Tatiana Talipova 1,2 , Katherina Terletska 3 , Vladimir Maderich 3 , Igor Brovchenko 3 , Kyung Tae Jung
Efim Pelinovsky 1,2 and Roger Grimshaw 5)
4 ,
1 Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Nizhny Novgorod,
Russia
2 Department of Applied Mathematics, Nizhny Novgorod Technical State University, Nigny Novgorod,
Russia
3 Department of Marine and River Systems, Institute of Mathematical Machine and System Problems,
Kiev, Ukraine
4 Korea Institute of Ocean Science & Technology, Ansan, Republic of Korea
5 Department of Mathematical Sciences, Loughborough University, Loughborough, UK
In this paper we extend the numerical study of the Maderich et al (2010) on the interaction of an interfacial solitary wave with a bottom step, considering (i) the energy loss of a solitary waves of both polarities interacting with a bottom step and (ii) features of 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.
Internal waves are well known to play a significant role in mixing processes and in the dissipation of mechanical energy in the ocean (Munk and Wunsch 1 ) These authors hypothesized that as well as along the ocean shelves, intense mixing occurs in regions with underwater mountains and banks, where intense internal waves are generated by the barotropic tide and are then broken down propagating over the non-uniform bottom relief with horizontally variable density stratification. There is a considerable literature on the transformation of nonlinear internal waves such as internal solitary waves (ISW) over horizontally variable background, and their consequent disintegration. The limiting case of adiabatic transformation of ISW over very smooth background is discussed in Grimshaw et al 2-4 and Helfrich and Melville.
5
On the other hand, n on-adiabatic ISW transformations over a bottom slope and at the shelf break due to critical points, instability, turbulence generation and mixing, and other processes were investigated in situ (Klymak and Moum, 6 Moum et al , 7,8 Bourgeault et al , 9 ) , theoretically
( Knickerbocker and Newell, 10
Grimshaw et al , 11,12 ), numerically (Vlasenko and Hutter, 13 Bourgeault et al, 14 Maderich et al 15 ) and in a number of laboratory studies (Helfrich and Melville 16 ; Michallet and
1

Ivey 17 Chen et al, 18,19 Cheng et al, 20 Gorodetska et al , 21 ). Often these processes were studied in the twolayer flow approximation. Important strongly nonlinear processes in ISW shoaling have been revealed and studied in field observations and laboratory and numerical experiments: formation of surging localised vortexes (“boluses”) transporting the dense water upslope (Helfrich, 22 Michallet and Ivey, 17
Klymak and Moum, 6 Bourgault et al , 14 Venayagamoorthy and Fringer, 23 Bourgault and Kelley 24 ;
Aghsaee et al , 25 . Maderich et al 15 ) and shear instability in the runup of ISW depression waves (Orr and
Mignerey, 26 Boegman et al, 27 Maderich et al 15 ). Energy losses of nonlinear internal waves in the shoaling process have been discussed including dissipation on the real ocean shelves (Lozovatskii et al 28 ; Lamb 29 ; Moum et al 30,31 ; Kelly et al 32 ; Shroyer et al 33 ). Another limiting case of ISW propagating over a bottom step was studied for a two-layer stratification theoretically by Grimshaw et al 34 and numerically using the fully nonlinear Navier-Stokes equations by Maderich et al.
35,36 In the paper by
Grimshaw et al 34 the study on the solitary wave transformation over a step was carried out using a weak nonlinear approximation (extended Korteweg – de Vries equation), without taking into account energy losses and mixing. It was shown that the bottom step height determines the reflection and transmission coefficients and in case when it is equal to the height of the lower layer, there is the full reflection of solitary wave and no energy passes over the step. The transformation of solitary wave with moderate amplitude and positive polarity over the bottom step was studied numerically using the fully nonlinear Navier-Stokes equations by Maderich et al 35 . The bottom step height in that study was chosen relatively small to compare with the asymptotic theory of Grimshaw et al 34 . With a larger bottom step and amplitude of the incident solitary wave the applicability of the weakly nonlinear theory is violated and it was then shown by Maderich et al 36 that for an incident solitary wave of large amplitude and negative polarity part of the energy is transformed into mixing and the formation of eddies in front of step with the Kelvin-Helmholtz (KH) instability after the step. Maderich et al 36 estimated the energy losses using the methodology developed by Shepherd, 37 Scotti et al 38 and Lamb, 29 and found a growth of energy losses with a decrease in the ratio of depth of lower layer ( h
2+
) behind the step to the incident wave amplitude ( a i
) in the range h
2 +
/ a i
from 3.3 to 1.25. In this paper we have extend these abovementioned studies considering (i) the interaction of the solitary waves of both polarities with a bottom step, (ii) the effect of varying the step height from a small fraction of total depth to the full depth, and (iii) features of ISW of large amplitude. The study is done by the numerical experiment simulating a tank of laboratory scale as in our previous works. The fully nonlinear Navier
- Stokes equations are solved as a two-dimensional problem. The short description of experiment setup is presented in Section 2. The results of numerical experiments for the interaction of an interfacial solitary wave with a step, and the consequent energy losses are discussed in Section 3. Our conclusions are summarized in Section 4.
The numerical model using the Navier – Stokes equations has been described in detail by Kanarska and Maderich 39 and Maderich et al 15 . A series of simulations are carried out in two spatial dimensions with a horizontal coordinate x and a vertical coordinate z . The tank of laboratory scale for the
2

