Prceedigs f The Thireeh (2003) Ieraial Offshre ad Plar Egieerig Cferece Hll, Hawaii, USA, May 25 30, 2003 Cpyrigh 2003 by The Ieraial Sciey f Offshre ad Plar Egieers ISBN 1 880653-60 5 (Se); ISSN 1098 6189 (Se) Uderwaer Ladslide Shape, Mi, Defrmai, ad Tsami Geerai Philip Was 1, Sepha T. Grilli 2 1 Applied Flids Egieerig, Ic. Lg Beach, Califria, USA 2 Ocea Egieerig, Uiversiy f Rhde Islad Narragase, Rhde Islad, USA ABSTRACT We sdy derwaer ladslide shape, mi ad defrmai a early imes. We prese experimeal ad merical resls i a search fr geeral behavirs f samigeic derwaer ladslides. Or experimeal wrk preses a gralar mass mvig ad defrmig dw a seep iclie. Or merical wrk ses a Bigham-like flid mdel (BING) simlae mre realisic derwaer ladslide mdels. We shw ha bh ladslide mi ad defrmai ca be described wih hereical predicis derived fr slid blcks. We fid ha i is pssible redce r resls i aalyical expressis fr shape, mi, ad defrmai f chere feares. KEY WORDS: Ladslide; sami; shape; mi; defrmai; legh; hickess. INTRODUCTION Uderwaer ladslides prese a challegig sami geerai prblem (Grilli ad Was, 1999, 2001; Grilli e al., 2002). O he e had, he derwaer ladslide ms be reaed wih care ad precisi as a acceleraig ad defrmig bdy. O he her had, he wave geerai prblem ivlves cmplicaed flid flws ad wave behavirs ha remai sbjecs f acive research. I his wrk, we prese fdameal aalyses f derwaer ladslide shape, mi, ad defrmai, wih he dersadig ha hese have a direc bearig he prblem f sami geerai. T be mre specific, we sdy he behavir f derwaer slides a early imes, as a reasable sarig pi i a larger research effr. The mechaics f mvig ad defrmig bdies is fe csidered wih "relaive mi", which decmpses a mvig bdy i he mi f he ceer f mass, ad he he mi ad/r defrmai ab ha same ceer f mass. Based his ccepal framewrk, a defrmig ladslide has a ceer f mass mi wih a give psii ad velciy. The ceer f mass mi may als deped defrmai. Hwever, whe we discss his ceer f mass mi, we are i way precldig ladslide defrmai, becase ha defrmai ccrs ab he ceer f mass, by defiii. Despie his fdameal ccep f relaive mi, here are few sami sdies ha have csidered he ceer f mass mi f defrmig derwaer ladslides. We prpse d s i his wrk. Whe aalyzig ceer f mass mi, i seems aral ha slid blck ladslides shld prvide a reasable sarig pi fr aalyses f defrmig ladslides. Oe advaage f dig his is ha he slid blck ceer f mass mi ca be well-defied frm labrary experimes. Besides, Was (1997) shwed hrgh labrary experime resls ha defrmig ladslides have a ceer f mass mi ha is similar i may regards ha f slid blcks. Therefre, he classical sami geerai wrk f Wiegel (1955) ad experimes ad aalyical wrk by Was (1998, 2000) shld apply ms derwaer ladslide samis, eve hgh ha wrk was based slely slid blcks. Cmpais by Was e al. (2000) sig he w-flid ladslide mdel f Imamra ad Imeaz (1995) prvide ye aher cfirmai ha he ceer f mass mi remais similar ha f a slid blck. These resls mivae he crre sdy, becase hey sgges ha he ceer f mass mi f a derwaer ladslide may be a rbs ad predicable qaiy, regardless f defrmai. Was e al. (2000) als fd ha ladslide defrmai ca cribe he ceer f mass mi. Ths, here is a slid blck cmpe ladslide ceer f mass mi, b here ca als be a secd cmpe ceer f mass mi brgh ab by ladslide defrmai. Basically, mass ca shif frward frm he se f a defrmig ladslide, ad his shif mves he ceer f mass frward. Bh cmpes f he al ceer f mass mi ca cribe sami geerai. Therefre, defrmai shld always be csidered fr sami geerai, alhgh is imprace remais prly assessed, i par becase ladslide shape evlves i ime. I his wrk, we sdy ladslide mi ad defrmai a early imes, which is dersd mea: while he ladslide is sill acceleraig ad while sami geerai is sill akig place. These w cdiis are redda becase e ca defie he drai f sami geerai by he characerisic drai f ladslide 364
