes,but also because it approaches deeper water. If V(t) is the velocity time history ofthe sliding mass (here meant as the time-dependent horizontal velocity component),then the Froude number rate can be written as:

dFrdt

=1c

dVdt

−Vc2

dcdt

=1c

dVdt

−Fr2 dcdx

=1c�dV

dt−

12

gFr2 dhdx�

=1c�dV

dt−

12

mgFr2�(46)

where use has been made of the definition of wave celerity (6) and of the Froudenumber, and where the constant gradient of the depth has been denoted by m. It isthen clear that the effective Froude number on a slope is less than that associatedwith a flat bottom (m=0), since in the last algebraic sum the second term tends todiminish the current value of Fr. For the velocity functions taken in these trials thisyields to effective Froude numbers close to 1, with the consequence that down-channel waves are considerably higher than those travelling up-channel. A relevantfeature is that synchronization of wavefronts corresponding to different bodyvelocity function is worse than in the flat-channel basin, as may be clearlyappreciated from snapshots of profiles given in Figure 6. Table 2 lists values ofenergy of the wave systems travelling up- and down-channel as well as of the totalwave energy. Comparing it to Table 1, it is realized that basic features areconserved. The constant-velocity body results to be the most effective in producingwaves, and down-slope energy propagation EPS

� is largely predominant. Differenceregard the percentage reduction computed for variable velocity functions, thatcomprises between 70 and 80 percent, instead of 60–70 percent, as well as the

Table 2

Wa6e energy for trials diagrammed in Figure 6, corresponding to the constant-slope channel. Normalization is performed o6er the total energy computed

numerically for the constant-speed body (Box function V0)

EPS�Velocity function EPS

�EPS�+EPS

�

V0=50 m/s 1 0.981 0.0190.739 0.720 0.019V1

0.715 0.697 0.018V2

0.0180.6900.708V3

V4 0.709 0.690 0.0190.0190.778V5 0.797


normalized upward-radiated amount of energy (about 2 10−2 against 5–6 10−2 ofTable 1), which may be explained in terms of the effective Froude numberassociated with the slope.

5. Numerical Solutions in a 2-D Water Basin

All experiments performed in the previous sections were subject to the restric-tion that both landslide and basin cross sections have no dependence on the lateralcoordinate y. However, it is worthwhile investigating water wave generation andpropagation when the above constraint is no longer imposed. A finite-width bodymoving in open ocean is such a case, which demands solving the governingequations in their full 2-D version (2). Since interest is focused on the fundamentalaspects of the produced wavefields, only the essential case of the finite-durationslide with constant speed in a flat basin will be considered here, for all the foregoingdiscussion in 1-D has taught that this very simple case is nonetheless an excellenttool to cast light on the process. The problem is solved numerically by means of thefinite-element code used for previous experiments. The body height is assumed topossess a finite width WS and rectangular cross section for any given axial position.Mathematically, its height hS(x, y, t) is described by

hS(x, y, t)=hS(x)�

H0�

y+WS

2�

−H0�

y−WS

2�n

(47)

where hS(x) is the same bump-like profile taken in all cases treated to date, andH0(j) is the Heaviside step function defined in (17). The body moves along the xaxis with constant speed. Due to the assumed symmetry of the body with respect tothe x axis, the solution is expected to possess the same symmetry, i.e., z(x, y, t)=z(x, −y, t), and v(x, y, t)=v(x, −y, t), which entails that the y component of thevelocity 6 vanishes on the axis: 6(x, 0, t)=0. Since this condition is also appropriatefor a rigid vertical wall, it is convenient to calculate the solution only in the positivey half basin, with the constraint of no water flow across the boundary placed aty=0. A sketch of the geometry is shown in the first panel of Figure 7. Body speedis supercritical with Froude number Fr$2.24. Wave elevation fields are portrayedin Figure 7 at intervals of 10 s, covering the entire slide duration T=30 s, as wellas the following 60 s where free propagation takes place. Figure 8 shows waveprofiles computed at the same snapshot times along the x axis, which is thesymmetry axis of the problem. Profiles are displayed in the left panels, whereas onthe right-side panels the corresponding 1-D waveforms are also plotted for com-parison.

Looking at the waveforms it can be observed that the resemblance between 1-Dand 2-D solutions is somewhat close. Indeed in both cases trough-crest systemstravel up- and down-channel, with leading trough and leading crest respectively,and with forward going waves being considerably higher. Figure 8 also reveals


however relevant differences that are fully evident in the sequence of wave fieldsdepicted in Figure 7. The first difference concerns wave height. 1-D profiles arecharacterized by larger wave amplitude than 2-D, with waves that do not change

Fig. 7.