computational experiments has two configurations shown in Fig. 1. The dimensions of tank are: the horizontal length L = 30 m, the length of the step L s
= 15 m whereas the depth of the tank, H , is varied in experiments. Details of parameters for the experiments are given in Table 1. The background salinity stratification at constant temperature of 20°C in the flume for both cases is modeled by two layers of thickness h
1
and h
2 ±
( =
1
+
2 ±
). Here h
2 − and h
2 +
are values of the lower layer thickness before and after the bottom step, respectively. The salinities of upper and bottom layers are S up
= 2 and
S bot
= 15, respectively. The vertical profile S ( z ) is approximated by
=
S up
+ S bot
S bot
− S up
2
−
2 tanh
⎛
⎝
( −
1
) dh
⎞
⎠
(1)
Fig. 1 Sketch of the numerical tank for simulation of ISW transformation over the bottom step: (a) for depression ISW, (b) for elevation ISW.
The initial interface thickness ( dh = 0.2 cm) is much less than the thickness of both layers. In the simulations we visualize the interface η as an isohaline with salinity equal 8.5 which is located at z = h
1 in undisturbed state.
Solitary waves are generated by the collapse of a mixed volume at the left-end side of the tank
(Maderich et al 34 ). Numerical experiments are carried out with the molecular values of kinematic viscosity ν = 1.1 10 − 6 m 2 s -1 and diffusivity of salt
= 10 − 9 m 2 s -1 . At the bottom and end walls nonslip boundary conditions are used. Spatial resolutions are 0.01 m × 0.0025 m for the case of depression
ISW and 0.01 m × 0.0015 m for the case of elevation ISW.
3

A total of 78 runs are performed, with about 12 to 14 runs for each incident wave amplitude. These runs cover a range of incident ISW with moderate and large amplitudes, a i
, for both elevation and depression types in a wide range of positive and negative heights of lower layer after step, h
2+
, counted at the undisturbed interface position. The sign “minus” for the height of lower layer after the step, h
2+
, means that the step height is more than the height of lower layer before the step, h
2 −
. Sections x l
and x r are the sections where the reflected and transmitted wave energy is estimated, and they are determined in Table 1 as the distance from the step point.
TABLE 1. The parameters of computational tank a i m h
1 m h
2- m
Range of h
2+ m x l m x r m
0.025 0.2 elevation ISW types
0.08 -0.11 – +0.06
0.042 0.2 0.08 -0.11 – +0.06
0.052 0.2 0.08 -0.11 – + 0.06
0.85 0.85
0.9
1.1
0.088 0.28 0.04 -0.22 – +0.037 0.6
0.9
1.1
0.6 depression ISW types
0.028 0.04 0.28 -0.035 – + 0.24 0.37 0.37
0.045 0.04 0.28 -0.035 – + 0.2
0.088 0.04 0.28 -0.035 – + 0.2
0.4
0.6
0.4
0.6
We present our results for the energy losses of interfacial solitary waves of both polarities incident at the bottom step for a wide range of step heights and for several incident wave amplitudes for elevation
ISW and depression. The balance of the energy after the wave has crossed the step is estimated following Maderich et al 36 . The pseudoenergy which is the sum of kinetic and available potential energy (Shepherd 37 ) of incident, transmitted and reflected waves is calculated using the energy fluxes of these waves at the sections x l and x r
. The method of estimation of available potential energy and energy fluxes had been discussed in (Scotti et al, 38 Lamb, 29 Kang and Fringer 40 ). The relative estimation of the energy loss is given by
E loss
=
PSE in
− PSE tr
− PSE ref
PSE in
, (2)
4

