On the internal soliton propagation over slopeshelf

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1 Journal of Physcs: Conference Seres PAPER OPEN ACCESS On the nternal solton propagaton over slopeshelf topography To cte ths artcle: Albert Sulaman 7 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Related content - Solton dynamcs n complex potentals Yanns Komns - Smulaton of Magnetostatc Wave Envelope Solton Propagaton n Yttrum Iron Garnet Flms Takuro Koke, Tomoyasu Ito, Tatsuya Hrao et al. - The Numercal Analyss of Solton Propagaton wth Plt-Step Fourer Transform Method Z B Wang, H Y Yang and Z Q L Ths content was downloaded from IP address on 7/7/8 at :55

2 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 On the nternal solton propagaton over slope-shelf topography Albert Sulaman Geostech Laboratory 8, Badan Pengkajan dan Penerapan Teknolog BPPT, Kawasan Puspptek Serpong, Tangerang Selatan 534, Indonesa E-mal: albertus.sulaman@bppt.go.d Abstract. Dynamcs and propertes of nternal solton propagaton over slope-shelf topography s nvestgated. We derve a nonlnear wave equaton based on the two-layer flud model, whch produce a varable-coeffcent perturbed Korteweg-deVres vp-kdv equaton. A specal soluton n term of sngle-solton wll be hghlghted.. Introducton Internalwave transmtsenergytotheocean nteror uptotheseafloor. Lkewaves ofseasurface, the nternal waves wll experence a transformaton of energy at a tme when approachng the beach. The transformaton of energy depends on the sea bottom topography. The depth of the sea wll change from deep to shallow wth vared forms. The most common s n the shape of a slope-shelf. The propagaton of nternal wave, especally nternal soltary waves ISW or nternal solton, from shelf break has remarkable propertes. The nternal waves would propagate nto much shallower water and transform to the elevaton waves [ 3]. The ISW propagaton that usually was descrbed by Korteweg de Vres KdV equaton wll change sgn n ts coeffcent. The comparson of observaton by usng satellte mage and n-stu experment showed the sgn s negatve n deep water and postve n shallow water. The wave transform from waves of depresson to waves of elevaton, for example the depresson propagatng from a deep part of a basn onto the shelf, wll break when the wave ampltude s larger than a half of the water depth mnus the undsturbed depth of the sopycnal of maxmum depresson [4 7]. In ths paper, we wll develop an analytc model of the ISW propagaton over slope-shelf topography. The two-layered flud model wll be used to derve KdV form. The specal soluton wll be used to examne the nature of dynamcs of ISW such as run-up and the breakng waves.. Dervaton of nonlnear nternal waves propagaton over slope-shelf topography The dervaton of ISW propagaton over uneven bottom has been done [4]. In ths paper, we wll derve another approach and show the lmtaton of prevous dervaton. We consder a two-layer flud bounded above by a rgd horzontal plane the rgd-ld approxmaton, z = h and below by a rgd horzontally-varyng boundary z = h x. Each layer conssts of ncompressble, nvscd and rrotatonal flud of a constant densty ρ for the upper layer and ρ for the lower layer. The free nterface between the layer s denoted by z = ηx,t. The geometry of boundary Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3. lcence. Any further dstrbuton of ths work must mantan attrbuton to the authors and the ttle of the work, journal ctaton and DOI. Publshed under lcence by IOP Publshng Ltd

3 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 Fgure. The boundaryproblem of two-layer flud wth rgd ld on the upper layer and uneven bottom on the lower layer. value problems s depcted n fgure. The dervaton of nternal solton wave equaton wll follow procedures as dscussed by [8]. We begn wth the equatons of moton and ther boundary values are gven by u +u u +w u w +u w +w w u + w = ρ p, = ρ p, =, 3 w z=h = 4 h x w z= h = u 5 w z=η = η +u η 6 p p z=η = ρ ρ gη 7 where u, w are orbtal veloctes for horzontal and vertcal respectvely, and ρ s the flud densty; here ndex takes the value for the upper layer and for the lower layer. Now, we ntroduce a scalng as follows: [t] = T, [x] = L, [z] = H, [u] = ǫc, [w] = ǫc H /L, [η] = ǫh and [p] = ρ ǫc, where ǫ s a small parameter and c s phase velocty. The phase speed s gven by c = ḡh wth ḡ = gρ ρ /ρ and H = h +h for constant h. Usng these scales, we can cast and n a nondmensonal form u +ǫu u +ǫw u w +ǫu w +ǫw w u + w = p = p δ 8 9 =

