Two-Layer Non-Hydrostatic Model for Generation and Propagation of Interfacial Waves
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摘要
When pycnocline thickness of ocean density is relatively small, density stratification can be well represented as a two-layer system. In this article, a depth integrated model of the two-layer fluid with constant density is considered,and a variant of the edge-based non-hydrostatic numerical scheme is formulated. The resulting scheme is very efficient since it resolves the vertical fluid depth only in two layers. Despite using just two layers, the numerical dispersion is shown to agree with the analytical dispersion curves over a wide range of kd, where k is the wave number and d the water depth. The scheme was tested by simulating an interfacial solitary wave propagating over a flat bottom, as well as over a bottom step. On a laboratory scale, the formation of an interfacial wave is simulated,which also shows the interaction of wave with a triangular bathymetry. Then, a case study using the Lombok Strait topography is discussed, and the results show the development of an interfacial wave due to a strong current passing through a sill.
When pycnocline thickness of ocean density is relatively small, density stratification can be well represented as a two-layer system. In this article, a depth integrated model of the two-layer fluid with constant density is considered,and a variant of the edge-based non-hydrostatic numerical scheme is formulated. The resulting scheme is very efficient since it resolves the vertical fluid depth only in two layers. Despite using just two layers, the numerical dispersion is shown to agree with the analytical dispersion curves over a wide range of kd, where k is the wave number and d the water depth. The scheme was tested by simulating an interfacial solitary wave propagating over a flat bottom, as well as over a bottom step. On a laboratory scale, the formation of an interfacial wave is simulated,which also shows the interaction of wave with a triangular bathymetry. Then, a case study using the Lombok Strait topography is discussed, and the results show the development of an interfacial wave due to a strong current passing through a sill.
When pycnocline thickness of ocean density is relatively small, density stratification can be well represented as a two-layer system. In this article, a depth integrated model of the two-layer fluid with constant density is considered,and a variant of the edge-based non-hydrostatic numerical scheme is formulated. The resulting scheme is very efficient since it resolves the vertical fluid depth only in two layers. Despite using just two layers, the numerical dispersion is shown to agree with the analytical dispersion curves over a wide range of kd, where k is the wave number and d the water depth. The scheme was tested by simulating an interfacial solitary wave propagating over a flat bottom, as well as over a bottom step. On a laboratory scale, the formation of an interfacial wave is simulated,which also shows the interaction of wave with a triangular bathymetry. Then, a case study using the Lombok Strait topography is discussed, and the results show the development of an interfacial wave due to a strong current passing through a sill.
引文
Bai,Y.F.,Yamazaki,Y.and Cheung,K.F.,2018.Convergence of multilayer nonhydrostatic models in relation to Boussinesq-type equations,Journal of Waterway,Port,Coastal,and Ocean Engineering,144(2),06018001-1-06018001-10.
Bai,Y.F.and Cheung,K.F.,2012.Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation,International Journal for Numerical Methods in Fluids,69(2),411-429.
Bijvelds,M.,2003.A Non-Hydrostatic Module for Delft3D-Flow,Technical Report,WL-Delft Hydraulics.
Casulli,V.and Stelling,G.S.,1998.Numerical simulation of 3Dquasi-hydrostatic.free-surface flows,Journal of Hydraulic Engineering,124(7),678-686.
Choi,W.and Camassa,R.,1999.Fully nonlinear internal waves in a two-fluid system,Journal of Fluid Mechanics,396,1-36.
Fang,K.Z.,Liu,Z.B.and Zou,Z.L.,2015.Efficient computation of coastal waves using a depth-integrated.non-hydrostatic model,Coastal Engineering,97,21-36.
Grimshaw,R.,Pelinovsky,E.and Talipova,T.,2007.Modelling internal solitary waves in the coastal ocean,Surveys in Geophysics,28(4),273-298.
Grimshaw,R.,Pelinovsky,E.and Talipova,T.,2008.Fission of a weakly nonlinear interfacial solitary wave at a step,Geophysical&Astrophysical Fluid Dynamics,102(2),179-194.
