PROPERTIES OF SURFACE AND INTERNAL SOLITARY WAVES. Kei Yamashita 1 and Taro Kakinuma 2

Transcription

1 PROPERTIES OF SURFACE AND INTERNAL SOLITARY WAVES Ke Yamashta and Taro Kaknuma Numercal solutons o surace and nternal soltary waves are obtaned through a new method where advecton equatons on physcal quanttes ncludng surace/nterace dsplacements and velocty potental are solved to nd convergent solutons by applyng the Newton-Raphson method. The nonlnear wave equatons derved usng a varatonal prncple are adopted as the undamental equatons n the present study. Surace and nternal soltary waves obtaned through the proposed method are compared wth the correspondng theoretcal solutons as well as numercal solutons o the Euler equatons to very the accuracy o solutons through the proposed method especally or nternal soltary waves o large ampltude progressng at a large celerty wth a lattened wave prole. Propertes o surace and nternal soltary waves are dscussed consderng vertcal dstrbuton o horzontal and vertcal velocty as well as knetc and potental energy. Numercal smulaton usng a tme-dependent model has also been perormed to represent propagaton o surace and nternal soltary waves wth permanent waveorms. Keywords: soltary wave; nternal wave; large ampltude; nonlnear wave equaton; advecton equaton INTRODUCTION Soltary waves o large ampltude play an mportant role n nearshore zones such that the characterstcs o surace soltary waves have been examned by many researchers e.g. Longuet-Hggns and Fenton (974). On the other hand n a lake or the ocean where densty stratcaton s well developed nternal soltary waves are observed (e.g. Stanton and Ostrovsky 998; Duda et al. 004). In partcular nternal waves o large ampltude have great eects on envronments through transportaton o nutrent salts varaton o water temperature etc. Internal soltary waves depend on not only the rato o wave heght or wavelength to water depth but also both the water-densty and layer-thckness ratos among layers whch makes t more dcult to understand the propertes o nternal waves than those o surace waves. In the present study numercal solutons o surace and nternal soltary waves are obtaned through a new method where we satsy advecton equatons on physcal quanttes ncludng surace/nterace dsplacements and velocty potental. Ths method s appled to solve the set o nonlnear wave equatons based on a varatonal prncple wthout any assumptons o wave nonlnearty and dsperson (Kaknuma 00). The numercal solutons are compared wth the correspondng theoretcal solutons o the KdV theory or the shallower verson o Cho and Camassa (999) as well as numercal solutons o the Euler equatons (Grue et al. 997). Propertes o surace and nternal soltary waves are also dscussed n consderaton o vertcal dstrbuton o horzontal and vertcal velocty as well as knetc and potental energy. NONLINEAR EQUATIONS FOR SURFACE/INTERNAL WAVES Nonlnear Equatons or Surace and Internal Waves Illustrated n Fg. s a schematc o a mult-layer system o nvscd and ncompressble luds represented as ( I) rom top to bottom. The thckness o the -layer s denoted by h (x) n stll water. None o the luds mx even n moton and the densty ρ (ρ < ρ < < ρ I ) s spatally unorm and temporally constant n each layer. Flud moton s assumed to be rrotatonal resultng n the exstence o velocty potental φ whch s expanded nto a power seres o vertcal poston z wth weghtngs as ( z t) ( x t) { z } z N 0 φ x () where N s number o terms or expanded velocty potental n the -layer; the sum rule o product s adopted or subscrpt ; the top ace o the -layer s descrbed by z 0 n the stll water condton. The nonlnear surace/nternal wave equatons derved by Kaknuma (00) based on a varatonal prncple are 0 0 { ( 0 ) } t t () ( 0 ) 0 Internatonal Research Insttute o Dsaster Scence Tohoku Unversty 468- Aramak Aza-Aoba Aoba-ku Myag Japan Graduate School o Scence and Engneerng Kagoshma Unversty --40 Kormoto Kagoshma Kagoshma Japan