Figure 7Water oscillation fields computed numerically at intervals of 10 s. The top panel is a sketch of the basin,which is a rectangle. Its bottom side is closed with no water flow allowed, while the other sides are openand fully transparent to water waves. The finite-element mesh is formed by triangles with equal size.Solid (dashed) contour lines are used for positive (negative) waves. Contour labels are in cm. The slideduration is supposedly 30 s long. The angle u formed by the straight wavefront and the direction of the

slide motion is shown in the panel corresponding to t=30 s.


Figure 8Wave profiles computed numerically along the x-axis, that is the symmetry axis for this problem. Plotson the left column refer to this case, those on the right are 1-D wave profiles calculated analytically bymeans of formulas (20)–(21) and (23)–(24). Each graph displays solutions computed at three different

times.


amplitude and shape after that separation of up- and down-travelling systems hasoccurred. The second difference concerns the number of waves. Wave systems consistonly of one positive and one negative wave in 1-D, while in 2-D they manifest aslonger oscillation trains where the initial two impulses are followed by othersmaller-amplitude waves. In particular, it is worth noticing that the advancing frontcan be seen as a system of two crests embedding an intermediate trough. Figure 7helps significantly explain the reason for these diverse behaviors. In 1-D wavefrontsare straight, and energy is transported either forward or backward, while in 2-D frontsare curvilinear and substantial energy radiation takes place even at other azimuths.Accordingly, wave amplitudes decrease as waves propagate, owing simply togeometrical spreading. Backgoing fronts are almost circular waves, while foregoingfronts have more complex geometry. Indeed the supercritical body speed assumedin this illustrative example causes wavefronts to be parallel to the slide front onlyin the proximity of the x axis, and to be nearly straight along a different directionat larger distances from it. The acute angle, say u, formed by the x axis, which issupposedly the direction of slide motion, and the wavefront, depends upon the bodyspeed and the free-mode phase celerity, and according to wave theory it may beestimated by means of the relation sin u=Fr−1. For this case, this yields u$29°(see panels corresponding to t=10–30 s). When the wavemaker no longer moves,circular wavefront radiation prevails, even in a forward direction. From a mathemat-ical point of view, 1-D cases may be seen as limiting cases of 2-D wavemakers asslide width WS goes to infinity. Practically, 1-D computations are reasonableapproximations of 2-D waves in the neighborhood of the central axis of the slide (xaxis) for initial times. Departure between these two solutions increases as long as theobserver moves away from the slide track and as long as time passes, since 2-D lateralradiation effects become increasingly more important.

The energy of the generated wavefield is shown in Figure 9 as a function of time.It has been computed numerically over the computational domain, which is therectangle sketched in the first panel of Figure 7, by means of integrals (4). The energygrows steadily during slide motion until it attains a maximum value in correspondencewith the slide arrest, after which it remains constant. Subsequently, energy drops nearlyto zero, due to the waves crossing the basin boundary as they radiate away from theirsource. At the final simulation time (t=200 s) only negligible numerical noise survivesin the interior of the basin. Numerical calculations of the cumulative energy flux acrossthe entire external boundary, performed by means of the defining relation (8), showsthat the energy function decrease is compensated by an equal increase of the fluxfunction. The energy growth of water waves at the generation phase can be comparedwith the energy function of the corresponding 1-D case, which is plotted in the sameFigure. At shorter times the two functions overlap, which means that 1-D and 2-Dwave generations are similar in this limit. Later, however, departure is drastic, forenergy saturates in 1-D, but it grows continuously in 2-D, and only body arrest preventsit from becoming increasingly larger. Therefore the final energy is an increasingfunction of, and is approximately linearly related to, the total slide duration. Observe


Figure 9Total wave energy for the 2-D case illustrated in Figures 7 and 8 computed numerically by means ofintegrals (4). The total simulation time is 200 s. Energy decrease is due to waves abandoning thecomputational domain. The cumulative energy flux across the boundary has been calculated byintegrating expression (8) over the open boundaries of the grid and then by integrating it over time inthe interval [0, t ]. Energy and flux functions are slightly corrected to account for undesired numericalenergy dissipation. Analytical total energy for the corresponding 1-D case is given for comparison. Allcurves are normalized to the total potential energy associated with a sudden vertical uplift of the sea

bottom as for Figure 4.

that the continuous energy growth may be mathematically associated with the factthat no steady solution exists for a finite-width wavemaker, this being a relevantdifference between 1-D and 2-D. Figure 10 illustrates the cumulative energy fluxdistribution as a function of the position over the basin boundary. The mostrelevant feature is the substantial peak that is located on the top boundary. Itcorresponds to the energy carried by the straight wavefront at the angle u$29°from the slide advancement direction (see Fig. 7). A second maximum of flux is dueto waves moving ahead along the x axis, while energy carried by backgoing wavesis comparatively much smaller. This signals a further remarkable difference between1-D and 2-D wave generations by underwater landslides.