Helfrich,K.R.and Melville,W.K.,1986.On long nonlinear internal waves over slope-shelf topography,Journal of Fluid Mechanics,167,285-308.
Hibiya,T.,2004.Internal wave generation by tidal flow over a continental shelf slope,Journal of Oceanography,60(3),637-643.
Kanarska,Y.,Shchepetkin,A.and McWilliams,J.C.,2007.Algorithm for non-hydrostatic dynamics in the regional oceanic modeling system,Ocean Modelling,18(3-4),143-174.
Lamb,K.G.,1994.Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge,Journal of Geophysical Research:Oceans,99(C10),843-864.
Lawrence,C.,Adytia,D.and Van Groesen,E.,2018.Variational Boussinesq model for strongly nonlinear dispersive waves,Wave Motion,76,78-102.
Lindzen,R.S.,1974.Stability of a Helmholtz velocity profile in a continuously stratified.infinite Boussinesq fluid-applications to clear air turbulence,Journal of Atmospheric Sciences,31(6),1507-1514.
Liu,Z.B.,Fang,K.Z.and Cheng,Y.Z.,2018.A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed,Journal of Fluid Mechanics,842(2),323-353.
Maderich,V.,Talipova,T.,Grimshaw,R.,Pelinovsky,E.,Choi,B.H.,Brovchenko,I.,Terletska,K.and Kim,D.C.,2009.The transformation of an interfacial solitary wave of elevation at a bottom step,Nonlinear Processes in Geophysics,16,33-42.
Maderich,V.,Talipova,T.,Grimshaw,R.,Terletska,K.,Brovchenko,I.,Pelinovsky,E.and Choi,B.H.,2010.Interaction of a large amplitude interfacial solitary wave of depression with a bottom step,Physics of Fluids,22(7),076602.
Magdalena,I.,2015.An Efficient Two-Layer Non-hydrostatic Numerical Model for Waves Propagation,Ph.D.Thesis,ITB.
Marshall,J.,Hill,C.,Perelman,L.and Adcroft,A.,1997.Hydrostatic.quasi-hydrostatic,and nonhydrostatic ocean modeling,Journal of Geophysical Research:Oceans,102(C3),5733-5752.
Maxworthy,T.,1979.A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge,Journal of Geophysical Research:Oceans,84(C1),338-346.
Murray,S.P.,Arief,D.,Kindle,J.C.and Hurlburt,H.E.,1990.Characteristics of circulation in an Indonesian Archipelago strait from hydrography,current measurements and modeling results,in:Pratt LJ.(ed.),The Physical Oceanography of Sea Strait,Springer,Dordrecht,318,3-23.
Nwogu,O.,1993.Alternative form of Boussinesq equations for nearshore wave propagation,Journal of Waterway,Port,Coastal,and Ocean Engineering,119(6),618-638.
Osborne,A.R.and Burch,T.L.,1980.Internal solitons in the Andaman Sea,Science,208(4443),451-460.
Pudjaprasetya,S.R.,2009.A study on internal solitary waves in lombok strait using the KdV Two-Layer Model,Far East J.of Oce.Res.,2(2),83-91.
Pudjaprasetya,S.R.and Tjandra,S.S.,2014.A hydrodynamic model for dispersive waves generated by bottom motion,in:Fuhrmann J.Ohlberger M,Rohde C.(eds.),Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects,Springer,Cham.
Pudjaprasetya,S.R.,Magdalena,I.and Tjandra,S.S.,2017.A nonhydrostatic two-layer staggered scheme for transient waves due to antisymmetric seabed thrust,Journal of Earthquake and Tsunami,11(1),1740002.
Stelling,G.and Zijlema,M.,2003.An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation,International Journal for Numerical Methods in Fluids,43(1),1-23.
Stelling,G.S.and Duinmeijer,S.P.A.,2003.A staggered conservative scheme for every Froude number in rapidly varied shallow water flows,International Journal for Numerical Methods in Fluids,43(12),1329-1354.
Susanto,R.D.,Mitnik,L.and Zheng,Q.N.,2005.Ocean internal waves observed in the Lombok Strait,Oceanography,18(4),80-87.
