NUMERICAL STUDIES OF INTERNAL SOLITARY WAVE GENERATION AND EVOLUTION BY GRAVITY COLLAPSE*

LIN Zhen-hua, SONG Jin-bao

Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China

Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Sciences, Qingdao 266071, China, E-mail: linzhenhuaouc@gmail.com

(Received December 22, 2011, Revised April 21, 2012)

NUMERICAL STUDIES OF INTERNAL SOLITARY WAVE GENERATION AND EVOLUTION BY GRAVITY COLLAPSE*

LIN Zhen-hua, SONG Jin-bao

Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China

Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Sciences, Qingdao 266071, China, E-mail: linzhenhuaouc@gmail.com

(Received December 22, 2011, Revised April 21, 2012)

In this study, an analysis on the internal wave generation via the gravity collapse mechanism is carried out based on the theoretical formulation and the numerical simulation. With the linear theoretical model, a rectangle shape wave is generated and propagates back and forth in the domain, while a two-dimensional non-hydrostatic numerical model could reproduce all the observed phenomena in the laboratory experiments conducted by Chen et al. (2007), and the related process realistically. The model results further provide more quantitative information in the whole domain, thus allowing an in depth understanding of the corresponding internal solitary wave generation and propagation. It is shown that the initial type of the internal wave is determined by the relative height between the perturbation and the environmental density interface, while the final wave type is related to the relative height of the upper and lower layers of the environmental fluid. The shape of the internal wave generated is consistent with that predicted by the KdV and EKdV theories if its amplitude is small, as the amplitude becomes larger, the performance of the EKdV becomes better after the wave adjusts itself to the ambient stratification and reaches an equilibrium state between the nonlinear and dispersion effects. The evolution of the mechanical energy is also analyzed.

gravity collapse, internal wave, energy analysis

Introduction

The internal waves are the gravity waves that occur within, rather than on the surface of, a stratified fluid. The density differences within the fluid are usually much smaller than the density of the fluid itself, thus the internal waves typically have much lower frequencies and larger amplitudes than their surface counterparts. They are ubiquitous in the ocean and the atmosphere, and may induce oscillations of the density interface, to alter the magnitude and the direction of acoustic transmissions; may also bring nutrients up from the deeper ocean into the euphotic layer and contributes to the primary production. The internal wave breaking is a major driver in the local mixing process.

The stratified fluid and the perturbation are nece-ssary to excite the Internal Solitary Waves (ISW). One common generation mechanism is the interaction of the tidal flow and the irregular bottom topography of the ocean floor[1-4]. Other generation mechanisms were also explored, for example, through river plumes[5], and the nonlinear steepening from an initial tilted density interface[6]. The ISW is of a depression type if the thickness of the lower layer is larger than that of the upper layer, after it passes the turning point where the thickness of the deeper layer is equal to that of the upper layer, the internal wave of depression gradually turns to one of an elevation type, finally, the ISW breaking occurs at coastal boundaries and contributes to the local mixing.

Various methods, e.g., the theoretical formulation[7], the in situ observation[8], the satellite image[9], the numerical model[10]and the laboratory experiment[11-15], were applied to study the ISW generation, propagation, interaction and breaking processes. Since it is often difficult to thoroughly investigate the characteristics of ISWs in field observations, and certain assumptions such as weak nonlinearity are commonly used in the theoretical formulation, some alternativeapproaches combined with laboratory tests to study the ISWs were developed[16,17]. Moreover, the laboratory tests are superior to other methods because of low cost, and good repeatability and reliability of results. However, unlike in oceanic scenarios, where the ISWs are mostly generated by stratified flows over an irregular topography, the gravity collapse is one of the widely used methods in the laboratory experiments to excite the ISWs. This method is simpler than others, such as those by the oscillating boundary or the movable body, and, therefore, is commonly used in the laboratory. Although the ISWs propagation and transformation properties were much studied via this generation mechanism[16], the ISWs generation and propagation by gravity collapse remains an interesting topics of research.

In the laboratory experiments conducted by Chen et al.[16], the ISWs are generated by gravity collapse in a two-layer stratified fluid system in a flume. The laboratory settings can be found in detail in their paper. The main conclusions are: (1) if the upper layer is thinner than the lower layer (i.e., the environmental condition of H1＜H2), the ISWs of depression type may be generated, on the other hand, the ISWs of elevation type are generated when H1＞H2, (2) the ISW in the wave flume maintains its soliton feature if the environment condition permits, otherwise, a train of wave oscillations occurs following the leading wave.

The conclusions of Chen et al.[16]are very instructive for later laboratory experiments on the ISW generation via gravity collapse mechanism, however, there are two apparent shortcomings inherent in their studies. Firstly, the density interface displacement at the generation site is determined via video snapshots using a patch of dye transfused at the interface. As the motion starts, it would not be easy to identify the interface because the dye is very diffusive and discontinuous. Secondly, there is no velocity field data measured in the experiment, which makes it impossible to discern the energy evolution characteristics of this process. The results presented in Chen et al.[16]are valuable from a qualitative point of view, but further studies are necessary to understand this ISW generation and propagation mechanism quantitatively. In this study, a high-resolution two-dimensional non-hydrostatic numerical model is used to fill this gap, which is based on the Reynolds Averaged Navier-Stokes equations. It should be stressed that the numerical method used in this study is different from that based on the theoretical simplified model[17-19], which generally contains certain assumptions, such as the inviscid fluid, weakly nonlinear and weakly dispersive etc., thus would provide more robust results.

The goal of the present paper is to better understand the ISW generation, propagation and reflection by the gravity collapse mechanism. This paper is arranged as follows. In Section 1 the mathematical problem is formulated. The model used is introduced in Section 2 and the numerical results are analyzed in Section 3. Finally conclusions are drawn in Section 4. The current study may provide some food of thought for future laboratory experiments on how to arrange the fluid with different densities to get the desired ISW type.