where PSE in
, PSE tr
and PSE ref
are the pseudoenergies of incident, transmitted and reflected waves respectively. To reduce viscous losses of energy in the reflected and transmitted waves, the cross sections before and after the step would be taken as close to the step edge as possible. So the energy of incident, transmitted and reflected wave are calculated approximately at the distance of the incident wavelength.
The losses of energy, δ E loss
, as a function of height of lower layer after step, h
2 +
, and incident wave amplitude, a i
, are shown in Fig. 2 and Fig. 3 for the positive and negative polarities of ISW, respectively. The energy losses are presented as percentages. On the left-hand panel the energy losses versus h
2+ are shown for various values of a i
, and on the right-hand panel this dependence is done versus h
2+
normalized by the incident wave amplitude, a i
. Various regimes of ISW interaction with the step are described.
Fig. 2. The energy loss δ E for the waves of elevation versus the height of bottom layer after the step h
2+
(a) and δ E versus ratio h
2+
/| a i
| (b) for various amplitudes of incident wave. 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 when the minimum Richardson number Ri 0.25
indicate presence of supercritical flow with the composite Froude number Fr 1 .
. The filled symbols
5

Fig. 3. The relative energy loss δ E for the waves of depression versus height of bottom layer after the step h
2+
(a) for various amplitudes of incident wave and δ E versus ratio h
2+
/| a
ISW interaction with step (I-V) separated by dashed lines are shown in the right panel. The halfshaded symbols indicate interaction with the minimum Richardson number
-
|(b). The regimes of
Ri 0.25
. The filled symbols indicate presence of supercritical flow with the composite Froude number Fr 1 .
Looking at the right-hand panels for ISW of both polarities in Figs. 2 and 3 it seems that the selfsimilarity of the curves takes place to the all of positive normalized heights of upper layer after step, h
2+
/| a i
| > 0.2
for elevation ISW and for h
2+
/| a i
| > 1.5 for depression ISW. We also see evident variability for negative values. However, the relative difference between transmitted and reflected waves Δ E given by
Δ E =
PSE tr
− PSE ref
PSE in
, (3) shows self-similarity for positive and negative h
2+
/| a i
| and for ISW of both polarities (Fig. 4). For large positive values h
2+
/| a i
| the loss of energy is relatively small. The incident energy is almost entirely transmitted over the step and the energy difference, Δ E , is close to 1. With thinning of lower layer over step two concurrent processes affect energy transformation: growing reflection of waves (Fig. 4) and growing dissipation (Figs 2 and 3). The maximal value of dissipation is about 50% and it is reached at h
2+
/| a i
| ≈ 0 for elevation ISW types and at h
2+
/| a i
| ≈ 1 for ISW depression types when the wave trough “touches” the step. At this maximum the reflected wave energy is approximately equal to the transmitted wave energy as seen in Fig. 4. With further increasing of the height of the step and decreasing of h
2+
the energy of the reflected wave increases and dissipation of energy at the step decreases. In the limiting case, h
2+
= h
1
, the ISW reflects from the vertical wall, however this process can be also accompanied by mixing.
6

Fig. 4. The relative energy difference Δ E 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 character of energy losses and relationship between transmitted and reflected wave energy allows us to distinguish five regimes for ISW interaction with step: the weak interaction (I), moderate interaction (II), strong interaction (III), transitional regime (IV) between splash on the step and complete reflection from the step and reflection regime (V). Consider regimes in more detail.
The weak interaction (I) is when ISW transforms over the step without any instability; the energy losses are related mainly with the viscous dissipation and bottom friction. It corresponds to values of h
2+
/ a i
> 0.75 for elevation ISW, as well as h
2+
/| a i
| > 3.1 for depression ISW. The energy losses are less than 10%. This regime was studied by Grimshaw et al, 33 Maderich et al 35,36 for ISW of both polarities.
The amplitudes and number of reflected and transmitted waves are well predicted by the theoretical model of Grimshaw et al 34 provided that the amplitude of the incident wave is less than the known
Gardner solitary wave limiting amplitude (Maderich et al 35,36 ).
The relative energy difference
Δ
E can be estimated from the coefficients of transmission ( tr
/ i
) and reflection ( A ref
/ A i
) derived from the linear theory (Grimshaw et al 34 ):
A
A t r i
= c
−
2 c
+
+ c
+
,
A ref
A i
= c c
−
−
−
+ c c
+
+
, (4) where A i
, A tr
, A ref
are the wave amplitudes for incident, transmitted and reflected waves, respectively; c
± is the speed of a linear long interfacial waves in the deep (-) and shallow (+) parts of the computational tank c
±
= g ʹ′ h h
±
+
2 ±
. (5)
Here g
= g
0
,
Δ ρ and ρ
0
are the density jump and undisturbed density of fluid, correspondingly, g is the gravity acceleration. The relative energy difference Δ
E is then
Δ E =
A 2 tr
−
A 2 ref
A i
2
.
(6)
As seen in Fig. 5a the relation (6) describes quite well the transformation of elevation ISW over relatively deep step h
2 +
/ a i
>1, however, at the shallower step it underestimates transmitted wave
7

