A numerical study of submarine-landslide-generated waves and run-up

Transcription

1 .98/ rspa A numerical sudy of submarine-landslide-generaed waves and run-up By Parick Lyne a n d Philip L.-F. L iu School of Civil and Environmenal Engineering, Cornell Universiy, Ihaca, NY 4853, USA Received 4 Ocober 2; acceped 8 February 22; published online 3 Sepember 22 A mahemaical model is derived o describe he generaion and propagaion of waer waves by a submarine landslide. The model consiss of a deph-inegraed coninuiy equaion and momenum equaions, in which he ground movemen is he forcing funcion. These equaions include full nonlinear, bu weak frequencydispersion, e ecs. The model is capable of describing wave propagaion from relaively deep waer o shallow waer. Simpli ed models for waves generaed by small sea oor displacemen or creeping ground movemen are also presened. A numerical algorihm is developed for he general fully nonlinear model. Comparisons are made wih a boundary inegral equaion mehod model, and a deep-waer limi for he deph-inegraed model is deermined in erms of a characerisic side lengh of he submarine mass. The imporance of nonlineariy and frequency dispersion in he wave-generaion region and on he shoreline movemen is discussed. Keywords: landslide sunamis; Boussinesq equaions; wave run-up. Inroducion In recen years, signi can advances have been made in developing mahemaical models o describe he enire process of generaion, propagaion and run-up of a sunami even (e.g. Yeh e al. 996; Geis 998). These models are based primarily on he shallow-waer wave equaions and are adequae for sunamis generaed by seismic sea oor deformaion. Since he duraion of he seismic sea oor deformaion is very shor, he waer-surface response is almos insananeous and he iniial waersurface pro le mimics he nal sea oor deformaion. The ypical wavelengh of his ype of sunami ranges from 2 o km. Therefore, frequency dispersion can be ignored in he generaion region. The nonlineariy is also usually no imporan in he generaion region, because he iniial wave ampliude is relaively small compared o he wavelengh and he waer deph. However, he frequency dispersion becomes imporan when a sunami propagaes for a long disance. Nonlineariy could also dominae as a sunami eners he run-up phase. Consequenly, a complee model ha can describe he enire process of sunami generaion, evoluion and run-up needs o consider boh frequency dispersion and nonlineariy. Tsunamis are also generaed by oher mechanisms. For example, submarine landslides have been documened as one of many possible sources for several desrucive 458, 2885{ c 22 The Royal Sociey

2 2886 P. Lyne and P. L.-F. Liu sunamis (Moore & Moore 984; von Huene e al. 989; Jiang & LeBlond 992; Tappin e al. 999; Keaing & McGuire 22). On 29 November 975, a landslide was riggered by a 7.2-magniude earhquake along he souheas coas of Hawaii. A 6 km srech of Kilauea s souh coas subsided 3.5 m and moved seaward 8 m. This landslide generaed a local sunami wih a maximum run-up heigh of 6 m a Keauhou (Cox & Morgan 977). More recenly, he devasaing Papua New Guinea sunami in 998 is hough o have been caused by a submarine landslide (Tappin e al. 999, 2; Keaing & McGuire 22). In erms of sunami-generaion mechanisms, wo signi can di erences exis beween submarine-landslide and coseismic sea oor deformaion. Firs, he duraion of a landslide is much longer and is in he order of magniude of several minues. Hence he ime-hisory of he sea oor movemen will a ec he characerisics of he generaed wave and needs o be included in he model. Secondly, he e ecive size of he landslide region is usually much smaller han he coseismic sea oor-deformaion zone. Consequenly, he ypical wavelengh of he sunamis generaed by a submarine landslide is also shorer, i.e. ca. { km. Therefore, he frequency dispersion could be imporan in he wave-generaion region. The exising numerical models based on shallow-waer wave equaions may no be suiable for modelling he enire process of submarine-landslide-generaed sunami (e.g. Raney & Buler 976; Harbiz e al. 993). In his paper, we shall presen a new model describing he generaion and propagaion of sunamis by a submarine landslide. In his general model, only he assumpion of weak frequency dispersion is employed, i.e. he raio of waer deph o wavelengh is small or O( 2 ) ½. Unil he pas decade, weakly dispersive models were formulaed in erms of a deph-averaged velociy (e.g. Peregrine 967). Recen work has clearly demonsraed ha modi caions o he frequency dispersion erms (Madsen & Sorensen 992) or expression of he model equaions in erms of an arbirarylevel velociy (Nwogu 993; Liu 994) can exend he validiy of he linear-dispersion properies ino deeper waer. The general guideline for dispersive properies is ha he `exended versions of he deph-inegraed equaions are valid for wavelenghs greaer han wo waer dephs, whereas he deph-averaged model is valid for lenghs greaer han ve waer dephs (e.g. Nwogu 993). Moreover, in he model presened in his paper, he full nonlinear e ec is included, i.e. he raio of wave ampliude o waer deph is of order one or = O(). Therefore, his new model is more general han ha developed by Liu & Earickson (983), in which he Boussinesq approximaion, i.e. O( 2 ) = O() ½, was used. In he special case where he sea oor is saionary, he new model reduces o he model for fully nonlinear and weakly dispersive waves propagaing over a varying waer deph (e.g. Liu 994; Madsen & Sch a er 998). The model is applicable for boh he impulsive slide movemen and creeping slide movemen. In he laer case, he ime duraion for he slide is much longer han he characerisic wave period. This paper is organized as follows. Governing equaions and boundary condiions for ow moions generaed by a ground movemen are summarized in he nex secion. The derivaion of approximae wo-dimensional deph-inegraed governing equaions follows. The general model equaions are hen simpli ed for special cases. A numerical algorihm is presened o solve he general mahemaical model. The numerical model is esed using available experimenal daa (e.g. Hammack 973) for one-dimensional siuaions. Employing a boundary inegral equaion model (BIEM), which solves for poenial ow in he verical plane, a deep-waer limi for waves generaed by