accelerai (Was, 1998, 2000). Becase we are ly ieresed i ladslide mi a early imes, i seems reasable csider a plaar iclie wih csa slpe. Las f all, i his wrk, we have chse fcs derwaer slides, i.e., a sbclass f derwaer ladslides, which is a mre geeral r mbrella erm. We will irdce a mrphlgical defiii f derwaer slides belw. Or research echiqe shld apply eqally well her classes f derwaer ladslides, alhgh we d edeavr prve his pi here. Isead, we wish esablish ha i is pssible describe derwaer slide shape, mi, ad defrmai i he firs place. ANALYTICAL CONSIDERATIONS Mahemaical r aalyical eqais eable sme predicis f derwaer slide shape ad mi. We wld like aalyze r sbseqe resls wihi he framewrk f aalyical eqais, becase hese eqais ffer relaiships bewee fdameal scales ad becase hey ca serve simplify cmplicaed phemea hrgh he se f simple qaiies. Uderwaer Slide Shape A his sage, i r csiderai f derwaer slide shape, we are ieresed i he grss feares prese a he iiiai f ierial failre. Whe viewed as a crss-seci frm he side, a derwaer slide ca be characerized by a iiial legh B parallel he slpe ad a maximm hickess T perpediclar he slpe. These w measres f size frm a dimesiless qaiy called he hickess legh rai T B ha helps qaify slide mrphlgy. We examied derwaer slide recrds (e.g., Prir ad Clema, 1979; Edgers ad Karlsrd, 1982; Schwab e al., 1993; Hamp e al., 1996) fr idicais f a ypical hickess legh rai. I appears as hgh derwaer slides are grped ard a mea hickess legh rai f rghly 0.01 wihi a facr f w, which implies lg ad hi mass failres. Aher mrphlgical feare f derwaer slides is ha he mass fe ravels sigifica disaces frm he headwall scar befre cmig res. This idicaes a rapid accelerai ad large raslaial velciies. A mre deailed mrphlgical classificai f a derwaer slide may deped specific mass failre prcesses ha we d csider here. Regardless, derwaer slides are sfficiely cmm be f ccer as sami hazards, perhaps cmprisig p half f all samigeic derwaer ladslides. Uderwaer Slide Mi Was (1997) csidered he frces acig derwaer slides i sme deail. Simplified versis f his aalysis prvidig reasable frce balaces appear i Pelivsky ad Pplavsky (1996) r Was (1998), wh describe a sefl ceer f mass mi valid a early imes. We smmarize hse resls here. If we assme a plaar iclie f agle θ fr he sake f r crre sdy, he he sli he balace f ieria, graviy, basal frici, ad hydrdyamic drag frces acig he eire slide blk becmes [ ( )] (1) s ()= s l csh i which s dees a characerisic disace f mi ad dees a characerisic drai f mi (Was, 1998, 2000). These w qaiies are fd frm he derwaer slide iiial accelerai a ad hereical ermial velciy a g = ( γ 1) ( siθ C csθ ) γ + C m ( γ 1) ( siθ C cs ) 4Bg θ accrdig = (3) 3C d s 2 a (4) a (5) where g is he accelerai f graviy, desiy, ρ b is he blk desiy, ρ w is he waer desiy, added mass cefficie, C is he drag cefficie, ad d (2) γ ρ b ρ w is he specific C m is he C is he Clmb frici cefficie. Eq. 1 is a cmm aalyical sli fr free bdies sbjeced graviy ad frm drag a high Reylds mbers. Eqs. 2-5 are specific a flly sbmerged, slid blck slide wih a apprximaely parablic prfile. Eq. 1 is mea slely capre he crrec iiial accelerai drig sami geerai, which ccrs a early imes < fllwig ierial failre. I fac, his crieri serves as r defiii f early ime. Give his sric ime csrai, r frce balace shld be sfficie crrecly represe slide ceer f mass mi. Frher, here shld be eed fr ceer f mass mi refieme if e is ccered ly wih sami hazards. Ideed, Fig. 1 shws ha he firs erm i he Taylr series f Eq. 1 ca prvide a reasably accrae descripi f slide mi drig < by sig ly he iiial accelerai frm Eq. 2. Eq. 3 demsraes ha ermial velciy icreases wih he sqare r f slide iiial legh. Hwever, he slide maximm hickess des appear explicily i Eqs. 2-5. Dimesiless Psii 2.0 1.5 1.0 0.5 Cmplee Sli Iiial Accelerai 0.0 0.0 0.5 1.0 1.5 2.0 Dimesiless Time Fig. 1: Cmparis f Eq. 1 wih he firs erm i is Taylr series We sgges he added mass cefficie C m 1.76 ad he drag cefficie C d 1.53 baied frm he experimeal wrk f Grilli e al. (2002) perfrmed wih a ellipsidal slide shape. Hwever, frm 365