2 COASTAL ENGINEERING 04 Fgure. Schematc o a mult-layer system. ( ) or 0 0 j P p g t j j j j j ρ (3) where 0 (xt) and (xt) are dsplacements o the lower and upper nteraces o the -layer respectvely; p 0 (xt) and p (xt) are pressures at the lower and upper nteraces o the -layer respectvely; g s gravtatonal acceleraton; ( ) k k k gh P ρ ρ ; s a partal derental operator n the horzontal plane.e. ( ) x y. In the present study the number o terms or expanded velocty potental n every layer s equal.e. N N. Nonlnear Equatons or Surace Waves wth a Free Water Surace Nonlnear eqatons or surace waves based on the varatonal prncple are obtaned by applyng Eqs. () and (3) to the one-layer problem wth a ree water surace where and j as ( ) { } ( ) 0 ζ ζ ζ ζ b b t (4) 0 ζ ζ ζ ζ g t (5) where the surace and bottom proles are descrbed by z ζ(x t) and z 0 b(x) respectvely. Nonlnear Equatons or Internal Waves n a Two-Layer System Covered wth a Fxed Horzontal Plate I we consder a two-layer system covered wth a xed horzontal plate where 0 and 0 b(x) then Eqs. () and (3) are reduced to [Upper layer] ( and j 0) ( ) 0 t (6) 0 0 ρ p g t (7) [Lower layer] ( and j ) ( ) { } ( ) 0 b b t (8)

3 COASTAL ENGINEERING 04 3 t g ρ { p ( ρ ρ ) gh } 0 (9) where the nterace prole s descrbed by z (xt); 0. Equatons (7) and (9) lead to t ρ ρ t or p 0 equals p. g ( h ) g ( h ) 0 Nonlnear Equatons or Permanent Waves Travelng at a Constant Celerty We consder waves progressng at a constant celerty wth no deormaton o waveorm such that the advecton equatons wrtten by F/t C F/x are satsed where C s the constant celerty; the physcal quanttes F are surace and nterace dsplacements velocty velocty potental etc. By substtutng the advecton equatons nto the nonlnear wave equatons presented above the tmedervatve terms are elmnated: or example the advecton equatons are substtuted nto Eqs. (4) and (5) we obtan nonlnear equatons or surace waves travelng at celerty C wthout deormaton o waveorm as Cζ ζ {( ζ b ) } ( ζ b ) 0 (0) () C ζ ζ ζ gζ 0. () NUMERICAL CALCULATION METHOD In the calculaton process o nonlnear equatons or permanent waves progressng at a constant celerty the Newton-Raphson method s appled to nd convergent solutons usng sutable ntal values wth lateral boundary condtons. In the present study numercal solutons o surace and nternal soltary waves are obtaned through ths method. The process to obtan numercal solutons o surace soltary waves usng Eqs. () and () s as ollows: Frst we get a set o soltary-wave solutons o the KdV theory.e. surace dsplacement ζ KdV celerty C KdV and velocty potental φ KdV. It should be noted that both the wave heght and wave celerty should be small enough n order to obtan more appropreate solutons. The weghtngs KdV or expanded velocty potental φ KdV can be evaluated by solvng the equaton o contnuty.e. Eq. () whch becomes a lnear equaton ater ζ KdV and C KdV are substtuted nto t. Second we solve Eqs. () and () usng ζ KdV and KdV as the ntal values n the Newton- Raphson method where the celerty C equals C C KdV to obtan the rst numercal solutons.e. ζ and. Thrd we solve Eqs. () and () usng ζ and as the ntal values n the Newton-Raphson method where the celerty C equals C C ε to obtan the next numercal solutons.e. ζ and. Note that ε /C can take a value between.0 5 and.0 3. By repeatng the process wth celerty C ncreased step by step we obtan numercal solutons o ζ and or surace soltary waves progressng at any celerty C. The method proposed above s expected to be applcable to not only Eqs. () and () but also Eqs. (6) (8) and (0) as well as other wave equatons such as Boussnesq-type equatons to obtan numercal solutons o surace/nternal soltary waves.

4 4 COASTAL ENGINEERING 04 (a) a/h and 0.6 (b) a/h 0.7 and 0.77 Fgure. Water surace proles o surace soltary waves. Fgure 3. Relatve celerty o surace soltary waves where C s0 s celerty o lnear surace waves n shallow water. SURFACE SOLITARY WAVES Comparson o Solutons Numercal solutons o surace soltary waves obtaned through the present method are compared wth the correspondng KdV solutons. The number o terms or expanded velocty potental N s equal to 3 4 or 5. The total length o the calculaton doman along the horzontal axs x L and the grd wdth or computaton x are 50h and 0.0h respectvely where h s stll water depth. Shown n Fg. are calculaton results o surace proles compared wth the correspondng KdV solutons. The surace proles obtaned through the present method are steeper than the KdV solutons around the peaks when the rato o wave ampltude to stll water depth a/h s large. In the smulaton the maxmum value o a/h s 0.77 when N > 3; t s dcult to generate such a soltary wave o large ampltude n usual hydraulc experments. Fgure 3 compares the relatve wave celerty C/C s0 obtaned usng the present method wth that o the KdV theory as well as the correspondng numercal soluton through Longuet-Hggns and Fenton (974) where C s celerty o surace soltary waves whle C s0 s celerty o lnear waves n shallow