3 z y x Submarine-landslide-generaed waves and run-up 2887 z a h h l Figure. Basic formulaional se-up. submarine slides is deermined for he deph-inegraed model. The imporance of nonlineariy and frequency dispersion is inferred hrough numerical simulaion of a large number of di eren physical se-ups. 2. Governing equaions and boundary condiions As shown in gure, ± (x ; y ; ) denoes he free-surface displacemen of a wave rain propagaing in he waer deph h (x ; y ; ). Inroducing he characerisic waer deph h as he verical lengh-scale, he characerisic lengh of he submarine slide region ` as he horizonal lengh-scale, `= p gh as he ime-scale, and he characerisic wave ampliude a as he scale of wave moion, we can de ne he following dimensionless variables, (x; y) = (x ; y p ) ; z = z gh ; = ; ` h ` and h = h h ; ± = ± a ; p = p ga (u; v) = (u ; v ) p gh ; w = w (= ) p gh ; (2.) in which (u; v) represens he horizonal velociy componens, w he verical velociy componen, and p he pressure. Two dimensionless parameers have been inroduced in (2.), which are = a ; = h : (2.2) h Assuming ha he viscous e ecs are insigni can, he wave moion can be described by he coninuiy equaion and Euler s equaions, i.e. ` 2 r u + w z = ; (2.3) u + u ru + 2 wu z = rp; (2.4) w + 2 u rw ww z = p z ; (2.5)

4 2888 P. Lyne and P. L.-F. Liu where u = (u; v) denoes he horizonal velociy vecor, r = (@=@ y) he horizonal gradien vecor, and he subscrip he parial derivaive. On he free surface, z = ± (x; y; ), he usual kinemaic and dynamic boundary condiions apply, w = 2 (± + u r± ) on z = ± ; (2.6 a) p = : (2.6 b) Along he sea oor, z = h, he kinemaic boundary condiion requires w + 2 u rh + 2 h = on z = h: (2.7) For laer use, we noe here ha he deph-inegraed coninuiy equaion can be obained by inegraing (2.3) from z = h o z = ±. Afer applying he boundary condiions (2.6), he resuling equaion reads Z ± r u dz + h H = ; (2.8) where H = ± + h: (2.9) We remark here ha (2.8) is exac. 3. Approximae wo-dimensional governing equaions The hree-dimensional boundary-value problem described in he previous secion will be approximaed and projeced ono a wo-dimensional horizonal plane. In his secion, he nonlineariy is assumed o be of O(). However, he frequency dispersion is assumed o be weak, i.e. O( 2 ) ½ : (3.) Using 2 as he small parameer, a perurbaion analysis is performed on he primiive governing equaions. The complee derivaion is given in Appendix A. The resuling approximae coninuiy equaion is h + ± + r (Hu ) ½ 2 r H ( 6 (2 ± 2 ± h + h 2 ) 2 z2 )r(r u ) µ + ( 2 (± h) z )r r (hu ) + h ¾ = O( 4 ): (3.2) Equaion (3.2) is one of hree governing equaions for ± and u. The oher wo equaions come from he horizonal momenum equaion (2.4) and are given in vecor

5 ± Submarine-landslide-generaed waves and run-up 2889 form as u + u ru + r± 2 2 z2 r(r u ) + z r r (hu ) + h ¾ ½ 2 + r (hu ) + h r r (hu ) + h µ r r (hu ) + h + (u rz )r r (hu ) + h µ + z r u r r (hu ) + h + z (u rz )r(r u ) ¾ + 2 z2 r[u r(r u )] + 2 ½ 2 r 2 ± 2 r u ± u r r (hu ) + h + ± r (hu ) + h ¾ r u rf 2 ± 2 [(r u ) 2 u r(r u )]g = O( 4 ): (3.3) Equaions (3.2) and (3.3) are he coupled governing equaions, wrien in erms of u and ±, for fully nonlinear weakly dispersive waves generaed by a sea oor movemen. We reierae here ha u is evaluaed a z = z (x; y; ), which is a funcion of ime. The choice of z is made based on he linear frequency-dispersion characerisics of he governing equaions (e.g. Nwogu 993; Chen & Liu 995). Assuming a saionary sea oor, in order o exend he applicabiliy of he governing equaions o relaively deep waer (or a shor wave), z is recommended o be evaluaed as z = :53h. In he following analysis, he same relaionship is employed. These model equaions will be referred o as FNL-EXT, for fully nonlinear `exended equaions. Up o his poin, he ime-scale of he sea oor movemen is assumed o be in he same order of magniude as he ypical period of generaed waer wave, w = `= p gh as given in (2.). When he ground movemen is creeping in naure, he ime-scale of sea oor movemen, c, could be larger han w. The only scaling parameer ha is direcly a eced by he ime-scale of he sea oor movemen is he characerisic ampliude of he wave moion. Afer inroducing he ime-scale c ino he ime derivaives of h in he coninuiy equaion (3.2), along wih a characerisic change in waer deph h, he coe cien in fron of h becomes w ; c where = h=h. To mainain he conservaion of mass, he above parameer mus be of order one. Thus w = = l p : (3.4) c c gh The above relaionship can be inerpreed in he following way. During he creeping ground movemen, over he ime period < c he generaed wave has propagaed a disance p gh. The oal volume of he sea oor displacemen, normalized by h, is l (= c ), which should be he same as he volume of waer underneah he generaed

6 289 P. Lyne and P. L.-F. Liu wave cres, i.e. p gh. Therefore, over he ground-movemen period, < c, he wave ampliude can be esimaed by (3.4). Consequenly, nonlinear e ecs become imporan only if de ned in (3.4) is O(). Since, by he de niion of a creeping slide, he value l =( c p gh ) is always less han one, fully nonlinear e ecs will be imporan for only he larges slides. The same conclusion was reached by Hammack (973), using a di eren approach. The imporance of he fully nonlinear e ec when modelling creeping ground movemens will be esed in x8. 4. Limiing cases In his secion, he general model is furher simpli ed for di eren physical condiions. (a) Weakly nonlinear waves In many siuaions, he sea oor displacemen is relaively small in comparison wih he local deph, and he sea oor movemen can be approximaed as h(x; y; ) = h (x; y) + h(x; y; ); (4.) in which is considered o be small. In oher words, he maximum sea oor displacemen is much smaller han he characerisic waer deph. Since he free-surface displacemen is direcly proporional o he sea oor displacemen, i.e. O(± ) = O( h), or much less han he sea oor displacemen in he case of creeping ground movemens, we can furher simplify he governing equaions derived in he previous secion by allowing O() = O( ) = O( 2 ) ½ ; (4.2) which is he Boussinesq approximaion. Thus he coninuiy equaion (3.2) can be reduced o ± + r (Hu ) + h 2 r ½h ( 6 h2 µ 2 z2 )r(r u ) ( 2 h + z )r r (h u ) + ¾ h = O( 4 ; 2 ; 2 ): (4.3) The momenum equaion becomes u + u ru + r± ½ 2 z2 r(r u ) + z r r (h u ) + ¾ h = O( 4 ; 2 ; 2 ): (4.4) These model equaions will be referred o as WNL-EXT, for weakly nonlinear `exended equaions. The linear version of he above will also be used in he following analysis, and will be referred o as L-EXT, for linear `exended equaions. I is also possible o express he approximae coninuiy and momenum equaions in erms of a deph-averaged velociy. The deph-averaged equaions can be derived using he same mehod presened in Appendix A. One version of he deph-averaged