r experiece, derwaer slide mi is pariclarly sesiive hese w cefficies (Was, 1998). We e ha r hydrdyamic drag frce depeds slide hickess ad widh, raher ha slide legh ad widh. Fricial frces alg he slip srface may r may be impra i derwaer slide mi, depedig he sedime ype, he waer pressre, he margi hisry, ad he ecic regime, ha all cribe failre. Give a effecive residal shear sregh S, we calclae a mea Clmb frici cefficie wih 3 S C = 2( ρ b ρ )gt csθ where we have ce agai assmed a parablic slide prfile. Eq. 6 irdces he slide maximm hickess i he ceer f mass mi. Cseqely, eve if sedime reais sigifica residal sregh, prgressively hicker slides experiece lesser resrai heir mi. The Clmb frici a lab scale will fe be very differe frm ha i he real wrld. O he e had, labrary scale slides will fe experiece prhibiive Clmb frici. O he her had, i wld be srprisig fid graviaial frcig mch larger ha basal frici fr a real derwaer slide eve. This arises becase he graviy ha acceleraes slides is a bdy frce, whereas he basal frici ha resrais slides is a srface frce, ad Eq. 6 idicaes ha slide vlme ypically icreases faser ha basal area. (6) diameer D f he paricles (i millimeers) is based mliple micrmeer readigs f hdreds f paricles. The average resls f all measremes are smmarized i Table 1. A maerial ladslide wih verical fr face f 85 mm ad hrizal p face f 85 mm was iiiaed by reracig a verical gae dw i a Lcie iclie f θ = 45 degrees. The gae mi i he iclie was rapid egh release he gralar mass i alms he same shape as whe i was iiially impded behid he gae. The dwward mi f he gae was ecessary miimize disrpi f bh he slide maerial ad he waer free srface. I is als impra e ha reracig he gae ms chage he sae f sress i he slide maerial regardless f gae shearig: he slide maerial ges frm beig sppred by a slid reaiig wall beig sppred. The gae was made f 0.4 mm hick sailess seel, ad prrded p 115 mm abve he iclie. The gae was reraced i ab 40 ms by a Nyl srig ceced by a plley sysem a fallig weigh. LABORATORY EXPERIMENT METHODS We repr a sbse f he experimeal wrk f Was (1997), which we refer he ieresed reader fr addiial deails. T be specific, we csider ly hree rials, carried wih hree differe gralar maerials: rial 82 carried wih 3 mm glass beads, rial 81 carried wih 3 mm seel sh, ad rial 86 carried wih 3 mm lead sh. Each derwaer slide sared wih he same riaglar shape ad blk vlme (Fig. 2), wih he primary differece csisig f he blk desiy. The ak widh ad legh are impra here. Similar experimes relaed sami geerai by derwaer slides have als bee cdced by Assier-Rzadkiewicz e al. (1997). Table 1. Gralar maerial prperies fr labrary experimes Maerial ρ s ρ b φ ψ D Glass 2563 1935 29 6 2.96 Seel 7954 5102 28 6 3.31 Lead 10727 7321 34 12 3.11 The slid desiy ρ s (i kg/m 3 ) was measred wih a specific graviy balace. Alms all spherical paricles had slid vlme fracis bewee 0.58-0.61 wih he excepi f 3 mm lead sh which had a slid vlme fraci f 0.65. The blk desiy ρ b (i kg/m 3 ) was calclaed wih he slid vlme fraci. The ieral frici agle φ ad iclie frici agle ψ were deermied experimeally fr he slide maerials. The ieral frici agle φ gvers failre hrgh he mass f slide maerial while he iclie frici agle ψ is relaed he Clmb frici f he slide maerial he Lcie iclie. If he ieral frici agle φ was beig deermied, he a radm layer f slide maerial was depsied a Lcie shee ad held i place by a hi layer f silice seala. If he iclie frici agle ψ was beig deermied, he he maerial was depsied direcly a Lcie shee ad esed by icliig he shee il failre. The mea Fig. 2: Seel sh slide