energy. At the same time the relations (5) describe transformation of depression ISW of small amplitude quite well even for the shallow step, but it overestimates transmitted wave energy (Fig. 5b) for waves of large amplitude. Such discrepancies can be explained by the appearance of elevation waves of finite amplitude through the step, and by instability and mixing in the depression waves.
Fig. 5. The relative difference Δ E between transmitted and reflected waves for the waves of elevation versus h
2+
(a) for waves of elevation at h
2-
/ h
1
=0.4 and (b) for wave of depression at h
2-
/ h
1
=7. Solid line is analytical estimation (6) for Δ E.
The moderate interaction (II) occurs when waves become unstable over the step. The energy losses lie here from 10% to 20%. To identify this regime the minimum of the Richardson number Ri for all runs was calculated. The Richardson number is
Ri = − g
0
∂
∂
z
/
⎛
⎜
⎝
∂
∂ u z
⎞
⎟
⎠
2
, (7) where u is the horizontal velocity. The boundary for linear stability of parallel stratified flow is
(Howard 41 ). The half-shaded symbols in Figs. 2b and 3b indicate interaction when the minimum Richardson number Ri 0.25
. The corresponding lower boundary for this regime is h
2+
/ a i
=
0.75 for elevation ISW, and h
2+
/| a i
| = 3.1 for depression ISW types. The transmitted wave becomes unstable and Kelvin-Helmholtz (KH) instability occurs just after the step. The instability was studied by Maderich et al 35 for depression waves and the KH billow parameters were discussed. The snapshots of salinity field for both polarities of incident waves with amplitude 8.8 cm passing through the step are shown in Fig. 6 for different h
2+
. The development of slow shear instability for the values
8

h
2+
/| a i
| close to the border between weak (I) and moderate (II) interaction zones is shown in Figs. 6 a
(case of h
2+
/| a i
| = 0.42), and b ( h
2+
/| a i
| = 2.7).
Formations of well developed KH instability in the mid of zone II for both types of the ISW are shown in Figs. 6 a ( h
2+
/| a i
| = 0.34) and b ( h
2+
/| a i
| = 2.3). When the parameter h
2+
/ a i
is close to the border between moderate (II) and strong (III) interaction regimes, the bolus features in the shape of transmitted wave in the elevation case (Fig. 6 a ( h
2+
/| a i
| = 0.23) and the small jet in the case of depression wave (Fig. 6 b ( h
2+
/| a i
| = 2) appears. The values of Ri in the wave crest or trough in all these runs were less of 0.1 that agree with estimations for shear induced breaking ISW (Maderich et al 36 ).
Fig. 6.
The salinity field in vicinity of the step shows the KH instability for incident elevation ISW (a) and depression ISW (b) with amplitude 8.8 cm and different h
2 +
/ a
− in the same time moments.
The strong interaction (III) of ISW with the step is the regime when the flow over the step is supercritical.
This regime is identified from condition that the composite Froude number Fr in step cross-section is more than 1 (Fig. 7). Here Fr is defined as
Fr 2
=
U
1
2 g h
1
+
U
ʹ′
2
2
2 +
, (8) where U
1 and U
2 are the layer averaged velocities in each layer. The supercritical flow causes a jet directed backward and downward for depression ISW (Fig. 8a). The jet formation and vortex mixing for depression ISW were studied by Maderich et al 36 . For incident wave of elevation the supercritical flow is directed forward. It results in the formation of vortex of dense fluid (bolus) propagating along
9

the step (Fig. 8 b). This regime corresponds to values -0.8 ≤ h
2+
/| a i
| ≤ 0.2 for elevation ISW and 0.47 ≤ h
2+
/| a i
| ≤ 2 for depression ISW types. Energy loss value in this regime exceeds 20% and tends to maximum about 50% both for waves of positive and negative polarities (Figs. 2 and 3). The relative energy difference between transmitted and reflected waves varies here from +0.6 to -0.6. The bolus intrusion into water over the step takes place when the depth of bottom layer over the step h
2+
tends to zero or a little bit less than zero for both kinds of incident solitary waves. This process for elevation
ISW is shown in Fig. 9 as snapshots of salinity field for various h
2+
in successive times.
Fig. 7. Maximal values of composite Froude number, elevation ISW types (a) and depression ISW types (b).
Fr max
, at the step versus ratio h
2+
/| a i
| for incident
Fig. 8.
Velocity vectors superimposed on the vorticity field in the vicinity of the step to show the formation of jet and vortexes in the wave of elevation (a) and in the wave of depression (b) with amplitude 8.8 cm.
The incident wave amplitude here is 4.2 cm. The run with h
2+
/| a i
| = 0.25 is intermediate between moderate (II) and strong (III) interactions of solitary wave with a bottom step. It demonstrates the transformation of incident solitary wave into a group of five secondary solitary waves of the same
10