7 Submarine-landslide-generaed waves and run-up 289 equaions will be employed in fuure secions, which is subjec o he resrain (4.2), and is given as ± + r (H u) + h = (4.5) and u + u r u + r± ½ 2 h2 r(r u) where he deph-averaged velociy is de ned as 6 h r r (h u) + ¾ h = O( 4 ; 2 ; 2 ); (4.6) u(x; y; ) = h + ± Z ± h u(x; y; z; ) dz: (4.7) This se of model equaions (4.5) and (4.6) will be referred o as WNL-DA, for weakly nonlinear deph-averaged equaions. (b) Nonlinear shallow-waer waves In he case ha he waer deph is very shallow or he wavelengh is very long, he governing equaions (3.2) and (3.3) can be runcaed a O( 2 ). These resuling equaions are he well-known nonlinear shallow-waer equaions in which he sea oor movemen is he forcing erm for wave generaion. This se of equaions will be referred o as NL-SW, for nonlinear shallow-waer equaions. 5. Numerical model In his paper, a nie-di erence algorihm is presened for he general model equaions, FNL-EXT. This model has he robusness of enabling slide-generaed surface waves, alhough iniially linear or weakly nonlinear in naure, o propagae ino shallow waer, where fully nonlinear e ecs may become imporan. The algorihm is developed for he general wo-horizonal-dimension problem; however, in his paper, only one-horizonal-dimension examples are examined. The srucure of he presen numerical model is similar o hose of Wei & Kirby (995) and Wei e al. (995). Di erences beween he model presened here and ha of Wei e al. exis in he added erms due o a ime-dependen waer deph and he numerical reamen of some nonlinear-dispersive erms, which is discussed in more deail in Appendix B. A high-order predicor-correcor scheme is used, employing a hird order in ime explici Adams{Bashforh predicor sep, and a fourh order in ime Adams{Moulon implici correcor sep (Press e al. 989). The implici correcor sep mus be ieraed unil a convergence crierion is sais ed. All spaial derivaives are di erenced o fourh-order accuracy, yielding a model ha is numerically accurae o ( x) 4, ( y) 4 in space and ( ) 4 in ime. The governing equaions (3.2) and (3.3) are dimensionalized for he numerical model, and all variables described in his and following secions will be in he dimensional form. Noe ha he dimensional equaions are equivalen o he non-dimensional ones wih = = and he addiion of graviy, g, o he coe cien of he leading-order free-surface derivaive in he momenum equaion (i.e. he hird erm on he lef-hand side of (3.3)). The predicor-correcor

8 2892 P. Lyne and P. L.-F. Liu equaions are given in Appendix B, along wih some addiional descripion of he numerical scheme. Run-up and rundown of he waves generaed by he submarine disurbance will also be examined. The moving-boundary scheme employed here is he echnique developed by Lyne e al. (22). Founded around he resricions of he high-order numerical wave-propagaion model, he moving-boundary scheme uses linear exrapolaion of free surface and velociy hrough he shoreline, ino he dry region. This approach allows for he ve-poin nie-di erence formulae o be applied a all poins, even hose neighbouring dry poins, and hus eliminaes he need of condiional saemens. In addiion o he deph-inegraed-model numerical resuls, oupu from a wodimensional (verical-plane) BIEM model will be presened for cerain cases. This BIEM model will be primarily used o deermine he deep-waer-accuracy limi of he deph-inegraed model. The BIEM model solves for inviscid irroaional ows and convers a boundary-value problem ino an inegral equaion along he boundary of a physical domain. Therefore, jus as wih he deph-inegraion approach, i reduces he dimension of he problem by one. The BIEM model used here solves he Laplace equaion in he verical plane (x; z), and, of course, is valid in all waer dephs for all wavelenghs. Deails of his ype of BIEM model, when used o model waer-wave propagaion, can be found in Grilli e al. (989), Liu e al. (992) and Grilli (993), for example. The BIEM model used in his work has reproduced he numerical resuls presened for landslide-generaed waves in Grilli & Was (999) perfecly. 6. Comparisons wih experimen and oher models As a rs check of he presen model, a comparison beween Hammack s (973) experimenal daa for an impulsive boom movemen in a consan waer deph is made. The boom movemen consiss of a lengh, l = 24:4 waer dephs, which is pushed verically upward. The change in deph for his experimen,, is., so nonlinear e ecs should play a small role near he source region. Figure 2 shows a comparison beween he numerical resuls using FNL-EXT, experimenal daa and he linear heory presened by Hammack. Boh he fully nonlinear model and he linear heory agree well wih experimen a he edge of he source region ( gure 2a). From gure 2b, a ime-series aken a 2 waer dephs from he edge of he source region, he agreemen beween all daa is again quie good, bu he deviaion beween he linear heory and experimen is slowly growing. The purpose of his comparison is o show ha he presen numerical model accuraely predics he free-surface response o a simple sea oor movemen. I would seem ha if one was ineresed in jus he wave eld very near he source, linear heory is adequae. However, as he magniude of he bed uphrus,, becomes large, linear heory is no capable of accuraely predicing he free-surface response, even very near he source region. One such linear versus nonlinear comparison is shown in gure 2 for = :6. The moion of he boom movemen is he same as in Hammack s case above. Immediaely on he ouskirs of he boom movemen, here are subsanial di erences beween linear and nonlinear heory, as shown in gure 2c. Addiionally, as he wave propagaes away from he source, errors in linear heory are more eviden. A handful of experimenal rials and analyic soluions exis for non-impulsive sea oor movemens. However, for he previous work ha made use of smooh obsacles, such as a semicircle (e.g. Forbes & Schwarz 982) or a semi-ellipse (e.g. Lee e

9 z /h (a) Submarine-landslide-generaed waves and run-up 2893 (b) (c) (d) z /h (g/h).5 - x/h 2 4 (g/h).5 - x/h Figure 2. (a), (b) Comparison beween Hammack s (973) experimenal daa (dos) for an impulsive sea oor uphrus of = :, FNL-EXT numerical simulaion (solid line), and linear heory (dashed line). (a) Time-series a x=h = ; (b) ime-series a x=h = 2, where x is he disance from he edge of he impulsive movemen. (c), (d) FNL-EXT (solid line) and L-EXT (dashed line) numerical resuls for Hammack s se-up, excep wih = :6. al. 989), he lengh of he obsacle is always less han.25 waer dephs, or > :8. Unforunaely, hese objecs will creae waves oo shor o be modelled accuraely by a deph-inegraed model. Was (997) performed a se of experimens where he le a riangular block free fall down a planar slope. In all he experimens, he fron (deep-waer) face of he block was seep, and in some cases verical. Physically, as he block ravels down a slope, waer is pushed ou horizonally from he verical fron. Numerically, however, using he deph-inegraed model, he dominan direcion of waer moion near he verical face is verical. This can be explained as follows. Examining he deph-inegraedmodel equaions, saring from he leading-order shallow-waer-wave equaions, he only forcing erm due o he changing waer deph appears in he coninuiy equaion. There is no forcing erm in he horizonal momenum equaion. Therefore, in he nondispersive sysem, any sea oor boom canno direcly creae a horizonal velociy. This concep can be furher illuminaed by he equaion describing he verical pro le of horizonal velociy, u(x; y; z; ) = u (x; y; ) + O( 2 ): (6.)