frames wih iled camera ad wave gage. The ilie is a 45 degrees ad he camera was iled by he same agle. All gralar maerial experimes were recrded wih a high-speed camera ha eabled shape, mi, ad defrmai be measred (Was, 1997). Fig. 2 prvides fr frames frm seel sh rial 81, where a small fraci f he seel sh had bee paied whie i rder dcme ieral slide defrmai, ad where he camera has bee iled 45 degrees make he iclie lk hrizal. The elapsed ime frm cmplee gae reraci is idicaed, ad a wave gage is visible abve he slide i he pper lef had crer. The firs frame reveals he slide mass mre r less i is iiial psii, ly slighly perrbed by gae reraci. Sbseqe frames reveal a rapid dwslpe spreadig alg wih ccmia hiig as he mass acceleraes. The maerial slide shape, ceer f mass mi, ad defrmai were fd by racig he slide lie frm he high-speed mvie frames. Fr a ypical slide, every hird r frh frame was raced wih he firs race ccrrig befre he rial bega. Fr regisry marks were spaced 101.6 mm (fr iches) apar alg he side f he iclie ad sme are visible Fig. 2. The black ad whie image was scaed a PICT file ad impred i NIH Image 1.61 sfware i rder prcess he image. The cerid calclaed by NIH Image fr a slide race assmed ha he mass was evely disribed hrgh he crss-secial prfile. Fr early imes ad fr maerials wih large paricles, he assmpi f a ifrm slid vlme fraci hrgh 366
he slide prfile appeared be valid. NUMERICAL EXPERIMENT METHODS Nmerical experimes are cveie becase f he ease wih which he ser ca crl mdel ips ad measre simlai ps. I addii, i is fe qicker ad cheaper explre a parameer space wih merical experimes ha i is d similar wrk wih labrary experimes. Hwever, a mber f chices ms be made, icldig he ype f mdel, he implemeai f he mdel, he base case f ieres, ad he size f he parameer space f ieres. Table 2. Ip parameers fr he merical experimes wih SI is Trial S ρ b T θ B 2 2000 1900 100 5 10000 4 1000 1900 100 5 10000 5 4000 1900 100 5 10000 6 2000 1600 100 5 10000 7 2000 2200 100 5 10000 8 2000 1900 75 5 10000 9 2000 1900 125 5 10000 10 2000 1900 100 4 10000 11 2000 1900 100 6 10000 1 200 1900 10 5 1000 3 20000 1900 1000 5 100000 The parameer space cvered apar frm r base case is bes csidered wih dimesiless qaiies. Amg her higs, Table 3 idicaes ha merical experimes 1 ad 3 are ideical r base case i his parameer space, eve hgh he slide legh chages by a rder f magide. Als, a qick calclai reveals ha graviaial frcig ( si θ ) is a leas a rder f magide greaer ha Clmb frici ( C csθ ). We ms assme ha all her mdel cefficies, icldig ay added mass r drag cefficies, remai csa. Las f all, we apprise he reader ha we ca r BING simlais f r labrary experimes, becase BING makes he small agle apprximai si θ θ, ad becase BING assmes a parablic iiial slide prfile, qie differe frm he iiial shape i Fig. 2. We iegrae ver he slide vlme fid he ceer f mass. LABORATORY EXPERIMENT RESULTS We fllw Was (1997) ad demsrae ha he aalyical frm describig maerial slide ceer f mass mi crrespds ha f a slid blck slide, ha is Eq. 1. Fig. 3 shws he ceer f mass psii f he 3 mm seel sh slide as a fci f ime alg wih a crve fi f Eq. 1. The crve fi has a r 2 vale f 0.9998. All her ceer f mass psii daa prdced similar crve fi resls. Give he accracy f he crve fi, we ca se he resls frm he crve fi exrac a characerisic disace s ad a characerisic drai f maerial slide mi. The secd ad hird clms i Table 4 smmarize he resls f he crve fis f slide ceer f mass psii alg he iclie as a fci f ime. We chse mdel derwaer slides wih BING versi 1.3, i par becase i is a widely sed ad cmparaively simple mdel wih which demsrae r echiqes (see Imra e al., 2000, 2001). We e here ha BING is csraied ly plaar slpes. Fr r implemeai, we chse he Herschel-Blkley csiive relai wih a expe f iy, which redces he familiar Bigham plasic behavir emplyed by he