polarity, with energy losses due to the KH instability just after the step. The number of secondary solitary waves according the asymptotic theory is 6 (Grimshaw et al 34 ), but numerical experiment gives only 5 secondary waves. The predicted amplitude of the leading secondary solitary wave is 6 cm whereas experiment result is about 5.7 cm. Nevertheless the value of the limiting Gardner soliton after step in this case is 2.9 cm, and leading wave in experiment is out of the applicability of the Gardner equation theory. The ISW after the step is well described by the Choi and Camassa 42 solution.
Next run with h
2+
/| a i
| = 0.12 (regime III) shows that the transformation process differs from the first case by the occurrence of supercritical flow ( Fr 2
≥ 1 ) at the step. The KH instability is strong and the leading secondary wave is also disturbed by KH billow. With the bolus propagation over the step two evident separate vortexes are formed with the tail behind of them and the vortex sheet.
The third run corresponds to h
2+
/| a i
| = 0. In this case bolus is smaller than in the previous run, nevertheless the leading vortex propagates for a long time and the vortex sheet after it dominates resulting in the strong mixing. The energy loss for this case tends to the maximum.
In the fourth run the step is above surface of interface, and h
2+
turns to negative values ( h
2+
/| a i
|=-0.12).
In this case flow is still supercritical (regime III), however, bolus becomes smaller, the leading vortex vanishes after some time and only the vortex sheet remains. This vortex sheet also is not so strong as in the third run and bolus is destroyed faster than in the third run.
Energy loss is decreased due to increase of amplitude of reflected wave. Processes of interaction of solitary wave of depression with the step, when the incident wave amplitude is -8.8 cm and the height of the bottom layer is changed, are presented in Fig. 10 for successive times.
Fig. 9 . The snapshots of the salinity field show the interaction with a step of elevation ISW for incident wave amplitude 4.2 cm 1) h
5) h
2+
/| a i
2+
/| a
| = -1 in successive times (a) – t i
| = 0.25; 2) h
= 0, (b) - t
2+
/| a i
| = 0.12; 3) h
= 13 c, (c) – t
2+
= 38 c.
/| a i
| = 0; 4) h
2+
/| a i
| = -0.12;
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The first run with h
2+
/| a i
| = 2 corresponds to the boundary between regimes II and III with jet formation and KH instability (see Fig. 10a), but without bolus-like phenomena. The second run with h
2+
/| a i
| = 1.35 (regime III) shows the generation of supercritical jet also and large vortexes just before the step. This case gives maximal energy loss about 50%. The secondary waves in reflected and transmitted waves are shown in Fig. 10 c. In more detail this case is considered by Maderich et al 36 .
The transitional regime (IV) is defined as ISW interaction with the step when height of step is large enough to prevent the formation of supercritical jet and supercritical bolus, however, disturbed flow occurs at the step resulting in mixing and dissipation. The reflected wave (-0.4 > Δ E > -0.8) is dominates. The lower boundary for this regime h
2+
/ a i
> -2 for elevation ISW types, and h
2+
/| a i
| > -0.75 for depression ISW types corresponds to a loss of energy about 10%. Unlike regimes I-III the selfsimilarity of the energy loss here is not achieved. Regime IV is well presented by the third run in
Fig.10 and it is the most interesting. The depth of the lower layer here is quite small, h
2+
= 1.8 cm, and h
2+
/| a i
| = 0.2. Amplitude of incident wave is -8.8 cm and it is a large amplitude. Transmitted wave of depression here is not really visible; it has so small amplitude and so large wavelength that it appears as a very slow interface inclination (see Fig. 11). As seen in Fig. 10b the incident wave almost reflects from the step ( Δ E = -0.8). This process is accompanied by KH instability and mixing. These motions near the step excite two secondary solitary-like waves of positive polarity, what are very clear in Fig.
11, which is a zoom of h
2+
/| a i
| = 0.2 case. The shape of the secondary wave in Fig. 11 is comparable with the shapes of the Gardner and Korteweg – de Vries solitary waves (Grimshaw et al 33 ). The fourth run ( h
2+
/| a i
| = 0) in Fig. 10 also refers to regime IV. The wave reflects with energy loss on mixing in result of KH instability but lower layer does not intrude on the step. The processes of interaction for waves of elevation are somewhat different. The fifth run in Fig. 9 ( h
2+
/| a i
| = -1) represents regime IV when reflection dominates ( Δ E = 0.6
), Froude number is subcritical (Fr = 0.63). However, the wave splash results in formation of small bolus propagating without KH instability.
Fig.10
. The snapshots of the salinity field show the interaction with a step of depression ISW for 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 in successive times (a) t = 0 (b) t = 10 s; (c) t = 22 s
12

