^{1,2}, Katherina Terletska

^{3}, Vladimir Maderich

^{3}, Igor Brovchenko

^{3}, Kyung Tae Jung

^{4}, Efim Pelinovsky

^{1,2,5}and Roger Grimshaw

^{6}

### Abstract

In this paper, we extend the numerical study of Maderich et al. [“Interaction of a large amplitude interfacial solitary wave of depression with a bottom step,” Phys. Fluids22, 076602 (Year: 2010)10.1063/1.3455984] on the interaction of an interfacial solitary wave with a bottom step, considering (i) the energy loss of solitary waves of both polarities interacting with a bottom step and (ii) features of the transformation of a large-amplitude internal solitary waves at the step. We show that the dependence of energy loss on the step height is not monotonic, but has different maximum positions for different incident wave polarities. The energy loss does not exceed 50% of the energy of an incident wave. The results of our numerical modeling are compared with some recent results from laboratory tank modeling.

This work was partially supported by the research project of KIOST (PE99164) and KISTI supercom center (K.T., V.M., I.B., K.J., and E.P.) and by grants of RFBR 12-05-00472 and 12-05-33070 (T.T). E.P. and T.T. received funding from the Federal Target Program “Research and scientific-pedagogical cadres of Innovative Russia” for 2009–2013.

I. INTRODUCTION

II. NUMERICAL EXPERIMENT SETUP

III. RESULTS FROM THE NUMERICAL MODELING

IV. CONCLUSIONS

### Key Topics

- Collisional energy loss
- 35.0
- Internal waves
- 19.0
- Kelvin Helmholtz instability
- 17.0
- Rotating flows
- 15.0
- Numerical modeling
- 9.0

## Figures

Sketch of the numerical tank for simulation of ISW transformation over a bottom step: (a) for a depression ISW, (b) for an elevation ISW.

Sketch of the numerical tank for simulation of ISW transformation over a bottom step: (a) for a depression ISW, (b) for an elevation ISW.

The energy loss δE for waves of elevation versus the height of the bottom layer after the step h 2+ (a) and δE versus ratio h 2+/|a i | (b) for various amplitudes of an incident wave. The regimes of ISW interaction with the step (I–V) separated by dashed lines are shown in the right panel. The half-shaded symbols indicate interaction when the minimum Richardson number . The filled symbols indicate presence of supercritical flow with the composite Froude number .

The energy loss δE for waves of elevation versus the height of the bottom layer after the step h 2+ (a) and δE versus ratio h 2+/|a i | (b) for various amplitudes of an incident wave. The regimes of ISW interaction with the step (I–V) separated by dashed lines are shown in the right panel. The half-shaded symbols indicate interaction when the minimum Richardson number . The filled symbols indicate presence of supercritical flow with the composite Froude number .

The relative energy loss δE for the waves of depression versus height of the bottom layer after the step h 2+ (a) for various amplitudes of the incident wave and δE versus ratio h 2+/|a −| (b). The regimes of ISW interaction with step (I–V) separated by dashed lines are shown in the right panel. The half-shaded symbols indicate interaction with the minimum Richardson number . The filled symbols indicate presence of supercritical flow with the composite Froude number .

The relative energy loss δE for the waves of depression versus height of the bottom layer after the step h 2+ (a) for various amplitudes of the incident wave and δE versus ratio h 2+/|a −| (b). The regimes of ISW interaction with step (I–V) separated by dashed lines are shown in the right panel. The half-shaded symbols indicate interaction with the minimum Richardson number . The filled symbols indicate presence of supercritical flow with the composite Froude number .

The relative energy difference ΔE in the waves of elevation (a) and depression (b) versus the ratio h 2+/|a −|. The regimes of ISW interaction with step are separated by dashed lines and shown by Roman numbers.

The relative energy difference ΔE in the waves of elevation (a) and depression (b) versus the ratio h 2+/|a −|. The regimes of ISW interaction with step are separated by dashed lines and shown by Roman numbers.