ladslide mdel f Jiag ad LeBld (1993). We simlaed all derwaer slides wih 276 des, a referece srai rae f 5, ad a ambie flid desiy f 1000 kg/m 3. Or base case crrespds rial 2 lised i Table 2, which als liss he ips fr he 10 her merical experimes. The base case cmprises a presmably realisic ad samigeic derwaer slide. This pariclar base case may yield derwaer slide behavirs ha are represeaive f eves wih csiderably differe size r shape. Table 3. Dimesiless parameers f he merical experimes Ceer f Mass Psii (cm) 14 12 10 8 6 4 2 Trial θ γ T B C 2 5 1.9 0.0100 0.0034 4 5 1.9 0.0100 0.0017 5 5 1.9 0.0100 0.0068 6 5 1.6 0.0100 0.0051 7 5 2.2 0.0100 0.0025 8 5 1.9 0.0075 0.0045 9 5 1.9 0.0125 0.0027 10 4 1.9 0.0100 0.0034 11 6 1.9 0.0100 0.0034 1 5 1.9 0.0100 0.0034 3 5 1.9 0.0100 0.0034 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Time (s) Fig. 3: Crve fi f Eq. 1 wih seel sh slide ceer f mass psii All maerial slides experieced a iiial drp i he ceer f mass heigh H perpediclar he iclie. I geeral, he ceer f mass drp was liear i ime H ( ) H U (7) fr sfficiely early imes, sch as hse csidered here. Fig. 4 shws he drp i heigh perpediclar he iclie as a fci f ime fr 367
he 3 mm seel sh slide. A liear crve fi ver all heigh measremes yields he dimesiless drp rae lised i Table 4. The liear crve fi has a r 2 vale f 0.9862. There is a s-shaped secdary srcre wihi he daa f Fig. 4 ha sggess a liear behavir a lger imes. The drp rae U appears deped he maerial desiy mre ha ay her maerial characerisic. I is he cmpe f graviaial frcig rmal he iclie ha is primarily respsible fr he spreadig f maerial slides a shr imes. The graviaial frcig is i r prprial he ladslide mass r drivig frce, ad herefre ms be relaed ceer f mass mi. Dimesiless Ceer f Mass Heigh 1.1 1.0 0.9 0.8 0.7 desiy γ. We have mdified Eq. 3 apprpriaely. Frm clms 5 ad 6 i Table 4, we fid ha he glass beads have characerisics f mi ab 30% larger ha prediced, ha he seel sh has similar characerisics f mi as hse prediced, ad ha lead sh has smaller characerisics f mi ha prediced. These resls idicae a cmplicaed mechaics wihi he slide mass prper. NUMERICAL EXPERIMENT RESULTS Fr r merical experimes wih BING, we als fid ha he ceer f mass mi f a defrmig slide cfrms Eq. 1. Fig. 5 shws a crve fi f ceer f mass psii as a fci f ime fr r base case. The crve fi has a r 2 vale f 0.9999. The crve fiig resls fr all merical experime are lised i Table 5. The fcial frm f he ceer f mass mi appears remai he same, wheher r a slide defrms, ad wheher resls are frm labrary r merical experimes f defrmig slides. We e he vale = 306 s frm Table 5 i rder prve ha he imes shw Fig. 5 d i fac spa r defiii f early imes. I addii, he fial experimeal pi shw Fig. 5, which ccrs a a ime >, crrespds he maximm slide velciy. Aferward, he ceer f mass begis slw dw. BING herefre predics sfficie ime fr cmplee sami geerai alg a plaar slpe. 30000 0.6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Time (s) Fig. 4: Crve fi f Eq. 7 wih seel sh slide ceer f mass heigh These experimes were cdced wih slides f similar size, similar shape, similar paricle diameers, ad similar frici cefficies he same 45 degree slpe. Therefre, he ly slide parameer wih ay appreciable variai was he blk desiy, r specific graviy. While ceer f mass mi eqais sch as Eqs. 2-5 already acc fr chages i blk desiy, we have ye acced fr ay iflece ha blk desiy may have slide defrmai. As we have already ed, slide defrmai ca iflece ceer f mass mi, ad herefre i ca mdify he characerisic disace s ad characerisic drai f mi a early imes. The riaglar iiial shape als mdifies Eq. 3 by a small merical facr (Was, 1998). Table 4. Acal measred (A) ad prediced hereical (P) slide ceer f mass mi, fr labrary experimes Maerial s (cm) (A) (s) (A) U H (A) s (cm) (P) (s) (P) Glass 17.1 0.324 0.150 13.4 0.251 