5 COASTAL ENGINEERING 04 5 Fgure 4. Relatve horzontal velocty at the locaton o crest where the subscrpts s b and m o horzontal velocty denote the values at the surace at the bottom and averaged vertcally respectvely. Fgure 5. Vertcal dstrbuton o relatve horzontal velocty at the locaton o crest o surace soltary waves. water.e. gh. The value o C/C s0 through the present method s almost equal to that by Longuet- Hggns and Fenton when N > 3 whch shows the hgh accuracy o the present results. Velocty at the Locaton o Crest Shown n Fg. 4 are three values o water partcle velocty at the locaton o crest u crest : u crests at the water surace u crestb at the bottom and u crestm.e. mean velocty dened as u ζ crest m ucrest dz. h a (3) h The dashed lne n Fg. 4 represents the theoretcal soluton o the lnear shallow water theory.e. C s0 a/h. Accordng to the gure the derence between u crests and u crestb becomes larger as a/h s larger. When a/h equals 0.77 u crests s.7 and. tmes as large as u crestm and u crestb respectvely. Fgure 5 shows vertcal dstrbuton o horzontal velocty at the locaton o crest u crest where a/h and The vertcal dstrbuton o u crest /u crestb shows larger curvature as a/h s larger. Such velocty dstrbuton o large curvature cannot be reproduced wth wave models consderng only weak dsperson (Fujma et al. 985). Wave Energy o Surace Soltary Waves Wave energy o surace soltary waves per unt wdth s evaluated by E L ζ s P gh L 0 3 ρgz dzdx ρ (4)

6 6 COASTAL ENGINEERING 04 Fgure 6. Potental energy E sp and knetc energy E sk o surace soltary waves. Fgure 7. Rato o derence between knetc and potental energy to total energy o surace soltary waves. E L ζ φ 3 ρ ( φ) dzdx ρ L h (5) z s K gh E E E (6) s sp sk where E sp E sk and E s are potental energy knetc energy and total energy respectvely. The relatonshp between wave energy and a/h s shown n Fg. 6. When N 3 the derence between the numercal solutons through the present method and the results by Longuet-Hggns and Fenton (974) becomes larger as a/h s larger. On the other hand when N 4 and 5 the wave energy through the present method s n good agreement wth that by Longuet-Hggns and Fenton owng to the accurate water surace dsplacement and velocty obtaned through the present method. I a/h > 0.77 however the solutons cannot be ound through the present method wth the calculaton condtons mentoned above even when N 5. As Fg. 7 ndcates the rato o derence between E sk and E sp to total energy (E sk E sp ) becomes larger as a/h s larger when a/h < INTERNAL SOLITARY WAVES Comparson o Solutons Numercal solutons o nternal soltary waves obtaned usng the present method are compared wth the correspondng solutons o the KdV theory as well as those through the shallower verson o Cho and Camassa (999) whch s a ully nonlnear theory or the assumpton o order s that O(a/h) and O((h /λ) 4 ) << where a h and λ are representatve values o wave ampltude water depth and wavelength respectvely. Introduced n Appendx are the solutons o nterace dsplacement and wave celerty presented by Cho and Camassa (999) or shallower condtons. The deeper verson derved by Cho and Camassa (999) s not appled n the present paper. In Fgs. 8 (a) (e) nterace proles obtaned through the present method are compared wth the KdV solutons as well as the results through the shallower verson o Cho and Camassa where h s