10 2894 P. Lyne and P. L.-F. Liu.2 (a) (b) - d h/d - d h/d L L 2 L L L s = L 2 = L L s = L 2 < L z x x Figure 3. Graphical de niion of he characerisic side lengh of a slide mass. The slide mass a ime is shown by he solid line, while he pro le a some ime > is shown by he dashed line. The negaive of he change in waer deph (or he approximae free-surface response in he non-dispersive equaion model) during he incremen is shown by he hick line ploed on z = :. Again, he changing sea oor boom canno direcly creae a horizonal velociy componen for he non-dispersive sysem. All of he sea oor movemen, wheher i is a verical or ranslaional moion, is inerpreed as sricly a verical moion, which can lead o a very di eren generaed wave paern. When adding he weakly dispersive erms, he verical pro le of he horizonal velociy becomes u(x; y; z; ) = u (x; y; ) ½ 2 2 z2 z 2 r(r u )+(z z )r r (hu )+ h ¾ +O( (6.2) Now, wih he higher-order dispersive formulaion, here is he forcing erm, rh, which accouns for he e ecs of a horizonally moving body. Keep in mind, however, ha his forcing erm is a second-order correcion, and herefore should represen only a small correcion o he horizonal velociy pro le. Thus, wih rapid ranslaional moion and/or seep side slopes of a submarine slide, he ow moion is srongly horizonal locally, and he deph-inegraed models are no adequae. In slighly di eren erms, le he slide mass have a characerisic side lengh, L s. A side lengh is de ned as he horizonal disance beween wo poins a =. This de niion of a side lengh is described graphically in gure 3. Figure 3a shows a slide mass ha is symmeric around is midpoin in he horizonal direcion, where he back (shallow-waer) and fron (deep-waer) side lenghs are equal. Figure 3b shows a slide mass whose fron side is much shorer ha he back. Noe ha for he slide shown in gure 3b, he side lenghs, measured in he 4 ):

11 Submarine-landslide-generaed waves and run-up 2895 z x x x c x r h (x) d () h c () D h h (x, ) b Figure 4. Se-up for submarine landslide comparisons. direcion parallel o he slope, are equal, whereas for he slide in gure 3a, he slide lenghs are equal when measured in he horizonal direcion. An irregular slide mass will have a leas wo di eren side lenghs. In hese cases, he characerisic side lengh, L s, is he shores of all sides. When L s is small compared o a characerisic waer deph, h, ha side is considered seep, or in deep waer, and he shallow-waer-based deph-inegraed model will no be accurae. For he verical face of Was s experimens, L s =, and herefore L s =h =, and he siuaion resembles ha of an in niely deep ocean. The nex secion will aemp o deermine a limiing value of L s =h where he deph-inegraed model begins o fail. 7. Limiaions of he deph-inegraed model Before using he model for pracical applicaions, he limis of accuracy of he dephinegraed model mus be deermined. As illusraed above, jus as here is a shorwave accuracy limi (wave should be a leas wo waer dephs long when applying he `exended model), i is expeced ha here is also a slide lengh-scale limiaion. By comparing he oupus of his model o hose of he BIEM model, a limiing value of L s =h can be inferred. The high degree of BIEM model accuracy in simulaing wave propagaion is well documened (e.g. Grilli 993; Grilli e al. 995). The comparison cases will use a slide mass ravelling down a consan slope. The slide mass moves as a solid body, wih velociy described following Was (997). This moion is characerized by a decreasing acceleraion unil a erminal velociy is reached. All of he solid-body moion coe ciens used in his paper are idenical o hose employed by Grilli & Was (999). Noe ha all of he submarine landslide simulaions presened in his paper are non-breaking.

12 2896 P. Lyne and P. L.-F. Liu z /d L s /h c = 6. (a) z /d z /d z /d L s /h c = 4.5 L s /h c = 3. L s /h c = 2.5 (b) (c) (d) D h/d.2. = (a) (b) (c) (d) x/b Figure 5. Free-surface snapshos for BIEM (solid line) and deph-inegraed (dashed line) resuls a (g=d ) =2 values of (a).6, (b) 2, (c) 3.6 and (d) 4. (e) The locaion of he slide mass in each of he four snapshos above. The se-up of he slide mass on he slope is shown in gure 4. The ime-hisory of he sea oor is described by µ µ x h(x; ) = h (x) 2 h xl () x xr () + anh anh ; (7.) S S where h is he maximum verical heigh of he slide, x l is he locaion of he anh in ecion poin of he lef side of he slide, x r is he locaion of he in ecion poin on he righ side, and S is a shape facor, conrolling he seepness of he slide sides. The lef and righ boundaries and seepness facor are given by x l () = x c () 2 b cos( ); x r() = x c () + :5 2 b cos( ); S = cos( ) ; where x c is he horizonal locaion of he cenre poin of he slide, and is deermined using he equaions governing he solid body moion of he slide. The angle of he slope is given by. The hickness of he `slideless waer column, or he baseline waer deph, a he cenre poin of he slide is de ned by h c () = h (x c ()) = h + d(). Wih a speci ed deph above he iniial cenre poin of he slide mass, d = d( = ), he iniial horizonal locaion of he slide cenre, x c ( = ), can be found. The lengh along he slope beween x l and x r is de ned as b, and all lenghs are scaled by b. (e)