Seel 29.5 0.271 0.492 29.6 0.262 Lead 38.4 0.295 0.836 70.6 0.408 These facs allw s relae he rai f bserved characerisics f mi s ad hse prediced by Eqs. 2-5, he specific Ceer f Mass Psii (m) 25000 20000 15000 10000 5000 0 0 100 200 300 400 500 Time (s) Fig. 5: Crve fi f Eq. 1 wih mdel base case ceer f mass psii The evli f maximm slide hickess i Fig. 6 als fllws a predicable paer, wih hickess drppig liearly i ime. The crve fi has a r 2 vale f 0.9927 acc f a secdary srcre wihi he daa. There is a csise s-shaped srcre ab he liear drp, wih he hickess decreasig slwer ha average a firs, he faser ha average, ad fially wih a idicai ha he decrease is slwig dw a laer imes. This behavir makes sese, becase we kw ha he hickess ms eveally reach a fiie asympe whe he slide cmes res, b his ccrs a laer imes ha hse shw. The behavir was ed Fig. 4 fr he gralar maerial slides ad may be geeral. The daa i Fig. 6 are a he same imes as hse i Fig. 5. 368
Maximm Thickess (m) 100 90 80 70 60 50 40 30 0 100 200 300 400 500 Time (s) Fig. 6: Liear crve fi f mdel base case maximm hickess i ime The se psii i Fig. 8 fllws he same paer as ha f ypical legh i Fig. 7, wih he se psii icreasig i direc prpri he disace raveled by he ceer f mass. The liear crve fi has a r 2 vale f 0.9998. The slpes fd frm he liear crve fis are all idicaed i Table 5. We e ha he slpe m hickess has is f legh ver ime becase f he ad hc are f ha specific crve fi. Table 5. Acal resls frm he merical experimes wih BING Trial s (m) (s) m m hickess legh m se 2 37330 306-0.168 0.738 1.32 4 66904 410-0.152 0.670 1.32 5 28928 282-0.173 0.820 1.33 6 34983 342-0.148 0.789 1.35 7 54213 348-0.176 0.713 1.32 8 33797 299-0.116 0.670 1.31 9 54395 374-0.211 0.808 1.36 10 37781 352-0.158 0.878 1.38 11 51286 332-0.170 0.647 1.30 1 2065 73-0.054 0.820 1.34 3 910610 1554-0.470 0.670 1.32 The isaaes slide legh is a difficl qaiy defie becase we d wa iclde slidified sedime i he slide ail. We defie a ypical slide legh based is vlme V ad maximm hickess T as B = 3V 2T by assmig ha he slide remais mre r less a parablic prfile. Or defiii has he advaage f reprdcig he crrec iiial slide legh eered i BING. Fig. 7 shws he ypical legh icreasig i direc prpri he disace raveled by he ceer f mass. The liear crve fi has a r 2 vale f 0.9992. Slides herefre pssess a direc relaiship bewee he ceer f mass mi ad defrmai ab he ceer f mass. 35000 Nse Psii (m) 50000 40000 30000 20000 30000 10000 Typical Legh (m) 25000 20000 0 0.0 5.0 10 3 1.0 10 4 1.5 10 4 2.0 10 4 2.5 10 4 3.0 10 4 Ceer f Mass Psii (m) Fig. 8: Liear crve fi f mdel base case se psii 15000 10000 0.0 5.0 10 3 1.0 10 4 1.5 10 4 2.0 10 4 2.5 10 4 3.0 10 4 Ceer f Mass Psii (m) Fig. 7: Liear crve fi f mdel base case ypical legh We fid ha r chse merical experimes prdce reasably csise vales fr he slide feares csidered here. T make r pi mre clearly, we prvide he meas ad sadard deviais fr all qaiies i Table 5, excldig rials 1 ad 3 which are ly similar fr dimesiless qaiies. We bai he characerisic disace f mi s = 44402 ± 12670 m, he characerisic drai f mi = 338 ± 39 s, he slpe m hickess = -0.164 ± 0.026 m/s, he slpe m legh = 0.748 ± 0.080, ad he slpe m se = 1.332 ± 0.025. I is clear ha ms qaiies have sadard errrs f 10% r less, idicaig a high degree f predicabiliy. This fac is all he mre 369