7 COASTAL ENGINEERING 04 7 Fgure 8. Interace proles o nternal soltary waves where h s thckness o the upper layer n stll water. total water depth and h s thckness o the upper layer n stll water; the densty rato ρ /ρ equals.0; the number o terms or expanded velocty potental N s equal to three. The nterace proles through the present method as well as those by Eq. (A-) o Cho and Camassa become latter as the trough s closer to the crtcal level whch cannot be passed by an nterace o a stable nternal wave (Funakosh and Okawa 986). In cases h /h equals 0. and 0.05 as shown n Fgs. 8 (d) and (e) respectvely the nterace proles by Eq. (A-) dsagree wth those through the present method. Shown n Fg. 8 () are nterace proles wth large ampltude very close to the derence between the crtcal level and the level o nterace n stll water where the results obtaned usng Eq. (A-) and the present method are n good agreement. Fgures 9 (a) (e) show the relatonshp between relatve wavelength λ I /h and rato o wave ampltude to upper layer thckness n stll water a /h where the wavelength s dened by ( λ h ) dx a. The relatve wavelength λ I /h through the KdV theory shows monotone decrease as I a /h becomes larger where t s assumed that O(a/h) O((h/λ) ). On the other hand the relatve wavelength λ I /h through the present method as well as that o the shallower verson o Cho and Camassa has a mnmal value whch means that the nterace prole s steeper as a /h s medum. The relatve wavelength λ I /h o the shallower verson o Cho and Camassa s larger than that through the present method when h /h s smaller and a /h s medum. In case h /h equals 0.05 as shown n Fg. 9 (e) the shallower verson o Cho and Camassa s not applcable to nternal soltary waves wth strong dsperson attrbuted to large O((h /λ) 4 ) when a /h s medum although O((h /λ) 4 ) s small when a /h s small enough or large enough resultng n the good agreement o the results

8 8 COASTAL ENGINEERING 04 Fgure 9. Relatonshp between relatve wavelength o nternal soltary waves and rato o ampltude to thckness o the upper layer n stll water. through the two models as shown n Fg. 8 (). Fgure 0 shows the relatonshp between relatve wave celerty C/C 0 and a /h where C 0 s celerty o lnear nternal waves n shallow water. In comparson wth the results obtaned through the present method the KdV theory overestmates C/C 0 except when a /h s small whle the shallower verson o Cho and Camassa overestmates t slghtly when a /h s medum. The relatonshp between hal o relatve wavelength 0.5λ I /h and rato o wave ampltude to ( lower layer thckness n stll water a/h s shown n Fg. where λ h ) dx a. The values o 0.5λ I /h through the present method are n good agreement wth the results obtaned by Grue et al. (997) where the Euler equatons were solved numercally. Velocty at the Locaton o Trough In Fg. nterace proles obtaned through the present method are drawn wth a crtcal level where the thckness o the upper and lower layers s 0.5 m and 0.6 m respectvely n stll water; the lud densty n the upper and lower layers s kg/m 3 and 0.0 kg/m 3 respectvely. The relatve ampltude o the nternal soltary waves a /h s 0.36 and.35; n the latter case the ampltude s close to the upper lmt o ampltude a max evaluated by hρ hρ a max (7) ρ ρ where the ampltude n the latter case s about 0.0 m whle the upper lmt s about 0.3 m. Shown n Fg. 3 are vertcal dstrbutons o u/c 0 at the locaton o trough o the nternal soltary waves shown n Fg..e. x/h 0 where the relatve horzontal velocty u/c 0 through the present I

9 COASTAL ENGINEERING 04 9 Fgure 0. Relatve celerty o nternal soltary waves where C 0 s celerty o lnear nternal waves n shallow water. Fgure. Relatonshp between hal o relatve wavelength o nternal soltary waves and rato o ampltude to thckness o the lower layer n stll water. method s compared wth the expermental data obtaned by Grue et al. (999). Accordng to Fg. 3 the numercal solutons through the proposed method are n harmony wth the expermental results.

10 0 COASTAL ENGINEERING 04 Fgure. Interace proles o two nternal soltary waves where ρ kg/m 3 ρ 0.0 kg/m 3 h 0.5 m and h 0.6 m. Fgure 3. Vertcal dstrbuton o relatve horzontal velocty at the locaton o trough o the nternal soltary waves shown n Fg.. Dstrbutons o relatve horzontal velocty u/c and relatve vertcal velocty w/c are shown n Fgs. 4 (a) and (b) respectvely where h /h 0. and a /h.4. The maxmum values o u/c and w/c are about 0.6 and 0. respectvely n the present case. Fgure 5 shows (u max u b )/u max where u max and u b are the values o horzontal velocty at the nterace and the upper boundary.e. the top ace respectvely n the upper layer at the locaton o trough. When a /a max s close to zero or one the value o (u max u b )/u max s around zero. The same s true o (u mn u b )/u mn as shown n Fg. 6 where u mn and u b are the values o horzontal velocty at the nterace and the bottom respectvely n the lower layer at the locaton o trough. Thus the vertcal dstrbuton o horzontal velocty s approxmately unorm n both the upper and lower layers when the ampltude s close to zero or the upper lmt a max. Wave Energy o Internal Soltary Waves Wave energy o nternal soltary waves per unt wdth s evaluated by E L 3 P gh ( ρ ρ ) g( h ) dx ( ρ ρ ) (8) L E ρ L 0 φ 3 K gh L h ( φ ) dzdx ( ρ ρ ) z (9)