Sutherland,B.R.,Keating,S.and Shrivastava,I.,2015.Transmission and reflection of internal solitary waves incident upon a triangular barrier,Journal of Fluid Mechanics,775,304-327.
Tjandra,S.S.and Pudjaprasetya,S.R.,2015.A non-hydrostatic numerical scheme for dispersive waves generated by bottom motion,Wave Motion,57,245-256.
Visser,W.P.,2004.On the Generation of Internal Waves in Lombok Strait through Kelvin-Helmholtz Instability,MSc Thesis,University of Twente,Twente.
Wessels,F.and Hutter,K.,1996.Interaction of internal waves with a topographic sill in a two-layered fluid,Journal of Physical Oceanography,26(1),5-20.
Yamazaki,Y.,Kowalik,Z.and Cheung,K.F.,2008.Depth-integrated.non-hydrostatic model for wave breaking and run-up,International Journal for Numerical Methods in Fluids,61(5),473-497.
Zhou,J.G.and Stansby,P.K.,1999.An arbitrary Lagrangian-Eulerianσ(ALES)model with non-hydrostatic pressure for shallow water flows,Computer Methods in Applied Mechanics and Engineering,178(1-2),199-214.
Zijlema,M.,Stelling,G.and Smit,P.,2011.SWASH:an operational public domain code for simulating wave fields and rapidly varied flows in coastal waters,Coastal Engineering,58(10),992-1012.
Bai,Y.F.and Cheung,K.F.,2012.Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation,International Journal for Numerical Methods in Fluids,69(2),411-429.
Bijvelds,M.,2003.A Non-Hydrostatic Module for Delft3D-Flow,Technical Report,WL-Delft Hydraulics.
Casulli,V.and Stelling,G.S.,1998.Numerical simulation of 3Dquasi-hydrostatic.free-surface flows,Journal of Hydraulic Engineering,124(7),678-686.
Choi,W.and Camassa,R.,1999.Fully nonlinear internal waves in a two-fluid system,Journal of Fluid Mechanics,396,1-36.
Fang,K.Z.,Liu,Z.B.and Zou,Z.L.,2015.Efficient computation of coastal waves using a depth-integrated.non-hydrostatic model,Coastal Engineering,97,21-36.
Grimshaw,R.,Pelinovsky,E.and Talipova,T.,2007.Modelling internal solitary waves in the coastal ocean,Surveys in Geophysics,28(4),273-298.
Grimshaw,R.,Pelinovsky,E.and Talipova,T.,2008.Fission of a weakly nonlinear interfacial solitary wave at a step,Geophysical&Astrophysical Fluid Dynamics,102(2),179-194.
Helfrich,K.R.and Melville,W.K.,1986.On long nonlinear internal waves over slope-shelf topography,Journal of Fluid Mechanics,167,285-308.
Hibiya,T.,2004.Internal wave generation by tidal flow over a continental shelf slope,Journal of Oceanography,60(3),637-643.
Kanarska,Y.,Shchepetkin,A.and McWilliams,J.C.,2007.Algorithm for non-hydrostatic dynamics in the regional oceanic modeling system,Ocean Modelling,18(3-4),143-174.
Lamb,K.G.,1994.Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge,Journal of Geophysical Research:Oceans,99(C10),843-864.
Lawrence,C.,Adytia,D.and Van Groesen,E.,2018.Variational Boussinesq model for strongly nonlinear dispersive waves,Wave Motion,76,78-102.
Lindzen,R.S.,1974.Stability of a Helmholtz velocity profile in a continuously stratified.infinite Boussinesq fluid-applications to clear air turbulence,Journal of Atmospheric Sciences,31(6),1507-1514.
Liu,Z.B.,Fang,K.Z.and Cheng,Y.Z.,2018.A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed,Journal of Fluid Mechanics,842(2),323-353.
Maderich,V.,Talipova,T.,Grimshaw,R.,Pelinovsky,E.,Choi,B.H.,Brovchenko,I.,Terletska,K.and Kim,D.C.,2009.The transformation of an interfacial solitary wave of elevation at a bottom step,Nonlinear Processes in Geophysics,16,33-42.