srprisig becase he differe ip parameers i Tables 2 ad 3 spa a variai f a leas 20-30%. Therefre, he merical mdel BING has acally redced he sesiiviy f r chse p resls relaive he ip parameers. DISCUSSION We divide r discssi i a examiai f r experimeal resls, ad he csiderais f sami geerai. Slide Shape, Mi, ad Defrmai Was (1997) ad Was e al. (2000) demsraed ha he ceer f mass mi f defrmig slides cld be described by Eq. 1 a early imes. We sgges i his wrk ha hese previs resls may be geeral, ad apply bh ceer f mass mi ad slide defrmai. The fac ha derwaer slides have a predicable ceer f mass mi sggess ha he frce balace sed derive Eq. 1 remais valid he scale f he eire slide, regardless f he ieral defrmais experieced lcally by he slide. Oe explaai lies i he bservai ha desely packed arrays f paricles several mlayers hick are bserved i Fig. 2 slidig alg he iclie i he blk f he slide (Was, 1997). A similar explaai is ha slide defrmai may aler appreciably he vales f he dyamical cefficies acig he slide mass. Therefre, slid blck vales f hese cefficies remai releva defrmig slides. Regardless, i appears pssible describe ceer f mass mi despie he mechaical deails ieral a ladslide mdel. Table 6. Rai f acal prediced (Eqs. 2-5) resls f merical experimes wih BING Trial s rai rai rai a rai 2 1.12 0.75 1.34 1.97 4 2.00 1.02 1.65 1.93 5 0.87 0.68 1.06 1.87 6 1.14 0.74 1.31 2.08 7 1.50 0.92 1.34 1.78 8 1.01 0.73 1.18 1.89 9 1.63 0.92 1.48 1.91 10 1.13 0.77 1.26 1.90 11 1.54 0.90 1.42 1.91 1 0.62 0.56 1.00 1.94 3 2.73 1.21 1.73 1.87 Cseqely, Eq. 1 ca be viewed as a caical descripi f slide ceer f mass mi a early imes. We have ye fid ay fdameal reas why slide hickess wld decrease liearly i ime a early imes. Hwever, we csider he liear drp i hickess be sfficiely accrae s as be sefl i sdies f sami geerai. I is clear ha his fcial frm ca remai valid a laer imes, s his is a geeral resl f slide dyamics. The fac ha slide legh icreases wih disace raveled sggess ha ceer f mass mi is he drivig frce fr slide defrmai. I rerspec, here seems be aleraive describe hw slide legh icreases, becase he ly her dyamic legh scale is he ceer f mass psii. The rai f hese w legh scales apparely frms a csa dimesiless qaiy, becase his is he ly way ha he dimesi f legh cld arise ha describes chages i he ypical slide legh. Cseqely, his resl may be fdameal slide defrmai a early imes, alhgh he merical cefficie may vary. A similar resl is fd fr he psii f he slide se, which spprs r cei, ce agai, ha he ceer f mass psii is a fdameal measre f slide defrmai. Table 6 prvides rais f acal prediced qaiies i rder es he sigificace f slid blck ceer f mass eqais ver he eire parameer space f he merical experimes. I geeral, we fid ha all f he rais are f rder iy, idicaig ha slid blck ceer f mass eqais ca acc fr ms firs-rder effecs prdced by BING. The characerisic disace f ceer f mass mi des deped defrmai ab he ceer f mass. The mea rai is 1.326 ± 0.362 a e sadard deviai. The characerisic drai f ceer f mass mi eds be smaller ha prediced, idicaig a mre rapid apprach a maximm velciy. The mea rai is 0.826 ± 0.330. Or maximm velciy resls are basically a cseqece f he scalig f slide velciy wih he sqare r f slide legh. Fr rial 1 we fid = max 26 m/s, fr rial 2 we fid =110 max m/s, ad fr rial 3 we fid max = 450 m/s. We ced ha her slis f New's eqais f mi will cfirm ha r slide velciies are crrec. Oce agai, he ceer f mass maximm velciy des deped slide defrmai. The mea rai f acal maximm hereical ermial velciy is 1.336 ± 0.172. Las f all, we fid ha he iiial accelerai f he ceer f mass has bee alms dbled acc f he rapid shifig f mass wards he fr f he slide a early imes. The mea rai is 1.916 ± 0.080. Was e al. (2000) fd a similar resl, ad sggesed ha he icreased accelerai may ehace sami geerai. Tsami Geerai Csiderais We have fcsed s far i his paper derwaer slide shape, mi, ad defrmai, wih he gal f applyig hese resls sami geerai. The available space permis a qaliaive csiderai ha bilds p he resls preseed abve. Despie hese limiais, we draw cclsis wih peially sigifica bearigs ladslide sami hazard assessme. Was (1997) fd experimeally ha here is sigifica chage i ceer f mass mi relaed defrmable slides, r is here ay appare chage i sami wavelegh assciaed wih defrmai. Defrmai ab he ceer f mass des appear aler he fdameal physics