13 z /d ver. shoreline movemen / d (a) L s /h c = 3.4 (b) Submarine-landslide-generaed waves and run-up (c) L s /h c = 2.4 L s /h c = (d) L s /h c = 3.4 L s /h c = L s /h c = (g/d ) /2 (g/d ) /2 (g/d ) / Figure 6. Time-series above he iniial cenre poin of he slide ((a), (c), (e)) and verical movemen of he shoreline ((b), (d), (f )) for a 2 slope and a slide mass wih a maximum heigh h = :. BIEM resuls are shown by he solid line, deph-inegraed resuls by he dashed line. (a), (b) d =b = :4; (c), (d) d =b = :6; and (e), (f ) d =b = :. For he rs comparison, a slide wih he parameer se = 6, d =b = :2 and h=b = :5 is modelled wih FNL-EXT and BIEM. Wih hese parameers, he characerisic horizonal side lengh of he slide mass, L s =b, is.7. L s is de ned as in gure 3 or, speci cally, he horizonal disance beween wo poins a is less han % of he value. Noe ha a 6 slope is roughly. Figure 5 shows four snapshos of he free-surface elevaion from boh models. The lowes panel in he gure shows he iniial locaion of he slide mass, along wih he locaions corresponding o he four free-surface snapshos. Iniially, as shown in gure 5a; b, where L s =h c = 6: and 4.5 respecively, he wo models agree, and hus are sill in he range of accepable accuracy of he deph-inegraed model. In gure 5c, as he slide moves ino deeper waer, where L s =h c = 3:, he wo models begin o diverge over he source region, and by gure 5d, he free-surface responses of he wo models are quie di eren. These resuls indicae ha in he viciniy of x=b = 5, he deph-inegraed model becomes inaccurae. A his locaion, h c =b = :5 and L s =h c = 3:4. Numerous addiional comparison ess were performed, and all indicaed ha he deph-inegraed model becomes inaccurae when L s =h c < 3{3.5. One more of he comparisons is shown here. Examining a 2 slope and a slide mass wih a maximum heigh h=b = :, he iniial deph of submergence, d =b, will be successively increased from.4 o.6 o.. The corresponding iniial L s =h c values are 3.4, 2.4 (e) ( f )

14 2898 P. Lyne and P. L.-F. Liu and.5, respecively. Time-series above he iniial cenre poin of he slide masses and verical shoreline movemens are shown in gure 6. The expecaion is ha he rs case (L s =h c = 3:4 iniially) should show good agreemen, he middle case (L s =h c = 2:4 iniially) marginal agreemen, and he las case (L s =h c = :5 iniially) bad agreemen. The ime-series above he cenre, gure 6a; c; e, clearly agree wih he saed expecaion. Various di eren z levels were esed in an aemp o beer he agreemen wih he BIEM-model resuls for he deeper waer cases, bu z = :53h provided he mos accurae oupu. Rundown, as shown in gure 6b; d; f, shows good agreemen for all he rials. The explanaion is ha he wave ha creaes he rundown is generaed from he back face of he slide mass. This wave sees a characerisic waer deph ha is less han h c, and hus his back face wave remains in he region of accuracy of he deph-inegraed model, whereas he wave moion nearer o he fron face of he slide is inaccurae. This feaure is also clearly shown in gure 5. Thus, if one was solely ineresed in he leading wave approaching he shoreline, he characerisic waer deph should be inerpreed as he average deph along he back face of he slide, insead of h c. The inaccurae elevaion waves creaed by he fron face of he moving mass could be absorbed numerically, such as wih a sponge layer, so ha hey do no e ec he simulaion. A guideline ha he deph-inegraed `exended model will yield accurae resuls for L s =h c > 3:5 is acceped. This resricion would seem o be more sringen han he `exended model frequency-dispersion limiaion, which requires ha he free-surface wave be a leas wo waer dephs long. In fac, he slide lengh-scale limiaion is more in line wih he dispersion limiaions of he deph-averaged (convenional) model. The limiaions of he various model formulaions, i.e. `exended and deph averaged, are discussed in he nex secion. 8. Imporance of nonlineariy and frequency dispersion Anoher useful guideline would be o know when nonlinear e ecs begin o play an imporan role. This can be deermined by running numerous numerical rials, employing he FNL-EXT, WNL-EXT and L-EXT equaion models. These hree equaion ses share idenical linear-dispersion properies, bu have varying levels of nonlineariy. The linear-dispersion limi of hese `exended equaions, for he rigid boom case, is near kh = 3, where k is he wavenumber. Nonlineariy, however, is only faihfully capured o near kh = : for he FNL-EXT model, and o an even lesser value for WNL-EXT (Gobbi e al. 2). The source-generaion accuracy limiaion of he model is such ha he side lengh of he landslide over he deph mus be greaer han 3.5. If he slide is symmeric in he horizonal direcion, which is he only ype of slide examined in his secion, hen he wavelengh of he generaed wave will be 2 3:5 h, or roughly kh =. Thus, up o he accuracy limi found in he previous secion, nonlineariy is expeced o be well capured. The FNL-EXT model will be considered correc, and any di erence in oupu compared o he oher models wih lesser nonlineariy would indicae ha full nonlinear e ecs are imporan. The imporance of nonlineariy will be esed hrough examinaion of various h=d combinaions, using he slide mass described in he previous secion. The value of h=d can be hough of as an impulsive nonlineariy, as his value represens he magniude of he free-surface response if he slide moion was enirely verical and insananeous. The procedure will be o hold he value h c = h c ( = ) = h + d

15 Submarine-landslide-generaed waves and run-up 2899 maximum depression / d slope = 3º, h c /b =.55 L s /h c = 3.5 (a) maximum run-down / d slope = 3º, h c /b =.55 L s /h c = 3.5 (b) maximum depression / d slope = 5º, h c /b =.55 L s /h c = 3.5 (c) maximum run-down / d slope = 5º, h c /b =.55 L s /h c = 3.5 (d) maximum depression / d maximum depression / d slope = 5º, h c /b =.55 L s /h c = 3.5 (e) ( f ) maximum run-down / d D h/d D h/d (g) (h) slope = 5º, h c /b =.5 L s /h c = 3 maximum run-down / d D D 2 2 h/d h/d slope = 5º, h c /b =.55 L s /h c = 3.5 slope = 5º, h c /b =.5 L s /h c = 3 Figure 7. Maximum depression above he iniial cenre poin of he slide mass and maximum rundown for four di eren rial ses. FNL-EXT resuls indicaed by he solid line, WNL-EXT by he dashed line and L-EXT by he doed line. consan for a given slope angle, while alering h and d. Two oupu values will be compared beween all he simulaions: maximum depression above he iniial cenre poin of he slide and maximum rundown. For all simulaions presened in his secion, x=b = :3 and p gh c =b = :3.

