The interaction of strongly The interaction of strongly nonlinear solitary waves with nonlinear solitary waves with

step-like bottomstep-like bottomtopographytopography

Kateryna TerletskaKateryna Terletska11,,Vladimir MaderichVladimir Maderich11, Igor Brovchenko, Igor Brovchenko11,, Kyung Tae JungKyung Tae Jung33

(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,

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FFour types of ISWsour types of ISWs can exists in the can exists in the three-layer fluid:three-layer fluid:

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Observations on internal wave Observations on internal wave of the second modeof the second mode

Dreadnought Dreadnought Bank in Bank in the Andaman Seathe Andaman Sea Vlasenko 2005Vlasenko 2005

Strait of Strait of GibraltarGibraltar Sannino et al 2011Sannino et al 2011Farmer and Armi, 1988Farmer and Armi, 1988

Strait of Messina Strait of Messina Alpers Alpers et al 1996et al 1996

South China SeaSouth China Sea:: YangYang et al 2011 NPG et al 2011 NPG YangYang et al 2009 JGR et al 2009 JGR Liu Liu et al 2013 CSRet al 2013 CSR

Knight InletKnight InletFarmer and Smith, 1980

MascareneMascarene Ridge in Ridge in the Indian Oceanthe Indian OceanKonyaev et al., 1995; Konyaev et al., 1995; SabininSabinin and Serebryany, 2005and Serebryany, 2005

LLakes Biwaakes BiwaSaggio and Imberger, 1998,Saggio and Imberger, 1998,2001; Antenucci et al., 2000; 2001; Antenucci et al., 2000; Boegman et al., 2003Boegman et al., 2003

Kinneret (IsraelKinneret (Israel)Saggio and Imberger, 1998,Saggio and Imberger, 1998,2001; Antenucci et al., 2000; 2001; Antenucci et al., 2000; Boegman et al., 2003Boegman et al., 2003

New Jersey shelfNew Jersey shelf Moum et al.,2008Moum et al.,2008Shroyer et al 2010Shroyer et al 2010

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Observations of the second baroclinic mode internal Observations of the second baroclinic mode internal solitary waves in solitary waves in the northernthe northern South China SeaSouth China Sea

Bathymetry of the area around the Luzon Strait

Isotherm observed during summerIsotherm observed during summer (24 June to 27 June 2005. ) (24 June to 27 June 2005. ) Y. J. Yang et al.Y. J. Yang et al.

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Second mode internal wave 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)

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Motivation:Motivation:

To study the properties of interaction of second To study the properties of interaction of second mode internal mode internal waves with awaves with a bottom featuresbottom features we we consider simple configuration of numerical tank consider simple configuration of numerical tank with a bottom stepwith a bottom step

ThisThis investigation investigation was inspired bywas inspired by the fact that the fact that there is insufficient understanding of thethere is insufficient understanding of the shoaling shoaling process of second mode ISWs from the deep part of process of second mode ISWs from the deep part of thethe ocean onto the shelf.ocean onto the shelf.

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3D non-hydrostatic free surface model NH-POM (*)3D non-hydrostatic free surface model NH-POM (*)

,0U

x

00 0

1,iU

U U

t xg

p

x x

UU

x x

g

The 3D equations of continuity, momentum and scalar transport in the Boussinesq approximation are:

,0U

x

(*)(*) Kanarska Y., Maderich V. (Ocean Dynamics 2003)

2j

jj j j j

Ut x x x x

, ,ix x y z are Cartesian coordinates, , ,iU U V W is velocity component,

p - is pressure deviation in the Boussinesq approximation, ρ′ - is density deviation

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H=0.46 m

Two series of experiments with different wavelengths were performed:

Numerical setupNumerical setup

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Classification of interaction regimes of Classification of interaction regimes of second mode with a stepsecond mode with a step

* 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

*Ratio of the lower layer depthover the step to incident wave amplitude (blocking parameter) was used to classified second mode

Weak interactionWeak interaction

Strong interactionStrong interaction -1<B<0.5-1<B<0.5

0.5<B<2.50.5<B<2.5

B>2.5B>2.5

Moderate interactionModerate interactionSSeveral regimes can be everal regimes can be identified:identified:

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Weak interaction: Weak interaction: generation of breather-like structuregeneration of breather-like structure (B = 3.2)(B = 3.2)

Breather–likeBreather–likestructurestructure

Density contours showing the evolving wave field

Second modeSecond mode

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Breather-like structure is well described by Breather-like structure is well described by breather solution of mKdV equationbreather solution of mKdV equation::

* Lamb, 1983

*

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ModerateModerate interaction: interaction:(B = 1.2)(B = 1.2)

1 mode wave1 mode wave2 mode wave2 mode wave

Density contours showing the evolving wave field

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Formation of the jetjet

Laboratory experiments from (Lapidievsky, 2013)

(interaction with the step in details)

Transient structureTransient structure appears after interaction with a

bottom sharp change:

ModerateModerate interaction: interaction:(B = 1.2)(B = 1.2)

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Strong interactionStrong interaction (( B = -0.2)B = -0.2)

Transmitted 1 modeTransmitted 1 mode Reflected 2 modeReflected 2 mode

Density contours showing the evolving wave field

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(case 1)Hovmöller diagram

Weak interactionWeak interaction Strong interactionStrong interaction

--1<B<0.51<B<0.50.5<B<2.50.5<B<2.5B>2.5B>2.5

Moderate interactionModerate interaction

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Hovmöller diagramWeak interactionWeak interaction Strong interactionStrong interaction

--1<B<0.51<B<0.50.5<B<2.50.5<B<2.5B>2.5B>2.5Moderate interactionModerate interaction

(case 2)

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The dependencies The dependencies ofof transformation coefficients transformation coefficients on on the parameter of blocking are similar for incident the parameter of blocking are similar for incident long and intermediate waves of long and intermediate waves of the the second modesecond mode

Weak Weak interactioninteraction

ModerateModerate interactioninteraction

Strong Strong interactioninteraction

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The dependencies The dependencies ofof phase velocity on phase velocity on the parameter the parameter of blocking are similar for incident long and of blocking are similar for incident long and

intermediate waves of intermediate waves of the the second modesecond mode

Weak Weak interactioninteraction

ModerateModerate interactioninteraction

Strong Strong interactioninteraction

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ConclusionsConclusions

1. 1. SSeveral regimes everal regimes have beenhave been identified based onidentified based on the parameter the parameter that is the that is the ratio of the lower layer ratio of the lower layer ddepthepth over the step to incidentover the step to incidentwave amplitudewave amplitude

2.2. NNew mechanism of the breather generation ew mechanism of the breather generation has been establishedhas been established iin then the frame of numerical modelingframe of numerical modeling.. IInternal wave-breather in the nternal wave-breather in the three-layerthree-layer stratificationstratification can occur due to interaction of the secondcan occur due to interaction of the second mode internal wave with abruptmode internal wave with abrupt changes of the bottom topographychanges of the bottom topography

3.The dependenciesThe dependencies for transformation coefficients on the blocking parameter are similar for incident long and intermediate wavesare similar for incident long and intermediate waves of the second mode.