1. Non-hydrostatic modelling of internal waves
Vladimir Maderich, Kateryna Terletska,
Igor Brovchenko
Institute of Mathematical Machine and System Problems, Kiev, Ukraine

2. Outlook Non-hydrostatic model POM-NH
Application of non-hydrostatic model to the internal waves problems
Internal solitary wave transformation over the bottom step
Frontal interaction of internal solitary waves of first and second modes.

3. Governing equations

4. Free surface equation in conservative form is used in the non-hydrostatic models

5. In the non-hydrostatic models the pressure P is decomposed on hydrostatic PH and non-hydrostatic Q components

6. Projection method

7. Pressure correction method

8. Non-hydrostatic baroclinic models using pressure decomposition

9. 3D non-hydrostatic free surface model NH-POM
Generalized vertical coordinate
Turbulence description
- DNS
- RANS
- LES
Mode splitting
-2D depth–integrated equations (external mode)
-3D equations (internal mode)
Decomposition of pressure and velocity fields into hydrostatic and non-hydrostatic components
Open- boundary conditions
-Radiation conditions
-Newtonian relaxation
*Kanarska, Maderich (2003) Ocean Dynamics, 53,
Maderich et al (2010) in book: Nonlinear internal waves in lakes. Springer,

10. Model Algorithm Poisson equation for Q P=Ph+Q P=Ph+Q
External mode
1 stage Free surface elevation and depth-averaged velocity
Explicit method
Semi-implicit method
2 stage Provisional velocity field Q=0
3 stage Non-hydrostatic velocity fields
Poisson equation for Q
Internal mode
Semi-implicit method
4 stage Scalar fields

11. ISWs impacting the flank of Ile-aux-Lievres Island
Echogram (Bourgault et al., 2007)
Modeling

12. Internal solitary wave transformation over the bottom step
depression ISW
for
elevation ISW
The parameters of computational tanks
Sketch of the numerical tank
dhfguhgjgfv

13. The energy loss δ E for the ISW versus ratio h 2+ /|ai |
ISW of elevation
ISW of depression

14. The relative energy difference ΔE versus the ratio h 2+ /|a- |
ISW of elevation
ISW of depression

15. The energy loss δ E for the ISW versus δ E versus ratio h 2+ /|ai |
ISW of elevation
ISW of depression

16. The salinity field in vicinity of the step shows the KH instability in moderate interaction regime
ISW of depression
ISW of depression
ISW of elevation

17. The energy loss δ E for the ISW versus δ E versus ratio h 2+ /|ai |
ISW of elevation
ISW of depression

18. Maximal values of composite Froude number Fr at the step
ISW of depression
ISW of elevation

19. Velocity vectors superimposed on the vorticity field in the vicinity of the step
ISW of elevation
ISW of depression
vortex and bolus formation
jet and vortex pair formation

20. The snapshots of the salinity field show the interaction with a step of elevation ISW

21. The snapshots of the salinity field show the interaction with a step of depression ISW

22. The energy loss δ E for the ISW versus δ E versus ratio h 2+ /|ai |
ISW of elevation
ISW of depression
reflection regime (V)

23. The salinity field snapshots show the KH billows appearance in the reflected from vertical wall ISW of large amplitude
elevation ISW
depression ISW

24. The energy loss δ E in the waves scattering on obstacle versus h 2+ /|ai| in comparison with laboratory experiments
ISW of elevation
ISW of depression

25. Frontal interaction of ISW of first mode
Sketch of computational basin

26. Interaction of ISW of moderate amplitude (a/h1=1.35)

27. Interaction of ISW of large amplitude (a/h1=2.17)

28. Spatio-temporal paths of interacted waves and energy transformationsEK
Kinetic and potential energy transformation EK EP EP
Moderate interaction
Strong interaction

29. Frontal interaction of ISW of second mode

30. Second mode internal wave propagation in pycnocline
ISW does not transport mass
Mass transporting ISW
Maderich et al. 2001

31. Classification of solitary wave of second mode

32. Wave speed vs wave amplitude
Solidon model
(Kozlov, Makarov, 1990)
Weakly nonlinear theory
(Benjamin, 1967)

33. Interaction of second mode ISW at a/h1=0.85
Contours of density
Dye

34. Interaction of second mode ISW at a/h1=1.7
Contours of density
Dye

35. Interaction of second mode ISW at a/h1=1.8
Contours of density
Dye

36. The energy loss δ E for the ISW versus ratio a/h

37. Publications
Maderich V., Brovchenko I., Terletska K., Hutter K. (2012) Numerical simulations of the nonhydrostatic transformation of basin-scale internal gravity waves and wave-enhanced meromixis in lakes. Ch. 4 in Hutter K. (Ed.) Nonlinear internal waves in lakes. Springer. Series: Advances in Geophysical and Environmental Mechanics, p
Maderich V., Talipova T., Grimshaw R., Terletska K., Brovchenko I., Pelinovsky E., Choi B.H. (2010) Interaction of a large amplitude interfacial solitary wave of depression with a bottom step. Physics of Fluids , 22, , doi: /
Maderich V., Talipova T., Grimshaw R., Pelinovsky E., Choi B.H., Brovchenko I., Terletska K., Kim D.C. (2009) The transformation of an interfacial solitary wave of elevation at a bottom step. Nonlinear Processes in Geophysics , 16,
Kanarska Y., Maderich V. (2003) A non-hydrostatic numerical model for calculating of free-surface stratified flows. Ocean Dynamics, 51, N3,
Maderich V., van Heijst G.J., Brandt A. (2001) Laboratory experiments on intrusive flows and internal waves in a pycnocline. J. Fluid Mechanics,. 432, p