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The interaction of strongly nonlinear solitary waves with step-like bottom topography Kateryna Terletska 1, 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 (1) 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,

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

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

Observations of the second baroclinic mode internal solitary waves in the northern South China Observations of the second baroclinic mode internal solitary waves in the northern South 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. 4

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

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

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

Numerical setup H=0. 46 m Two series of experiments with different wavelengths were performed: Numerical setup H=0. 46 m Two series of experiments with different wavelengths were performed: 8

Classification of interaction regimes of second mode with a step Ratio of the lower 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 Several regimes can be identified: * Weak interaction B>2. 5 Moderate interaction 0. 5

Weak interaction: generation of breather-like structure (B = 3. 2) Density contours showing the Weak interaction: generation of breather-like structure (B = 3. 2) Density contours showing the evolving wave field Breather–like Second mode structure 10

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

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

Moderate interaction: (B = 1. 2) (interaction with the step in details) Formation of 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: 13 Laboratory experiments from (Lapidievsky, 2013)

Strong interaction ( B = -0. 2) Density contours showing the evolving wave field Strong interaction ( B = -0. 2) Density contours showing the evolving wave field Reflected 2 mode Transmitted 1 mode 14

Hovmöller diagram (case 1) Weak interaction Moderate interaction Strong interaction B>2. 5 0. 5<B<2. Hovmöller diagram (case 1) Weak interaction Moderate interaction Strong interaction B>2. 5 0. 5

Hovmöller diagram (case 2) Weak interaction B>2. 5 Moderate interaction 0. 5<B<2. 5 Strong Hovmöller diagram (case 2) Weak interaction B>2. 5 Moderate interaction 0. 5

The dependencies of transformation coefficients on the parameter of blocking are similar for incident 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 17

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

Conclusions 1. Several regimes have been identified based on the parameter that is the Conclusions 1. Several regimes have been identified based on the parameter that is the ratio of the lower layer depth over the step to incident wave amplitude 2. New mechanism of the breather generation has been established in the frame of numerical modeling. Internal wave-breather in the three-layer stratification can occur due to interaction of the second mode internal wave with abrupt changes 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. 19