Fig.11 Zoom of Fig. 10 for the third run (right panel) in various time moment and comparison of wave after step with the KdV and Gardner solitons (left panel) corresponding to snapshots in right panel
The reflection regime (V) is when the height of step is high enough to result in full reflection of the
ISW wave. The energy losses are small ( δ E less than 10-15%) for both wave types. In this regime energy losses depend on the wave amplitude: small and moderate incident waves reflect without the
KH instability. The reflection of waves of large amplitudes is accompanied by KH instability and mixing. Fig. 12 shows the reflected from vertical wall elevation ISW and depression with initial amplitude of 8.8 cm that corresponds to the parameter h
2+
/| a i
| values -3.7 and -0.7, respectively. In both cases KH instability appears and the energy losses exceed 7 %.
13

Fig.12
The salinity field snapshots show the KH billows appearance in the reflected from vertical wall elevation ISW (a) and depression (b) with initial amplitude of 8.8 cm.
We compare the results of our modeling with laboratory tank measurements of ISW energy losses in interaction of ISW in two-layer flows with the bottom obstacles of different shapes. Such investigations were done by Wessels and Hutter 43 and Chen 44 . These comparisons are presented in Fig.
13. The left panel of Fig. 13 shows the energy loss estimated by Wessels and Hutter 43 for interaction of elevation ISWs with a triangular obstacle, and right panel shows energy loss for depression ISWs interacting with obstacles of circular and triangular shape (Chen 44 ; Wessels and Hutter 43 ) and our numerical simulation results for the bottom step. The comparison shows good similarity between energy loss for the interaction of ISW of both polarities with steps and relatively steep obstacles despite the difference in geometry. The important features of the wave transformation for a strong interaction with an obstacle (Chen 44 ) are similar to the bottom step: formation of bolus for the elevation ISW and jets and vortices for the depression ISW. We conclude that the interaction of an internal wave with a bottom step describes the characteristic processes of the wave transformation for steep obstacles in general.
14

Fig.13 The energy loss δ E in the waves scattering on obstacle versus h
2+
/| a i
| for incident wave of elevation (a) and incident wave of depression (b) in comparison with laboratory experiments in twolayer flow by Wessels and Hutter 43 and Chen 44 .
The interaction of an internal solitary wave with a bottom step is studied to estimate the energy loss of an incident internal solitary wave. It is studied numerically in a computing tank in the approximation of two-layer flow within the full Navier - Stokes equations.
Five different regimes of internal solitary wave interaction were identified within the full range of ratios of height of bottom layer after the step to the incident wave amplitude : (I) weak interaction, when wave dynamics can fully described by weakly nonlinear theory (Grimshaw et al 34 ) (II) moderate interaction when wave breaking mechanism over the step is mainly shear instability; (III) strong interaction when supercritical flow in the step vicinity results in backward jet and vortices for depression waves and in a forward moving vortex (bolus) transporting dense fluid on the step; (IV) transitional regime of interaction at the step height between splash on the step and complete reflection from the step; reflection regime (V) when almost all energy transfers to the energy of reflected wave.
The mechanism of KH instability takes place for reasonable amplitude waves of both depression and elevation during interaction with the bottom step for all regimes except regime I. For this two-layer flow the energy loss due to an internal solitary wave interacting with the bottom step does not exceed
50% of the energy of the incident wave. The maximum of energy loss an elevation incident wave is reached when the ratio of the height of bottom layer after the step to incident wave amplitude, h
2+
/ a , equals zero. For an incident depression wave this ratio in maximum of energy loss is close to one.
Self-similarities of the energy loss versus the ratio of the height of upper layer after the step to incident wave amplitude take place for the values, h
2+
/ a > -0.75 for elevation ISW and for h
2+
/ a i
> 0.5 for depression ISW. It is shown that incident depression ISW in the transitional regime reflects with the formation of secondary solitary waves of opposite polarity after the step. Finally, the numerical
15

