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The interaction of strongly nonlinear solitary waves with step-like bottom topography

The interaction of strongly nonlinear solitary waves with step-like bottom topography. Kateryna Terletska 1 , Vladimir Maderich 1 , Igor Brovchenko 1 , Kyung Tae Jung 3. Institute of Mathematical Machines and System Problems NASU, Marine and River Systems Modelling Department, Kiev,Ukraine,

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The interaction of strongly nonlinear solitary waves with step-like bottom topography

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  1. The interaction of strongly nonlinear solitary waves with step-like bottomtopography Kateryna Terletska1,Vladimir Maderich1, Igor Brovchenko1, Kyung Tae Jung3 • Institute of Mathematical Machines and System Problems NASU, Marine and • River Systems Modelling Department, Kiev,Ukraine, • (2)Korea Institute of Ocean Science and Technology, Ansan, South Korea,

  2. Four types of ISWs can exists in the three-layer fluid:

  3. Observations on internal wave of the second mode New Jersey shelfMoum et al.,2008 Shroyer et al 2010 Lakes Biwa Saggio and Imberger, 1998, 2001; Antenucci et al., 2000; Boegman et al., 2003 Knight Inlet Farmer and Smith, 1980 South China Sea: Yang et al 2011 NPG Yang et al 2009 JGR Liu et al 2013 CSR Strait of Gibraltar Sannino et al 2011 Farmer and Armi, 1988 Strait of Messina Alpers et al 1996 Dreadnought Bank in the Andaman Sea Vlasenko 2005 MascareneRidge in the Indian Ocean Konyaev et al., 1995; Sabininand Serebryany, 2005 Kinneret (Israel) Saggio and Imberger, 1998, 2001; Antenucci et al., 2000; Boegman et al., 2003

  4. Observations of the second baroclinic mode internal solitary waves in the northernSouth China Sea Bathymetry of the area around the Luzon Strait Isotherm observed during summer (24 June to 27 June 2005. ) Y. J. Yang et al.

  5. Reflection of the mode 1 ISW Second mode internal wave generation: Interaction of mode 1 ISW with a sill Generation by intrusion in the interface layer (Horn et al 2001, JFM) (Maderich et al 2001, JFM ) ( Vlasenko andHutter 2001 NPG)

  6. Motivation: This investigation was inspired bythe fact that there is insufficient understanding of theshoaling process of second mode ISWs from the deep part of theocean onto the shelf. To study the properties of interaction of second mode internal waves with abottom features we consider simple configuration of numerical tank with a bottom step

  7. The 3D equations of continuity, momentum and scalar transport in the Boussinesq approximation are: 3D non-hydrostatic free surface model NH-POM (*) are Cartesian coordinates, is velocity component, p - is pressure deviation in the Boussinesqapproximation, ρ′ - is density deviation (*)Kanarska Y., Maderich V. (Ocean Dynamics 2003)

  8. Numerical setup H=0.46 m Two series of experiments with different wavelengths were performed:

  9. Classification of interaction regimes of second mode with a step Ratio of the lower layer depth over the step to incident wave amplitude (blocking parameter) was used to classified second mode * Weak interaction B>2.5 Several regimes can be identified: Moderate interaction 0.5<B<2.5 Strong interaction -1<B<0.5 *Talipova T., Terletska K., Maderich V., Brovchenko I., Jung K.T., Pelinovsky E., Grimshaw R., (2013) Internal solitary wave transformation over a bottom step: loss of energy, Physics of Fluids 25

  10. Weak interaction: generation of breather-like structure (B = 3.2) Density contours showing the evolvingwave field Breather–like structure Second mode

  11. Breather-like structure is well described by breather solution of mKdV equation: * * Lamb, 1983

  12. Moderate interaction: (B = 1.2) Density contours showing the evolvingwave field 2 mode wave 1 mode wave

  13. Moderate interaction: (B = 1.2) (interaction with the step in details) Formation of the jet Transient structure appears after interaction with a bottom sharp change: Laboratory experiments from (Lapidievsky, 2013)

  14. Strong interaction(B = -0.2) Density contours showing the evolvingwave field Reflected 2 mode Transmitted 1 mode

  15. Hovmöller diagram (case 1) Strong interaction Weak interaction Moderate interaction B>2.5 -1<B<0.5 0.5<B<2.5

  16. Hovmöller diagram (case 2) Moderate interaction Weak interaction Strong interaction -1<B<0.5 0.5<B<2.5 B>2.5

  17. The dependencies of transformation coefficients on the parameter of blocking are similar for incident long and intermediate waves of the second mode Strong interaction Moderate interaction Weak interaction

  18. The dependencies ofphase velocity on the parameter of blocking are similar for incident long and intermediate waves of the second mode Strong interaction Weak interaction Moderate interaction

  19. Conclusions 1. Several regimes have beenidentified based on the parameter that is the ratio of the lower layer depthover the step to incident wave amplitude 2.New mechanism of the breather generation has been established in theframe of numerical modeling.Internal wave-breather in the three-layerstratificationcan occur due to interaction of the second mode internal wave with abruptchanges of the bottom topography 3.The dependencies for transformation coefficients on the blocking parameter are similar for incident long and intermediate waves of the second mode.

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