4 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 where δ = H /L. The boundary condtons can be wrtten n the nondmensonal form, w z=α = h x w z=α H = u w z=ǫη = η +ǫu η 3 p p z=ǫη = η 4 where h = α H, h x = α H, H = H/H and H = Hx = h +h x. Furthermore, we assume that ǫ = Oδ and δ = ǫδ wth δ = O [8]. By usng 8 4, we wll obtan the dynamcs of nternal wave η descrbng by D η+ǫd η+ǫ =, 5 where D and D are dfferental operators. The problem to be solved s the boundary condton at z = η. It means that we stll do not know the poston of water partcle at the nterface. Ths can be obtaned by applyng the Taylor expanson of each term that depends on z around z =. Applyng the Taylor expanson to 3 and usng the contnuty equaton yeld Comparng wth 3 we have The boundary condton 4 can be wrtten as p η = p p z= +ǫη p w z=ǫη = w z= ǫη u +Oǫ. 6 w z= = η +ǫ ηu +Oǫ. 7 z= +Oǫ. 8 We also defne a shear cross-nterface as ū = u u z=ǫη and ts Taylor expanson as follows u ū = u u z= +ǫη u +Oǫ. 9 We develop varables p, u, w and η n a seres, n whch ǫ serves as small parameter: u w η p = u w η p +ǫ u w η p z= +Oǫ. By usng ths expanson, the equatons of moton 8, 9 and become u + p p +ǫδ u +ǫ u w + w +ǫ +ǫ p +ǫu u +ǫw u +ǫ p +Oǫ =, +Oǫ =, + w +Oǫ =, 3 u 3

5 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 and the boundary condtons,, 7, 7 and 9 gve w z=α +ǫw z=α +Oǫ =, 4 w z=α H +ǫw z=α w z= +ǫw z= η η +ǫη = p p H +ǫu ǫ η z= +ǫ p p h x +Oǫ =, 5 η u ǫ +Oǫ =, 6 z= +ǫη p p u ū +ǫū = u u z= +ǫ u u z= +ǫη u z= z= +Oǫ, +Oǫ, where we have assumed that the varaton of bottom topography acts as a perturbaton of soltary wave, whch means h x = ǫ h x. At the lowest order, ǫ, the equatons of moton and ther boundary condtons read u + p w z= η =, p =, =, η = u + w p p 7 8 =, w z=α =, w z=α H =, 9 z=, ū = u u z=. 3 We see that, n the lowest order, u and p do not depend on vertcal coordnate z. Wrtng down the frst and the second of 9 nto ther components, we get u Subtractng the two equatons yelds + p =, u + p = 3 ū + η =. 3 Next, we ntegrate the contnuty equaton,.e, the thrd equaton of 9, wth respect to vertcal coordnate wth the upper and lower bounds. For the upper layer, By usng the boundary condton, we get For the lower layer, u u α +w z=α w z= =. 33 α u η =. 34 α H +w z= w z=α H=. 35 4

6 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 By usng the boundary condton and multplyng 34 by α H, and addng up the results we get ū αα H + α H α η =. 36 Fnally, by usng 34, we get η c η =, 37 where c = αα H/α H α. When h x = h wth h beng constant and H =, we have the phase speed of lnear long wave for two-flud system. It s well known that 37 descrbes lnear long waves, wth both left and rght wards propagatng. Its general soluton can be wrtten as η = Fξ +Gξ +, where ξ = x c t represents the rght propagaton, and ξ + = x+c t, the left propagaton. Now, we wll arrve at the crucal blackpont, namely the frst order. At the frst order, ǫ, we have u w δ u +u u + p + p = 38 = 39 + w =, 4 for the equatons of moton note that we have used the fact that u,p / = and w z=α =,w z= = η + η = p p w z=α H = u η u z=,ū = h x 4 4 u u z=, 43 for the boundary condtons. Before we work further wth the frst order, we requre the followng relatons u α H = α H αū, u α = α H +αū. 44 These relatons can be obtaned from the zeroth order. By dong ntegraton from α to z and usng 34 for the upper layer, and smlarly dong ntegraton from α H to z and usng boundary condton of the zeroth order for the lower layer, we get w α z η α z = α, w z = H z η α H. 45 We wrte the horzontal momentum equaton, 38, for each layer, and then subtract and use boundary condton of the frst order and 44, we get ū +Λū ū + η =, 46 5

7 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 where Λ = α α H /α H α. Takng dfferental operaton wth respect to z for the horzontal momentum equaton and to x for the vertcal momentum equaton, followed by subtractng these results and takng ntegraton wth respect to t yeld w δ = ū 47 By substtutng 45 and takng ntegral operaton wth respect to z, we obtan u = δ αz η α z u = δ α H +u α Hz z η z= 48 +u z=. 49 Same as the lowest order, ntegratng the contnuty equaton 4 wth respect to z and usng 48 gve δ α 3 η +α u 3 z= +w z=α w z= =. 5 By usng the frst order of boundary condton and 4, we have η η u + In the same manner, we get the lower layer as follows [ ] δ 3 α H η +α Hu z= By usng 4 and 4, we arrve at η + η u δ H α 3 +u h α u z= 3 δ α 3 η =. 5 +w z= w z=α H=. + α H u z= + α u H z= η δ 3 α H 3 η =. 5 By assumng that H h and consderng that the varyng depth s a perturbaton of nternal waves, then we can gnore the dervatve of H; thus by substtutng 44, we get η + η u + α α H +αū h u + α H z= δ 3 α H 3 η =. 53 Multplyng 5 by α H, 53 wth α, followed by addng them up and usng 43, we obtan α+ α H η +α α H ū + α α H +αū h ū η u η +α + α H δ 3 α H α+ α H 3 η =. 54 6