16 29 P. Lyne and P. L.-F. Liu Table. Characerisics of he simulaions performed for he nonlineariy es slope se no. (deg) hc =b Ls =hc Figure 7 shows he oupu from four ses of comparisons, whose characerisics are given in able. Figure 7a; b show he depression above he cenre poin and he rundown for se, gure 7c; d for se 2, gure 7e; f for se 3 and g- ure 7g; h for se 4. Examining he maximum depression plos for ses {3, i is clear ha he rends beween he hree ses are very similar, wih FNL-EXT predicing he larges depression and L-EXT predicing he smalles. The di erence beween FNL-EXT and WNL-EXT is solely due o nonlinear-dispersive erms, which are of O( 2 ), while he di erence beween WNL-EXT and L-EXT is caused by he nonlinear-divergence erm in he coninuiy equaion and he convecion erm in he momenum equaion, which are of O(). The relaive di erences in he maximum depression prediced beween FNL-EXT and WNL-EXT are roughly he same as he di erences beween WNL-EXT and L-EXT for ses, 2 and 3. Therefore, in he source region, for L s =h c values near he accuracy limi of he `exended model (near 3.5), he nonlinear-dispersive erms are as necessary o include in he model as he leading order nonlinear erms. As he L s =h c value is increased, he slide produces an increasingly longer (shallow-waer) wave. Frequency dispersion plays a lesser role, and hus he nonlinear-dispersive erms become expecedly less imporan. This can be seen in he maximum depression plo for se 4. For his se, L s =h c = 3, and he FNL-EXT and WNL-EXT resuls are nearly indisinguishable. Inspecing he maximum rundown plos for ses, 2 and 3, i seems ha he rends beween he hree di eren models have changed. Now, WNL-EXT predics he larges rundown, while L-EXT predics he smalles. I is hypohesized ha he documened over-shoaling of WNL-EXT (Wei e al. 995) cancels ou he lesser wave heigh generaed in he source region compared o FNL-EXT, leading o rundown heighs ha agree well beween he wo models. As he slope is decreased, he error in he L-EXT rundown predicion increases. This is aribued o a longer disance of shoaling before he wave reaches he shoreline. As he slope is decreased, while h c is kep consan, he horizonal disance from he shoreline o he iniial cenre poin of he slide increases. The slide lengh is roughly he same for he hree ses, herefore he generaed wavelengh is roughly he same. Thus, wih a lesser slope, he generaed wave shoals for a greaer number of wave periods. During his relaively larger disance of shoaling, nonlinear e ecs, and in paricular he leading-order nonlinear e ecs, accumulae and yield large errors in he linear (L-EXT) simulaions. This rend is also eviden in he rundown plo for se 4. Also noe ha in se 4, where he nonlinear-dispersive erms are very small, he FNL-EXT and WNL-EXT rundowns are idenical.

17 Submarine-landslide-generaed waves and run-up 29 max. dep./ max. dep. WNL-EXT maximum depression / d slope = 5º D h/b =.5 L s /b =.85 (a) (b) z/d L s /h c = 3.5 (c).92 maximum rundown / d max. run-down/ max. rundown WNL-EXT slope = 5º D h/b =.5 L s /b = L s /h c (d) (e) shoreline movemen / d ( f ) L s /h c = ime * (g/d ) /2 Figure 8. Maximum depression above he iniial cenre poin of he slide mass (a) and maximum rundown (d) for a se of numerical simulaions on a 5 slope. (b), (e) The maximum depression and maximum rundown scaled by he corresponding values from he WNL-EXT model. Time-series comparisons for Ls =hc = 3:5 showing he free-surface elevaion above he cenre poin (c) and verical shoreline movemen (f ) are given on he righ. WNL-EXT resuls indicaed by he solid line, WNL-DA by he dashed line and NL-SW by he doed line. A deep-waer limi has been deermined for he `exended model (L s =h c > 3:5), bu i would also be ineresing o know he limis of applicabiliy of he dephaveraged (WNL-DA) and shallow-waer (NL-SW) models. The only di erences

18 292 P. Lyne and P. L.-F. Liu beween hese hree models (he weakly nonlinear `exended, weakly nonlinear deph averaged and nonlinear shallow waer) are found in he frequency-dispersion erms he nonlinear erms are he same. The esing mehod o deermine he deep-waer limis of he various model ypes will be o x boh a slope of 5 and a slide mass, wih h=b = :5 and L s =b = :85, while incremenally increasing he iniial waer deph above he cenre poin of he slide, d. Figure 8 shows a summary of he comparisons of he hree models. Figure 8a; d show he maximum free-surface depression measured above he iniial cenre poin of he slide and he maximum rundown for various L s =h c combinaions. WNL-EXT soluions are indicaed by solid lines, WNL- DA by dashed lines and NL-SW by he doed lines. Also shown in gure 8b; e are he maximum depression and rundown resuls from WNL-DA and NL-SW relaive o he resuls from WNL-EXT, hereby more clearly depicing he di erences beween he models. These gures show WNL-EXT and WNL-DA agreeing nearly exacly, while he errors in NL-SW decrease wih increasing L s =h c. The NL-SW resuls do no converge wih he WNL-EXT resuls unil L s =h c & 5. Figure 8c; f are ime-series of he free-surface elevaion above he iniial cenre poin of he slide and he verical movemen of he shoreline for he case of L s =h c = 3:5, respecively. Di erences beween NL-SW and WNL-EXT are clear, wih NL-SW under-predicing he free surface above he slide, bu over-predicing he rundown due o over shoaling in he non-dispersive model. The only signi can di erence beween he WNL-EXT and WNL-DA resuls come afer he maximum depression in gure 8c, where WNL-DA predics an oscillaory rain following he depression. These resuls indicae ha o he deep-waer limi ha WNL-EXT was shown o be accurae, WNL-DA is accurae as well. As menioned previously, alering he level on which z is evaluaed in he `exended model does no increase he deep-waer accuracy limi for slide-generaed waves. In summary, he nonlinear-dispersive erms are imporan for slides near he deep-waer limi (L s =h c = 3:5) whose heighs, or h=d values, are large (greaer han.4). For shallow-waer slides (L s =h c > ), he nonlinear-dispersive erms are no imporan near he source, even for he larges slides. The `exended formulaion of he deph-inegraed equaions does no appear o o er any bene s over he deph-averaged formulaion in regards o modelling he generaion of waves in deeper waer. The `exended model would be useful if one was ineresed in modelling he propagaion of shallow-waer slide-generaed waves ino deeper waer, which is no he focus of his paper. The shallow-waer-wave equaions are only valid for slides in very shallow waer, where L s =h c & Conclusions A model for he creaion of fully nonlinear long waves by sea oor movemen, and heir propagaion away from he source region, is presened. The general fully nonlinear model can be runcaed, so as o only include weakly nonlinear e ecs, or model a non-dispersive wave sysem. Rarely will fully nonlinear e ecs be imporan above he landslide region, bu he model has he advanage of allowing he slide-generaed waves o become fully nonlinear in naure, wihou requiring a ransiion among governing equaions.