Maderich,V.,Talipova,T.,Grimshaw,R.,Terletska,K.,Brovchenko,I.,Pelinovsky,E.and Choi,B.H.,2010.Interaction of a large amplitude interfacial solitary wave of depression with a bottom step,Physics of Fluids,22(7),076602.
Magdalena,I.,2015.An Efficient Two-Layer Non-hydrostatic Numerical Model for Waves Propagation,Ph.D.Thesis,ITB.
Marshall,J.,Hill,C.,Perelman,L.and Adcroft,A.,1997.Hydrostatic.quasi-hydrostatic,and nonhydrostatic ocean modeling,Journal of Geophysical Research:Oceans,102(C3),5733-5752.
Maxworthy,T.,1979.A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge,Journal of Geophysical Research:Oceans,84(C1),338-346.
Murray,S.P.,Arief,D.,Kindle,J.C.and Hurlburt,H.E.,1990.Characteristics of circulation in an Indonesian Archipelago strait from hydrography,current measurements and modeling results,in:Pratt LJ.(ed.),The Physical Oceanography of Sea Strait,Springer,Dordrecht,318,3-23.
Nwogu,O.,1993.Alternative form of Boussinesq equations for nearshore wave propagation,Journal of Waterway,Port,Coastal,and Ocean Engineering,119(6),618-638.
Osborne,A.R.and Burch,T.L.,1980.Internal solitons in the Andaman Sea,Science,208(4443),451-460.
Pudjaprasetya,S.R.,2009.A study on internal solitary waves in lombok strait using the KdV Two-Layer Model,Far East J.of Oce.Res.,2(2),83-91.
Pudjaprasetya,S.R.and Tjandra,S.S.,2014.A hydrodynamic model for dispersive waves generated by bottom motion,in:Fuhrmann J.Ohlberger M,Rohde C.(eds.),Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects,Springer,Cham.
Pudjaprasetya,S.R.,Magdalena,I.and Tjandra,S.S.,2017.A nonhydrostatic two-layer staggered scheme for transient waves due to antisymmetric seabed thrust,Journal of Earthquake and Tsunami,11(1),1740002.
Stelling,G.and Zijlema,M.,2003.An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation,International Journal for Numerical Methods in Fluids,43(1),1-23.
Stelling,G.S.and Duinmeijer,S.P.A.,2003.A staggered conservative scheme for every Froude number in rapidly varied shallow water flows,International Journal for Numerical Methods in Fluids,43(12),1329-1354.
Susanto,R.D.,Mitnik,L.and Zheng,Q.N.,2005.Ocean internal waves observed in the Lombok Strait,Oceanography,18(4),80-87.
Sutherland,B.R.,Keating,S.and Shrivastava,I.,2015.Transmission and reflection of internal solitary waves incident upon a triangular barrier,Journal of Fluid Mechanics,775,304-327.
Tjandra,S.S.and Pudjaprasetya,S.R.,2015.A non-hydrostatic numerical scheme for dispersive waves generated by bottom motion,Wave Motion,57,245-256.
Visser,W.P.,2004.On the Generation of Internal Waves in Lombok Strait through Kelvin-Helmholtz Instability,MSc Thesis,University of Twente,Twente.
Wessels,F.and Hutter,K.,1996.Interaction of internal waves with a topographic sill in a two-layered fluid,Journal of Physical Oceanography,26(1),5-20.
Yamazaki,Y.,Kowalik,Z.and Cheung,K.F.,2008.Depth-integrated.non-hydrostatic model for wave breaking and run-up,International Journal for Numerical Methods in Fluids,61(5),473-497.
Zhou,J.G.and Stansby,P.K.,1999.An arbitrary Lagrangian-Eulerianσ(ALES)model with non-hydrostatic pressure for shallow water flows,Computer Methods in Applied Mechanics and Engineering,178(1-2),199-214.
Zijlema,M.,Stelling,G.and Smit,P.,2011.SWASH:an operational public domain code for simulating wave fields and rapidly varied flows in coastal waters,Coastal Engineering,58(10),992-1012.