modeling of ISW interacting with a bottom step agrees well with results of laboratory experiments for internal wave transformation over steep obstacles.
This work was partially supported by the research project of KIOST (PE98743) and KISTI supercom center (K.T., V.M., I.B., K.J.) and by grant of RFBR 12-05-00472, RS and LMS (T. T). E.P. received funding from the Federal Target Program "Research and scientific-pedagogical cadres of Innovative
Russia" for 2009-2013.
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REFERENCES
1 W. Munk, and C. Wunsch, “Abyssal recipes II: energetic of tidal and wind mixing,” Deep-Sea Res . I ,
45 , 1977 (1998).
2 R, Grimshaw, E, Pelinovsky, T, Talipova and A, Kurkin, “Simulation of the transformation of internal solitary waves on oceanic shelves,” J. Phys. Oceanogr , 34 , 2774 (2004).
3 R. Grimshaw, E. Pelinovsky and T. Talipova, ”Modeling internal solitary waves in the coastal ocean”, Surv. in Geophys.
, 28, 273 (2007).
4 R. Grimshaw, E. Pelinovsky, T. Talipova and O. Kurkina, “Internal solitary waves: propagation, deformation and disintegration,” Nonlinear Processes in Geophys.
, 17 , 633 (2010).
5 K. R. Helfrich, and W. K. Melville, “Long nonlinear internal waves,” Annu. Rev. Fluid Mech ., 38 ,
395 (2006).
6 J. M. Klymak, and J. N. Moum, “Internal solitary waves of elevation advancing on a shoaling shelf”,
Geophys. Res. Lett ., 30 , 2045 (2003).
7 J. N. Moum, D. M. Farmer, W. D. Smyth, L. Armi, & S. Vagle, “Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf”. J. Phys. Oceanogr . 33 , 2093 (2003).
8 J. N.
Moum, D. M. Farmer, E. L. Shroyer, W. D. Smyth, and L. Armi, “Dissipative losses in nonlinear internal waves propagating across the continental shelf.” J. Phys. Oceanogr ., 37, 1989
(2007). .
9 D. Bourgault, D. E. Kelley, P. S. Galbraith, “Observations of a large-amplitude internal wave train and its reflection off a steep slope”, J. Phys. Oceanogr ., 41 , 586 (2011).
10 C. J. Knickerbocker, and A. C. Newell, “Shelves and the Korteweg-de Vries equation,” J . Fluid
Mech ., 98 , 803 (1980).
11 R. Grimshaw, E. Pelinovsky and T. Talipova, “Solitary wave transformation due to a change in polarity”. Stud. Appl. Math ., 101 , 357 (1998).
12 R. Grimshaw, E. Pelinovsky and T, Talipova, “Solitary wave transformation in a medium with signvariable quadratic nonlinearity and cubic nonlinearity,” Physica D , 132 , 40 (1999).
13
V. Vlasenko K. Hutter, “Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography,” J. Phys. Oceanogr . 32 , 1779 (2002).
14 D. Bourgault, M. D. Blokhina, R. Mirshak, and D. E. Kelley, “Evolution of a shoaling internal solitary wavetrain,” Geophys. Res. Lett ., 34 , L03601 (2007).
15 V. Maderich, I.Brovchenko, K. Terletska, K.Hutter, “Numerical simulations of the nonhydrostatic transformation of basin-scale internal gravity waves and wave-enhanced meromixis in lakes”. (in
Nonlinear internal waves in lakes, Hutter K. (Ed.), Springer. Series: Advances in Geophysical and
Environmental Mechanics, 2012).
16 K. R. Helfrich, and W. K. Melville, “On long nonlinear internal waves over slope-shelf topography”.
J. Fluid Mech , 167 , 285 (1986).
17 H. Michallet, & G. N. Ivey, “Experiments on mixing due to internal solitary waves breaking on uniform slopes,” J. Geophys. Res . 104 , 13467 (1999).
18 C.Y. Chen, J.R.C. Hsu, H.H. Chen, C.F. Kuo, M.H. Cheng,“Laboratory observations of internal solitary wave evolution on steep and inverse uniform slopes,” Ocean Eng . 34 , 157 (2007a)
17