6. Conclusions

The analyses carried out in this paper have been purposedly focused on simple1-D and 2-D experiments, for it is believed that ideal cases can better serve to gainproper knowledge of the essential features of wave generation by moving underwa-ter bodies. An appropriate schematization for a 1-D slide has been considered to bea rigid body, moving with variable velocity during a finite interval of time over a


Figure 10Distribution of the energy flux computed by integrating expression (8) over time in the interval [0, 200s] for each boundary node of the grid. The graph abscissa is the position on the open boundary of thebasin measured clockwise, starting from the left bottom corner. The passages from left to top side andfrom top to right side of the basin are marked by vertical lines. The same normalization used for Figure

9 is applied here.

slope. The effects of these assumptions have been examined separately, startingfrom a rigid body moving with constant velocity on a flat ocean, a problemadmitting a steady-state solution. Thereafter finite-duration T has been imposed,which admits analytical solutions for wave and water energy, enabling the readystudy of the dependence from the relevant dynamic parameters. Variable-velocityfunctions for the body and constant bottom slope have been handled by numericalmeans. It has been shown that 1) two wave systems are generated travelling forward(slide direction) and backward; 2) these systems are both formed by a trough anda crest; 3) the trough precedes the crest in the backgoing wave train and 6ice 6ersafor the other; 4) waveforms depend greatly upon landslide profile; 5) waveamplitudes depend mainly on the Froude number, the larger waves travellingforward; 6) most of the energy is radiated forward; 7) the amount of energy storedin the water body depends on slide duration; 8) if slide duration is sufficiently long,the energy reaches a saturation plateau, unless the body moves at critical speed, inwhich case the energy could grow unlimited with time, and its maximum is attainedat the end of the slide forcing interval; 9) the leading regressing trough and theleading advancing crest if Fr\1 (or the trough following the leading crest ifFrB1), are mainly produced by the accelerating body at the initial phase of slidemotion; 10) the other two impulses are mainly produced by the decelerating bodyin the final phase of the slide motion; 11) body velocity changes play a key role inwater waves production and energy transfer; 12) a variable-velocity body generatesbroader waves; 13) all above characteristics are only slightly affected by wave


propagation on non-flat bottom, such as a constant-slope incline; 14) the expectedmodifications regarding wave propagation are amplitude increases for waves climb-ing the incline, accompanied by wavelength contraction; 15) varying depth impliesthe concept of local phase celerity and consequently of local Froude number.

Wave generation in 2-D has been studied by means of a single example, that isa finite-length, finite-width body moving during a finite interval of time withconstant supercritical speed in an open flat ocean. The solution has been computednumerically. The analysis of this case has been a valuable asset to draw thefollowing observations: 16) wave profiles along the slide axis are somewhat close tothose computed for the corresponding 1-D approximation; 17) the back-radiatingwave system is formed mainly by a leading trough and a following positive wave,with prevailing circular radiation; 18) the advancing front is chiefly formed by aleading crest, a following trough, and then a second crest, and expectedly a seriesof other smaller oscillations; 19) during body motion, the advancing front is mainlystraight with direction forming an angle with the slide track depending uponFroude number if Fr\1; 20) after the slide end, circular wave radiation alsodominates the advancing system; 21) transfer of energy from body to waves is verysimilar to 1-D and 2-D only at the initial stage of body motion; 22) while body issliding with constant speed, wave energy grows steadily in 2-D and reaches nosaturation level; 23) propagation of energy corresponding to the straight wavefrontis predominant if Fr\1.

Similarities between 1-D and 2-D wave generations are relevant, nonethelessdifferences have been shown to be very substantial. This paper has investigated 1-Dcases with full detail, however 2-D solutions have only been touched and not deeplyexamined, and deserve to be the subject of future research and publications.

Acknowledgements

The authors acknowledge that the present work has been funded partially by theGNV (Gruppo Nazionale di Vulcanologia) of Italian CNR (Consiglio Nazionaledelle Ricerche) and partially by MURST (Ministero dell’Universita e della RicercaScientifica e Tecnologica).