11 COASTAL ENGINEERING 04 (a) Dstrbuton o relatve horzontal velocty (b) Dstrbuton o relatve vertcal velocty Fgure 4. Dstrbuton o relatve velocty where h /h 0. and a /h.4; C s celerty o nternal soltary waves. E E E (0) P K where E P E K and E are avalable potental energy knetc energy and total energy o nternal soltary waves. Accordng to Fg. 7 whch shows the relatve energy derence.e. the rato o derence between knetc energy and avalable potental energy to total energy (E K E P )/(E K E P ) the relatve energy derence s large when h /h s small or a /a max s medum. On the other hand the wave ampltude s close to zero or the upper lmt a max then the relatve energy derence s small; hence t ollows that nternal soltary waves o large ampltude show the characterstc smlar to that o lnear waves n terms o the relatve energy derence whch s zero or lnear waves although the lnear theory makes no menton o such nternal waves o large ampltude. NUMERICAL SIMULATION OF SOLITARY WAVES TRAVELING OVER A HORIZONTAL BOTTOM Propagaton o a Surace Soltary Wave Numercal smulaton o propagaton o surace/nternal soltary waves over a horzontal bottom has also been perormed usng the tme-dependent numercal model (Yamashta et al. 03). In the calculaton process o wave propagaton the tme development s carred out by applyng mplct schemes smlar to that o Nakayama and Kaknuma (00). A progressve surace soltary wave o large ampltude s depcted n Fg. 8 wth an almost permanent waveorm obtaned by gvng numercal solutons through the method proposed above as the ntal condtons to solve Eqs. (4) and (5). The calculaton condtons are as ollows: a/h 0.66 N 3 x/h 0. and C s0 t / x where a h N and C s0 are ampltude stll water depth number o terms or expanded velocty potental and celerty o lnear surace waves n shallow water respectvely; x and t are grd wdth and tme-step nterval or computaton respectvely. Propagaton o an Internal Soltary Wave Shown n Fg. 9 s a progressve nternal soltary wave obtaned by solvng Eqs. (6) (8) and (0) wth the ntal condtons through the proposed method. The calculaton condtons are as ollows:

12 COASTAL ENGINEERING 04 Fgure 5. Rato o derence between horzontal veloctes at the nterace and top ace n the upper layer at the locaton o trough where a max dened by Eq. (7) s the upper lmt o ampltude o nternal soltary waves. Fgure 6. Rato o derence between horzontal veloctes at the nterace and bottom n the lower layer at the locaton o trough where a max dened by Eq. (7) s the upper lmt o ampltude o nternal soltary waves. Fgure 7. Rato o derence between knetc energy and avalable potental energy to total energy where a max dened by Eq. (7) s the upper lmt o ampltude o nternal soltary waves. ρ /ρ.0 h /h 0. a /h.4 N 3 x/h 0. and C 0 t / x 0.08 where ρ and ρ are lud densty n the upper and lower layers respectvely; h and h are thckness o the upper layer n stll water and total water depth respectvely; a N and C 0 are ampltude number o terms or expanded velocty potental and celerty o lnear nternal waves n shallow water respectvely; x and t are grd wdth and tme-step nterval or computaton respectvely. The waveorm o the nternal soltary wave s almost permanent wth large ampltude n the neghborhood o the upper lmt o ampltude.e. a max.49h such that the nterace prole shows a lattend trough.

13 COASTAL ENGINEERING 04 3 Fgure 8. Tme varaton o water surace prole o a progressve surace soltary wave. Fgure 9. Tme varaton o nterace prole o a progressve nternal soltary wave where ρ /ρ.0; h /h 0.. Fgure 0. Tme varaton o nterace prole n an overtakng process o two nternal soltary waves where ρ /ρ.0; h /h 0.; a /h.4 and 0.3. Interacton between Internal Soltary Waves Nonlnear nteracton between nternal soltary waves s also smulated as shown n Fg. 0 where the ntal ampltude o these two waves a s.4h and 0.3h ; the other calculaton condtons are the same as those n the above-descrbed smulaton o propagaton o the nternal soltary wave. In the overtakng process the larger nternal wave decreases ts ampltude owng to nonlnear nteracton wth the smaller one.