The relative difference ΔE between transmitted and reflected waves for waves of elevation versus h 2+ (a) for waves of elevation at h 2−/h 1 = 0.4 and (b) for a wave of depression at h 2−/h 1 = 7. The solid line is an analytical estimate (6) for ΔE.

The relative difference ΔE between transmitted and reflected waves for waves of elevation versus h 2+ (a) for waves of elevation at h 2−/h 1 = 0.4 and (b) for a wave of depression at h 2−/h 1 = 7. The solid line is an analytical estimate (6) for ΔE.

The salinity field in the vicinity of the step shows KH instability for an incident elevation ISW (a) and depression ISW (b) with amplitude 8.8 cm and different h 2+/|a −| at the same time moments.

The salinity field in the vicinity of the step shows KH instability for an incident elevation ISW (a) and depression ISW (b) with amplitude 8.8 cm and different h 2+/|a −| at the same time moments.

Maximal values of composite Froude number, Fr max , at the step versus ratio h 2+/|a i | for incident elevation ISW type (a) and depression ISW type (b).

Maximal values of composite Froude number, Fr max , at the step versus ratio h 2+/|a i | for incident elevation ISW type (a) and depression ISW type (b).

Velocity vectors superimposed on the vorticity field in the vicinity of the step to show the formation of jet and vortices in the wave of elevation (a) and in the wave of depression (b) with amplitude 8.8 cm.

Velocity vectors superimposed on the vorticity field in the vicinity of the step to show the formation of jet and vortices in the wave of elevation (a) and in the wave of depression (b) with amplitude 8.8 cm.

Snapshots of the salinity field show the interaction with a step of an elevation ISW for an incident wave amplitude 4.2 cm (1) h 2+/|a i | = 0.25; (2) h 2+/|a i | = 0.12; (3) h 2+/|a i | = 0; (4) h 2+/|a i | = −0.12; (5) h 2+/|a i | = −1 at successive times (a) −t = 0, (b) −t = 13 c, (c) −t = 38 c.

Snapshots of the salinity field show the interaction with a step of an elevation ISW for an incident wave amplitude 4.2 cm (1) h 2+/|a i | = 0.25; (2) h 2+/|a i | = 0.12; (3) h 2+/|a i | = 0; (4) h 2+/|a i | = −0.12; (5) h 2+/|a i | = −1 at successive times (a) −t = 0, (b) −t = 13 c, (c) −t = 38 c.

Snapshots of the salinity field show the interaction with a step of a depression ISW for an incident wave of amplitude 8.8 cm (1) h 2+/|a i | = 2; (2) h 2+/|a i | = 1.36; (3) h 2+/|a i | = 0.2; (4) h 2+/|a i | = 0 at successive times (a) t = 0; (b) t = 10 s; (c) t = 22 s.

Snapshots of the salinity field show the interaction with a step of a depression ISW for an incident wave of amplitude 8.8 cm (1) h 2+/|a i | = 2; (2) h 2+/|a i | = 1.36; (3) h 2+/|a i | = 0.2; (4) h 2+/|a i | = 0 at successive times (a) t = 0; (b) t = 10 s; (c) t = 22 s.

Zoom of Fig. 10 for the case 3 (left panel) at various time moments and a comparison of the wave after the step with the KdV and Gardner solitons (right panel) corresponding to snapshots in the left panel.

Zoom of Fig. 10 for the case 3 (left panel) at various time moments and a comparison of the wave after the step with the KdV and Gardner solitons (right panel) corresponding to snapshots in the left panel.

Salinity field snapshots show KH billows appearing in the reflected wave from a vertical wall; ISW of elevation (a) and depression (b) with initial amplitude of 8.8 cm.

Salinity field snapshots show KH billows appearing in the reflected wave from a vertical wall; ISW of elevation (a) and depression (b) with initial amplitude of 8.8 cm.

Energy loss δE for wave scattering by an obstacle versus h 2+/|a i | for an incident wave of elevation (a) and an incident wave of depression (b), in comparison with laboratory experiments for two-layer flow by Wessels and Hutter 43 and Chen. 44

## Tables

Parameters of the computational tank.

Parameters of the computational tank.

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