19 Submarine-landslide-generaed waves and run-up 293 A high-order nie-di erence model is developed o numerically simulae wave generaion by sea oor movemen. The numerical generaion of waves by boh impulsive and creeping movemens agrees wih experimenal daa and oher numerical models. A deep-waer accuracy limi of he model, L s =h c > 3:5, is adoped. Wihin his limiaion, he `exended formulaion of he deph-inegraed equaions shows no bene over he `convenional deph-averaged approach near he source region. Leading-order nonlinear e ecs were shown o be imporan for predicion of shoreline movemen, and he fully nonlinear erms are imporan for only he hickes slides wih relaively shor lengh-scales. Alhough only one-horizonal-dimension problems are examined in his paper, slides in wo horizonal dimensions have been analysed by he auhors, bu, due o paper lengh limiaions, will be presened in a fuure publicaion. As a nal remark, i is noed ha predicion of landslide sunamis in real cases is subjec o he large uncerainy inheren in knowing he ime-evoluion of a landslide. Exensive eld research of high-risk sies is paramoun o reducing his uncerainy. The research repored here is parly suppored by grans from he Naional Science Foundaion (CMS-95283, CTS and CMS ) and a subconrac from he Universiy of Puero Rico. The auhors hank Ms Yin-yu Chen for providing he numerical resuls based on her BIEM model. Appendix A. Derivaion of approximae wo-dimensional governing equaions In deriving he wo-dimensional deph-inegraed governing equaions, he frequency dispersion is assumed o be weak, i.e. O( 2 ) ½ : (A ) We can expand he dimensionless physical variables as power series of 2, X f = w = n= X n= 2n f n (f = ± ; p; u); (A 2) 2n w n : (A 3) Furhermore, we will assume he ow is irroaional. Zero horizonal voriciy yields he u = ; u = rw : (A 5) Consequenly, from (A 4), he leading-order horizonal velociy componens are independen of he verical coordinae, i.e. u = u (x; y; ): (A 6)

20 294 P. Lyne and P. L.-F. Liu Subsiuing (A 2) and (A 3) ino he coninuiy equaion (2.3) and he boundary condiion (2.7), we collec he leading-order erms as r u + w z = ; h < z < ± ; (A 7) w + u rh + h = on z = h: (A 8) Inegraing (A 7) wih respec o z and using (A 8) o deermine he inegraion consan, we obain he verical pro le of he verical velociy componens, w = zr u r (hu ) h : (A 9) Similarly, inegraing (A 5) wih respec o z, wih informaion from (A 8), we can nd he corresponding verical pro les of he horizonal velociy componens, u = 2 z2 r(r u ) zr r (hu ) + h + C (x; y; ); (A ) in which C is a unknown funcion o be deermined. Up o O( 2 ), he horizonal velociy componens can be expressed as u = u (x; y; ) ½ z2 r(r u ) zr r (hu ) + h ¾ + C (x; y; ) + O( 4 ); h < z < ± : (A ) Now, we can de ne he horizonal velociy vecor, u (x; y; z (x; y; ); ), evaluaed a z = z (x; y; ), as ½ 2 u = u + 2 z2 r(r u ) z r r (hu )+ h ¾ + C (x; y; ) + O( 4 ): (A 2) Subracing (A 2) from (A ), we can express u in erms of u as ½ 2 u = u 2 z2 z 2 r(r u ) + (z z )r r (hu ) + h ¾ + O( 4 ): (A 3) Noe ha u = u + O( 2 ) has been used in (A 3). The exac coninuiy equaion (2.8) can be rewrien approximaely in erms of ± and u. Subsiuing (A 3) ino (2.8), we obain ½ H + r (Hu ) 2 r H ( 6 (2 ± 2 ± h + h 2 ) 2 z2 )r(r u ) µ + ( 2 (± h) z )r r (hu ) + h ¾ = O( 4 ); (A 4) in which H = h + ±. Equaion (A 4) is one of hree governing equaions for and u. The oher wo equaions come from he horizonal momenum equaion (2.4). However, we mus

21 ± Submarine-landslide-generaed waves and run-up 295 nd he pressure eld rs. This can be accomplished by approximaing he verical momenum equaion (2.5) as p z = 2 (w + 2 u rw + 2 w w z ) + O( 4 ); h < z < ± : (A 5) We can inegrae he above equaion wih respec o z o nd he pressure eld as µ z p = ± ½ (z2 2 ± 2 )r u + (z ± ) r (hu) + h + 2 (z2 2 ± 2 )u r(r u ) + (z ± )u r r (hu ) + h + 2 (2 ± 2 z 2 )(r u ) 2 + (± z) r (hu ) + h ¾ r u 4 ) + O( (A 6) for h < z < ±. We remark here ha (A ) has been used in deriving (A 6). To obain he governing equaions for u, we rs subsiue (A 3) and (A 6) ino (2.4) and obain he following equaion, up o O( 2 ), u + u ru + r± ½ z2 r(r u ) + z r r (hu ) + h ¾ ½ + 2 z z r(r u ) + r r (hu ) + h ¾ ½ 2 + r (hu ) + h r r (hu ) + h µ r r (hu ) + h + (u rz )r r (hu ) + h µ + z r u r r (hu ) + h + z (u rz )r(r u ) ¾ + 2 z2 r[u r(r u )] + 2 ½ 2 r 2 ± 2 r u ± u r r (hu ) + h + ± r (hu ) + h ¾ r u + 3 ½ ¾ 2 r 2 ± 2 [(r u ) 2 u r(r u )] = O( 4 ): (A 7) Equaions (A 4) and (A 7) are he coupled governing equaions, wrien in erms of u and ±, for fully nonlinear weakly dispersive waves generaed by a submarine landslide.