14 4 COASTAL ENGINEERING 04 CONCLUSIONS The new method has been proposed to obtan numercal solutons o surace and nternal soltary waves where we solve advecton equatons on physcal quanttes ncludng surace/nterace dsplacements and velocty potental to nd convergent solutons by applyng the Newton-Raphson method. The nonlnear wave equatons derved usng the varatonal prncple were adopted as the undamental equatons. The surace and nternal soltary waves obtaned through the proposed method were compared wth the correspondng theoretcal solutons as well as the numercal solutons o the Euler equatons to very the accuracy o the solutons through the present method especally or the nternal soltary waves o large ampltude travelng at the large celerty wth the lattened wave prole. In the cases o surace soltary waves the maxmum value o the rato o wave ampltude to stll water depth was equal to 0.77 wth the present calculaton condtons. The surace prole obtaned usng the present method was steeper than that through the KdV theory around the peak when the rato o wave ampltude to stll water depth was large. The vertcal dstrbuton o horzontal velocty at the locaton o crest showed larger curvature as the rato o wave ampltude to stll water depth was larger. In the cases o nternal soltary waves the dstrbuton o horzontal velocty was approxmately unorm n both the upper and lower layers when the ampltude was close to zero or the upper lmt o ampltude o nternal soltary waves. When the ampltude was close to zero or the upper lmt the relatve derence between the knetc and potental energy was small whch meant that the nternal soltary waves o large ampltude showed the characterstc smlar to that o lnear waves n terms o the relatve energy derence. The progressve surace and nternal soltary waves were also numercally smulated usng the tme-dependent model resultng n the propagaton o waves wth the almost permanent waveorms. In the overtakng process o the two nternal soltary waves o derent ampltude the larger wave decreased ts ampltude owng to the nonlnear nteracton wth the smaller one. APPENDIX The solutons o spatal dervatve o nterace dsplacement / x and wave celerty C n twolayer systems presented by Cho and Camassa (999) or shallower condtons are ( ρ ρ ) ( a )( a ) ( a ) 3g x C ( ρh ρh ) * a * ( ρh ρ h ) ρh (A-) h h (A-) ρ h a± q ± q q (A-3) 4 C C C C q h h (A-4) g C q h h (A-5) C0 0 ( h a)( h a) ( C g ) (A-6) h h a gh h 0 ( ρ ρ ) 0 (A-7) ρh ρh where g s gravtatonal acceleraton; ρ and ρ are lud densty n the upper and lower layers respectvely; h and h are thckness o the upper and lower layers n stll water respectvely; a s wave ampltude; C 0 s celerty o lnear nternal waves n shallow water. REFERENCES Cho W. and R. Camassa Fully nonlnear nternal waves n a two-lud system Journal o Flud Mechancs

15 COASTAL ENGINEERING 04 5 Duda T. F. J. F. Lynch J. D. Irsh R. C. Beardsley S. R. Ramp C.-S. Chu T. Y. Tang and Y.-J. Yang Internal tde and nonlnear nternal wave behavor at the contnental slope n the northern south Chna Sea IEEE Journal o Oceanc Engneerng Fujma K. C. Goto and N. Shuto Accuracy o nonlnear dspersve long wave equatons Coastal Engneerng n Japan Funakosh M. and M. Okawa Long nternal waves o large ampltude n a two-layer lud Journal o the Physcal Socety o Japan Grue J. H. A. Frs E. Palm and P. O. Rusas A method or computng unsteady ully nonlnear nteracal waves Journal o Flud Mechancs Grue J. A. Jensen P.-O. Rusas and J. K. Sveen Propertes o large-ampltude nternal waves Journal o Flud Mechancs Kaknuma T. 00. A set o ully nonlnear equatons or surace and nternal gravty waves Proceedngs o 5 th Internatonal Conerence on Computer Modellng o Seas and Coastal Regons WIT Press Longuet-Hggns M. S. and J. D. Fenton On the mass momentum energy and crculaton o a soltary wave. II Proceedngs o the Royal Socety o London. Seres A Nakayama K. and T. Kaknuma. 00. Internal waves n a two-layer system usng ully nonlnear nternal-wave equatons Internatonal Journal or Numercal Methods n Fluds Stanton T. P. and L. A. Ostrovsky Observatons o hghly nonlnear nternal soltons over the contnental shel Geophyscal Research Letters Yamashta K. T. Kaknuma and K. Nakayama. 03. Shoalng o nonlnear nternal waves on a unormly slopng beach Proceedngs o 33 rd Internatonal Conerence on Coastal Engneerng ASCE waves.7 3 pages.