described by Wiegel (1955) r Was (1998, 2000). Cseqely, slid blck mi is represeaive f defrmig slide mi, ad slid blck sami geerai shld als be represeaive f defrmig slide sami geerai. Or research dae sggess ha defrmai ab he al ceer f mass mi appears s far have lile samigeic imprace. Tha is, he precise shape f a slide may be a majr facr i sami geerai, alhgh research is ciig his maer. The basic isse is ha a decrease i hickess ver ime will decrease sami geerai, whereas a icrease i he iiial accelerai will icrease sami geerai, bh acig a he same ime. Wih hese cmpeig effecs, i is clear hw sami amplide wld be affeced by defrmai. Sme sami geerai mdels (e.g., Jiag ad LeBld 1992, 1993, 1994) have emphasized slide defrmai, b crre sers seem skip "relaive mi" algeher ad assme ha he defrmai iself is fdameally samigeic. The prblem here is ha he waer wave mdel f Jiag ad LeBld (1992, 1993, 1994) will always make defrmai lk impra, becase he shallw waer wave assmpi frces he free srface mimic verical bm accelerais. Cseqely, a mre sphisicaed sami geerai mdel may be eeded reslve hese isses. 370
Uderwaer ladslide velciies i he rage f 50-70 m/s are cmm vales fr desrcive ladslides, b i is als re ha smaller ladslides are mre cmm ha larger eves. We have shw ha researchers eed be carefl, becase maximm velciy depeds ladslide size. This fac eables slides f iiial legh f rder O(10 km) achieve maximm velciies ha are sigificaly greaer ha cmmly acceped. A ypical lg wave celeriy i he pe cea is 220 m/s, ad a derwaer slide wih his velciy prvides a isaaes Frde mber f iy (Fr=1). Or aalyses sgges ha sch a cdii ca be achieved i are. I geeral, whe a mvig bdy maches he wave speed, wave amplides ca be expeced grw rapidly (Tii ad Brlcci, 2000). Hwever, we are ms ieresed i he far-field behavir f sch a wave. We csider r maximm velciy resl alg wih he wrk f Be- Meahem ad Rsema (1972) as well as ha f Iwasaki (1997) i rder describe sami prpagai i a qaliaive maer. Uder he cdii f Frde mber clse iy, ladslide samis wld be capable f beamig fcsed eergy i he far field as if he waer waves were blles rigiaig frm a ladslide rifle. I herefre appears s as if ladslide samis ca smeimes geerae rasceaic hazards. CONCLUSIONS We have preseed experimeal ad merical resls ha sgges geeral behavirs f samigeic derwaer slide shape, mi ad defrmai a early imes. These behavirs have direc cseqeces he samigeic peial f derwaer slides. Wih a srprisig degree f csisecy ad accracy, we dcme mi ad defrmai ha ca be described ad scaled wih hereical predicis derived fr slid blcks. Or experimeal wrk preses a gralar mass mvig ad defrmig dw a seep iclie. Or merical wrk ses BING simlae mre realisic derwaer slide mdels. Regardless f he sysem beig sdied r he echiqes emplyed, we fid ha i is pssible redce r resls i simple aalyical expressis f slide shape, mi, ad defrmai. We d develp pariclar aalyical expressis fr his wrk, which is msly preseed i ablar frm, alhgh crve fis f r resls are readily perfrmed. Isead, he echiqe ha we develped is presmably geeral i are. We cjecre ha alms all experimeal r merical wrk derwaer slides ca be redced i a similar maer, perhaps wih differe crve fi resls r merical cefficies. I his wrk, we sacrifice he precisi leared hrgh redcive mdels i rer fr apprximae descripis f chere slide feares. Becase r expressis capre ly specific chere feares f derwaer slides, here will always be a rle fr mre redcive experimes ad simlais. ACKNOWLEDGEMENTS PW received sppr fr dcral sdies frm he US Naial Sciece Fdai hrgh gra CMS-9523414 Fred Raichle a he Califria Isie f Techlgy. The remaider f his wrk was sppred by he US Naial Sciece Fdai der research gra CMS-0100223 made STG a he Uiversiy f Rhde Islad. REFERENCES Assier-Rzadkiewicz, S, Marii, C, ad Heirich, P (1997). 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