Appendix A

Appendix A is devoted to the comparison between the energy of a landslide-generated tsunami and the energy of a tsunami caused by an earthquake. For thesake of simplicity, we will refer to the simple 1-D case of a tsunami propagating inan ocean with a flat floor treated in the main text. To favor comparison, let usassume that the sea bottom deformation due to the earthquake has the same


dimensions and shape as the landslide body, i.e. that it has the same width W anda longitudinal shape that can be mathematically described by the same function hS.We will further assume that the bottom deformation transmits instantaneously tothe free water surface, keeping the same shape, and accordingly, that the initialconditions for the water waves are:

z(x, 0)=hS(x),

u(x, 0)=0.

Under these hypotheses the solution to the shallow water equations (2a, 2b) is givenby:

z(x, t)=12

(hS(x−ct)+hS(x+ct)), (A1a)

u(x, t)=12

gc

(hS(x−ct)−hS(x+ct)) (A1b)

which is a system of two free waves propagating forward and backward. The energyassociated with these waves may be derived very easily and, if use is made of thesame notation as adopted in the main text, can be expressed as follows:

EP(t)=14

rgW(R(0)+R(2ct)), (A2a)

EK(t)=14

rgW(R(0)−R(2ct)), (A2b)

Equake=EP(t)+EK(t)=12

rgWR(0), (A2c)

where Equake is the total energy of the waves produced by the earthquake that isconstant in time. Figure 11a shows the potential and kinetic energy vs. time duringthe wave propagation. At the initial time the potential energy has its maximumvalue and the kinetic energy is null. As time elapses, potential energy is partiallyconverted into kinetic, with the total energy conserving its initial value. After thetime interval T1=LS/2c, the two waves separate completely and both energy termsreach their final value.

EP(t)=EK(t)=0.25rgWR(0), t\T1. (A3)

Let us now compare the total energy Equake computed above with the total energy,say Eslide, of a tsunami generated by a landslide moving across the sea bottom witha constant velocity V, corresponding to a Froude number Fr, for a certain periodT. For the sake of simplicity, we can surmise that T is sufficiently long so that Eslide

attains its maximum possible value, that in the case of Fr"1 is given by eq. (29),while in the case of Fr=1 is infinitely large. Under these conditions, the ratio


Figure 11a) Potential and kinetic energy normalized over the total energy of an earthquake-generated tsunami.The water displacement considered as an initial condition is hS, given by eq. (26). Notice that exactequipartition of the total wave energy into potential and kinetic occurs after the full separation of thefree waves travelling backward and forward has taken place. b) Ratio h of the earthquake-generated

tsunami total energy Equake over the landslide-generated tsunami energy Eslide vs. Froude number.

h=Equake/Eslide results to be a function of only the landslide Froude number and isgiven by:

h=Equake

Eslide

=(Fr2−1)2

2Fr2(Fr2+1). (A4)

The plot of h vs. Fr is shown in Figure 11b. It is straightforward to see that thereis only one real positive value Fr0= (5−2)1/2:0.486 for which h equals unity,and accordingly Eslide=Equake. In the interval [0, Fr0], the ration h exceeds unitywith EslideBEquake while in the range of larger Froude numbers the energyinequality is reversed, i.e., Eslide\Equake. In the special case Fr=1, the ratio isvanishingly small, since Eslide is infinitely large. Therefore, the value Fr0 could beused to evaluate the efficiency of tsunami generation by earthquakes vs. landslides.In a sense, it can be interpreted as the relative efficiency per unit area of the source.Landslides occurring in coastal areas where the sea is shallow (hB1000 m) andslopes are very steep are typically characterized by large Froude numbers (Fr\Fr0), and consequently they are more efficient than earthquakes in producing


energetic tsunamis. Conversely, landslides taking place in deep waters over gentleslopes attain lower velocities and small Froude numbers, and happen to be lessefficient than earthquakes as tsunami sources. It is important to remember that theabove considerations hold, provided that the tsunami source region associated withthe earthquake has the same size as the landslide. Typically, tsunamigenic earth-quakes involve a submarine area much larger than landslides, and accordingly thetsunamis they induce possess considerably more energy, however the ratio h

introduced above can nevertheless be useful in estimating the local efficiency of thegeneration process. For example, if a large earthquake generates a tsunami and atthe same time triggers a landslide in coastal waters that in turn induces a secondtsunami, the total energy possessed by the tsunami due to the earthquake isexpected to be order of magnitudes larger than the total energy possessed by thelandslide-induced tsunami. Yet the latter might be locally substnatially moreenergetic than the former, since according to the eq. (A4), it could be characterizedby a larger amount of energy per unit area.

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(Received November 30, 1998, accepted June 7, 1999)

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Energy of Water Waves Induced by Submarine Landslides

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