19 C.Y. Chen, J.R.C. Hsu, M.H. Cheng, H.H. Chen, C.F. Kuo, ”An investigation on internal solitary waves in a two-layer fluid: propagation and reflection from steep slopes,” Ocean Eng . 34 , 171
(2007b).
20 M.-H. Cheng, J. R.-C. Hsu, Ch.-Y. Chen, Ch.-W. Chen, “Modelling the propagation of an internal solitary wave across double ridges and a shelf-slope,” Environ.
Fluid Mech . 9 , 321 (2009).
21 N. Gorodetska, V. Nikishov, K. Hutter, “Laboratory modeling on transformation of large-amplitude internal waves by topographic obstructions,” (in Nonlinear internal waves in lakes, Hutter K. (Ed.),
Springer. Series: Advances in Geophysical and Environmental Mechanics, 107 (2012)).
22 K. R. Helfrich, “Internal solitary wave breaking and run-up on a uniform slope,” J. Fluid Mech. 243 ,
133 (1992).
23 S. K. Venayagamoorthy, and O. B. Fringer, “Numerical simulations of the interaction of internal gravity waves with a shelf break,” Phys. Fluids , 18 , 076603, (2006).
24 D. Bourgault and D. E. Kelley, “On the reflectance of uniform slopes for normally incident interfacial solitary waves,” J. Phys. Oceanogr ., 37 , 1156 (2007).
25 P. Aghsaee, L. Boegman, and K G. Lamb, “Breaking of shoaling internal solitary waves,” J . Fluid
Mech . 659 , 289 (2010).
26 M. H. Orr and P. C. Mignerey, “Nonlinear internal waves in the South China Sea: Observation of the conversion of depression internal waves to elevation internal waves,” J. Geophys. Res. 108 , 3064,
(2003).
27 L. Boegman, G.N. Ivey, J. Imberger, “The degeneration of internal waves in lakes with sloping topography,” Limnol Oceanogr . 50 , 1620 (2005).
28 I. D. Lozovatskii, E. G. Morozov, and V. G. Neiman, “Energy Decay of Internal Tidal Waves
Generated near Submarine Ridges,” Doklady Earth Sciences , 375 , 1335 (2000)..
29 K. Lamb, “Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography,” Cont. Shelf Res. 27 , 1208 (2007).
30 J. N.
Moum, D. M. Farmer, E. L. Shroyer, W. D. Smyth, and L. Armi, “Dissipative losses in nonlinear internal waves propagating across the continental shelf,” J. Phys. Oceanogr ., 37, 1989
(2007).
31 J. N. Moum, J. M. Klymak, J. Nash, A. Perlin, and W. D. Smyth, “Energy transport by nonlinear internal waves,” J. Phys. Oceanogr ., 37, 1968 (2007)
32
S. M. Kelly, J. D. Nash, and E. Kunze, “Internal-tide energy over topography,” J. Geophys.Res
., 115 ,
C06014 (2010).
33 E. L. Shroyer, J. N. Moum, and J. D. Nash, “Energy transformations and dissipation of nonlinear internal waves over New Jersey’s continental shelf,” Nonlin. Processes Geophys ., 17 , 345 (2010).
34 R. Grimshaw, E. Pelinovsky, and T. Talipova, “Fission of a weakly nonlinear interfacial solitary wave at a step,” Geophys. Astrophys. Fluid Dyn.
102 , 179 (2008).
35 V.
Maderich, T. Talipova, R. Grimshaw, E. Pelinovsky, B.H. Choi, I. Brovchenko, K. Terletska and
D.C. Kim, “The transformation of an interfacial solitary wave of elevation at a bottom step,”
Nonlinear Proc. Geoph.
, 16 , 33 (2009).
36 V. Maderich, T. Talipova, R. Grimshaw, K. Terletska, I. Brovchenko, E. Pelinovsky, and B. H.
Choi, “Interaction of a large amplitude interfacial solitary wave of depression with a bottom step,”
Phys. Fluids , 22 , 076602 (2010) .
37 T. G. Shepherd, “A unified theory of available potential-energy,” Atmos.- Ocean . 31 , 1 (2006).
18

38 A. Scotti, R. Beardsley, and B. Butman, “On the integration of energy and energy fluxes of nonlinear internal waves: An example from Massachusetts Bay,” J. Fluid Mech. 561 , 103 (2006).
39 D. Kang and O. Finger, “On the calculation of available potential energy in internal wave fields,” J.
Phys. Oceanog ., 40 , 2539 (2010).
40 Y. Kanarska and V. Maderich, “A non-hydrostatic numerical model for calculating of free-surface stratified flows,” Ocean Dyn. 53 , 176 (2003).
41 L. N. Howard, “Note on a paper by John W. Miles,” J. Fluid Mech. 10 , 509 (1961
) .
42 W. Choi and R. Camassa, “Fully nonlinear internal waves in a two-fluid system,” J. Fluid Mech.
396 , 1 (1999).
43 F. Wessels, K. Hutter, “Interaction of internal waves with a topographic sill in a two-layered fluid,”
J. Phys. Oceanogr .,, 26, 5 (1996).
44 C.Y. Chen, “An experimental study of stratified mixing caused by internal solitary waves in a two layered fluid system over variable seabed topography,” Ocean Eng.
34, 1995 (2007).
19