8 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 Multplyng by and usng the boundary condton of the frst order and assumng the second dervatve of h can be gnored, we then get η c η +Λ η η h + 4 η η 4 =, 55 where = α /α H α and = δ α Hα+ α Hc /3α H α. Because the functon of η also meets the wave equaton 37, then the dervatve wth respect to x can be splt as follows η = c c η. 56 Usng the operator we wrte c +c η Λ c c η η c c h η c c 3 η 3 =, 57 wth ntegraton constant beng zero; ths yelds +c η Λc η η h c η 3 η c 3 =. 58 Fnally, after thorough effort, we get evoluton equaton for η as η +c η Λ η η c 3 η c 3 h c η =. 59 Ths s called the varable-coeffcent perturbed Korteweg-deVres vp-kdv equaton. 3. Specal soluton of ISW propagaton over slope-shelf topography Before we solve 59, let us wrte the equaton n the form of η +c η +γ η η +γ 3 η +Fxη =, 6 3 where γ = Λ/c, γ = /c and Fx = /c h /. The sngle-solton soluton of the equaton s well-known when Fx = and the coeffcents are constant. It has been shown that by usng transformaton equaton 6 reduces to [9] Fx x ψ = exp[ dx dx ]η,τ =,ς = τ t, 6 c c ψ τ +δ τψ ψ ς +δ τ 3 ψ ς 3 =, 6 where δ = γ /c exp[ F/c dx] and δ = γ /c 3. Now, the coeffcents are only functon of τ alone. An analytc soluton based on slowly varyng topography approxmaton has been done 7

9 Conference on Theoretcal Physcs and Nonlnear Phenomena 6 IOP Publshng IOP Conf. Seres: Journal of Physcs: Conf. Seres do :.88/ /856//3 by several researcher [9]. The slowly varyng topography means that the topography undulaton s larger than soltary wave wavelength. The soluton can be expressed as follows ψ = ψ τ, ς+ǫψ τ, ς+ ǫ, 63 where τ = ǫτ and ς = ǫς. The zeroth order leads to sngle-solton soluton gven by [4] ψ ς, τ = A sech [γ ς V τ], 64 where A = 3V /δ and γ = δ A /δ. The frst order satsfes the thrd-order ordnary dfferental equaton wth ψ coeffcent. By usng the conservaton of mass and momentum, the soluton s gven by [ ] ψ ς, τ = A sech A ς V τ, 65 where A = V /4+A +A +A V / and V = A +A. Another way to fnd the analytc soluton of varable-coeffcent KdV equaton has been developed by expanson methods based on the Rccat ordnary dfferental equaton. The soluton s gven by [] where Vτ = τ ψx,t = ψ sech[γς Vττ], 66 δ τ δ τ dτ ψ δ dτ + 3 dτ Wth the specal form of the topography, then soluton could be obtaned., Concluson Propagaton of nternal soltary wave n two layer flud have been derved. We obtan the perturbed varable coeffcent Kortweg and de Vres equaton. By usung slowly varyng bottom topography the solton soluton can be obtaned. Acknowledgments Ths work s partally funded by DIPA PTPSW-BPPT for fscal year 6, Rsetpro 3 and Paul Center for Theoretcal Studes, ETH Zurch 3. References [] Fu K H, Wang Y H, Laurent L S, Smmons H and Wang D P Cont. Shelf Res. 37 [] Holloway P E, Pelnovsky E, Talpova T and Barnes B 997 J. Phys. Oceanogr [3] Prtchard M and Weller R A 5 J. Geophys. Res. C: Oceans C3 [4] Helfrch K R and Melvlle W K 6 Annu. Rev. Flud Mech [5] Shshkna O D, Sveen J K and Grue J 3 Nonlnear Processes Geophys. 743 [6] Helfrch K R and Melvlle W K 986 J. Flud Mech [7] Vlasenko V and Hutter K J. Phys. Oceanogr [8] Gerkema T and Zmmerman J T F 8 An ntroducton to nternal waves lecture notes Royal Netherlands Insttute for Sea Research, Texel [9] Grmshaw R 7 Soltary Waves Propagatng over Varable Topography Berln, Hedelberg: Sprnger Berln Hedelberg pp 5 64 [] Latf M S A Commun. Nonlnear Sc. Numer. Smul