22 ² ± ± 296 P. Lyne and P. L.-F. Liu Appendix B. Numerical scheme To simplify he predicor-correcor equaions, he velociy ime derivaives in he momenum equaions are grouped ino he dimensional form, U = u + 2 (z2 ± V = v + 2 (z2 ± 2 )u xx + (z ± )(hu) xx ± x[± u x + (hu) x ]; (B ) 2 )v yy + (z ± )(hv) yy ± y[± v y + (hv) y ]; (B 2) where subscrips denoe parial derivaives. Noe ha his grouping is di eren from ha given in Wei e al. (995). The grouping given above in (B ) and (B 2) incorporaes nonlinear erms, which is no done in Wei e al. These nonlinear ime derivaives arise from he nonlinear-dispersion erms r[± (r (hu ) + h =)] and r( 2 ± 2 r u ), which can be reformulaed using he relaion µ r r (hu ) + h µ = r r (hu ) + h µ r ± r (hu ) + h r( 2 ± 2 r u ) = r( 2 ± 2 r u ) r(± ± r u ): The auhors have found ha his form is more sable and requires less ieraions o converge for highly nonlinear problems, as compared o he Wei e al. formulaion. The predicor equaions are n+ i;j = ² n i;j + 2 (23En i;j 6Ei;j n + 5Ei;j n 2 ); (B 3) U n+ i;j = U n i;j + 2 (23F n i;j 6F n i;j + 5F n 2 i;j ) + 2(F ) n i;j 3(F ) n i;j + (F ) n 2 i;j ; (B 4) V n+ i;j where = V n i;j + 2 (23Gn i;j 6G n i;j + 5G n 2 i;j ) + 2(G ) n i;j 3(G ) n i;j + (G ) n 2 i;j ; (B 5) E = h [(± + h)u] x [(± + h)v] y F = + f(h + ± )[( 6 (± 2 ± h + h 2 ) + f(h + ± )[( 6 (± 2 ± h + h 2 ) 2 z2 )S x + ( 2 (± h) z )T x ]g x 2 z2 )S y + ( 2 (± h) z )T y ]g y ; (B 6) 2 [(u2 ) x + (v 2 ) x ] g± x z h x z h x + (± h ) x [E(± S + T )] x [ 2 (z2 ± 2 )(us x + vs y )] x [(z ± )(ut x + vt y )] x 2 [(T + ± S)2 ] x ; (B 7) F = 2 (± 2 z 2 )v xy (z ± )(hv) xy + ± x[± v y + (hv) y ]; (B 8) G = 2 [(u2 ) y + (v 2 ) y ] g± y z h y z h y + (± h ) y [E(± S + T )] y [ 2 (z2 ± 2 )(us x + vs y )] y [(z ± )(ut x + vt y )] y 2 [(T + ± S)2 ] y ; (B 9) G = 2 (± 2 z 2 )u xy (z ± )(hu) xy + ± y[± u x + (hu) x ] (B )

23 ² ± Submarine-landslide-generaed waves and run-up 297 and S = u x + v y ; T = (hu) x + (hv) y + h : (B ) All erms are evaluaed a he local grid poin (i; j), and n represens he curren ime-sep, when values of ±, u and v are known. The above expressions (B 6){(B ) are for he fully nonlinear problem; if a weakly nonlinear or non-dispersive sysem is o be examined, he equaions should be runcaed accordingly. The fourh-order implici correcor expressions for he free-surface elevaion and horizonal velociies are n+ i;j = ² n i;j + 24 (9En+ i;j + 9Ei;j n 5Ei;j n + Ei;j n 2 ); (B 2) Ui;j n+ = Ui;j n + n+ (9F 24 i;j + 9Fi;j n 5Fi;j n + Fi;j n 2 ) + (F ) n+ i;j (F ) n i;j ; (B 3) V n+ i;j = V n i;j + 24 (9Gn+ i;j + 9G n i;j 5G n i;j + G n 2 i;j ) + (G ) n+ i;j (G ) n i;j : (B 4) The sysem is solved by rs evaluaing he predicor equaions, hen u and v are solved via (B ) and (B 2), respecively. Boh (B ) and (B 2) yield a diagonal marix afer nie di erencing. The marices are diagonal, wih a bandwidh of ve (due o ve-poin nie di erencing), and an e cien LU decomposiion can be used. A his poin in he numerical sysem, we have predicors for ±, u and v. Nex, he correcor expressions are evaluaed, and again u and v are deermined from (B ) and (B 2). The relaive errors in each of he physical variables is found, in order o deermine if he implici correcors need o be reieraed. This relaive error is given as w n+ w n+ w n+ ; (B 5) where w represens ±, u and v, and w is he previous ieraions value. The correcors are recalculaed unil all errors are less han 4. Noe ha, ineviably, here will be locaions in he numerical domain where values of he physical variables are close o zero, and applying he above error calculaion o hese poins may lead o unnecessary ieraions in he correcor loop. Thus i is required ha u; v a ; p gh > 4 for he corresponding error calculaion o proceed, where a is deermined from equaion (3.4) for a creeping slide. For he model equaions, linear sabiliy analysis gives ha < x=2c, where c is he wave celeriy in he deepes waer. Noe ha when modelling highly nonlinear waves, a smaller is usually required for sabiliy. In his analysis, = x=4c produced sable and convergen resuls for all rails. For he numerical exerior boundaries, wo ypes of condiions are applied: re ecive and radiaion. The re ecive, or no- ux, boundary condiion for he Boussinesq equaions has been examined by previous researchers (Wei & Kirby 995), and heir mehodology is followed here. For he radiaion, or open, boundary condiion, a sponge layer is used. The sponge layer is applied in he manner recommended by Kirby e al. (998). Run-up and rundown are modelled wih he `exrapolaion moving-boundary algorihm described in Lyne e al. (22).

24 298 P. Lyne and P. L.-F. Liu Nomenclaure s ± a wave ampliude b lengh along he slope beween x l and x r for he anh slide c wave celeriy d deph of waer above he cenre poin of he slide, funcion of ime d iniial deph of waer above he cenre poin of he slide, i.e. a = g graviy h characerisic waer deph or baseline waer deph, funcion of space h waer deph pro le, funcion of space and ime h he changing par of he waer deph pro le ((h h )= ) h c baseline waer deph a he cenre poin of he slide ( h + d) h c iniial baseline waer deph a he cenre poin of he slide ( h + d ) H oal waer deph (h + ± ) l characerisic horizonal lengh-scale of he submarine slide L characerisic horizonal side lengh of he submarine slide p deph-dependen pressure S shape facor for anh slide ime c ime-scale of sea oor moion w ypical period of wave generaed by a speci ed sea oor moion u, v, w deph-dependen componens of velociy in x, y, z u, v magniude of horizonal velociy componens u, v evaluaed on z u, v deph-averaged horizonal velociy componens u horizonal velociy vecor, (u; v) x c, y c horizonal coordinaes of he midpoin of he sea oor movemen x l, x r locaions of he lef and righ in ecion poins for he anh slide pro le z arbirary level on which he `exended equaions are derived scaled characerisic change in waer deph due o sea oor moion ( h=h ) h characerisic, or maximum, change in waer deph due o sea oor moion ime-sep in numerical model x, y space seps in numerical model nonlineariy parameer (a=h ) r horizonal gradien vecor densiy of waer slope angle frequency-dispersion parameer (h =l ) free-surface displacemen