Fig.1 Schematic representation of the initial condition of the gravity collapse scenario with the main region in the left and the perturbation region in the right. The dashed line marks the initial interface distribution, and the dashdotted line marks the final equilibrium interface position

1. Mathematical formulation

In the present study, the effects of the earth’s rotation is neglected and the system is subject to only a single forcing event. The schematic representation of the system is shown in Fig.1. The flow is driven by the baroclinic pressure gradient that results from a density step. The interface displacement for a twolayer stratified fluid is governed by the linear non-dispersive wave equation under the rigid lid approximation and the Boussinesq approximation[6],

where the horizontal coordinate axis x is fixed along the bottom boundary (positive rightward), the vertical axis z is positive upward, t is the time, (,)xtη is the vertical displacement of the density interface,is the linear long-wave speed,is the reduced gravity, and1H and2H are the depths of the upper and the lower layers, respectively.

For the two superposed incompressible fluids with different densities, the no-flux boundary conditions apply on two ends, thus it is required that the horizontal velocity vanishes and the fluid interface remains perpendicular to the boundary[6],

The initial conditions consist of a piecewise distribution of the interface displacement and no initial motion,

where1f and2f are constants representing the interface heights at different positions,0x is the position of the gate. The general solution for the interface displacement is found in the following form,

wherenk are the coefficients in the Fourier cosine series,

The general analytical result shown above involves the infinite summation of trigonometrical functions, and certain procedures are required to simplify it to facilitate a direct understanding of what the solution represents. For simplicity, in the following we assume L/2＜x0＜L, then Eq.(4) can be simplified as

The detail of the mathematical transformation is given in appendix. It should be noted that the above solution is only valid for 0＜t＜(L-x0)/c0. Similar tactics can be used to derive the alternative forms of Eq.(4) within other parts of the period 0＜t＜2L/c0. The solutions for one period are summarized as follows

Now the solution (4) is reformulated into a group of piecewise spatial functions in different time ranges, allowing a direct and accurate identification of what kinds of motions they represent. Without loss of generality, we further assume that f1＞f2, Figs.2(a)-2(h) shows the temporal evolution of the density interface displacement for a whole period. From the figure it can be seen that from the initial condition 2, two opposite directional interface motions with the same amplitude are generated as shown in Fig.2(b), in Fig.2(c) it is shown that, after the right limb reaches the right boundary, it is reflected and then moves leftward, a rectangle type wave is formed with a constant propagation speed0c, as inferred from Eqs.(7), the amplitude of the wave is equal to a half of the initial density interface depth difference across the step position0x. When the left limb reaches the left boundary, it is reflected and moves rightward as shown in Fig.2(d). Finally, the two branches meet at L-x0and then the inversion process continues. It is worth noting that from the initial state, as shown in Fig.2(a), to the steady wave propagation state, as shown in Fig.2(c), a half of the available potential energy is lost, which means that for this theoretical model, the kinetic energy of the system is equal to the Available Potential Energy (APE) when the wave is formed.

Fig.2 Temporal evolution of the density interface obtained by the analytical solution

The above solution shows the case for L/2＜x0＜L, a similar method can be used to obtain the solution for 0＜x0＜L/2. It can also readily be inferred that for x0=L/2, a standing wave solution is obtained, however, in the current work this scenario is not further analyzed, since the propagation waves are desired in almost all laboratory experiments.

2. The numerical model

In this section, a two-dimensional non-hydrostatic model is constructed to simulate the ISWs generated by the gravity collapse. Our main objective here is to use the laboratory observations made by Chen et al.[16]to validate our numerical results, and the model results in turn could provide more reliable information,such as the temporal and spatial distributions of the interface displacements, the horizontal and vertical velocity fields across the whole domain, for a better understanding of the ISW generation, propagation and reflection processes.

The model amounts to solving the two-dimensional Reynolds Averaged Navier-Stokes (RANS) equations for an incompressible Newtonian fluid with the Boussinesq approximation,

the tracer transport equation and the equation of state,

where (,)xz are the horizontal and vertical coordinates and (,)uw are the associated velocity vector, p and ρ are the pressure and density fields, respectively,0ρ is the reference density,tν is the turbulent viscosity coefficient, S is the salinity andsσ is the turbulent Schmidt number, g is the gravitational acceleration in the vertical direction. The standard kε- turbulence model is used to describe the sub-grid process.

Table 1 Initial layer depths for different numerical experiments

Fig.3 The temporal evolution of the interface displacement for Case 1

Fig.4 The temporal evolution of the interface displacement in Case 2

Fig.5 The temporal evolution of the interface displacement for Case 3

Fig.6 ISW generation processes for Case 1 from 0 s to 7 s, the time interval is 1 s

The model settings in our numerical simulations are shown in Fig.1, similar to those in laboratory experiments performed by Chen et al.[16], but with some differences based on numerical considerations. The steel-framed wave flume used by Chen et al.[16]is 12 m long, with a cross section of 0.7 m high by 0.5 m wide. The flume has a slope to facilitate the study of its interaction with the internal waves, which is beyond the scope of the present study, and the total water depth is 0.4 m for all their laboratory experiments, thus our numerical model domain is set to be 6 m long and 0.4 m high. The salinity of the upper layer is 0 psu (fresh water) and 39 psu (brine water) for the lower layer, the corresponding densities are about 999 kg/m3and 1 029 kg/m3for the upper and lower layers, respectively, producing a stratified fluid environment of the Boussinesq parameter (the ratio of density difference and reference density) of 0.03, a typical value in the ocean environment[16]. For all our model studies,0x is set to be 5.7 m, the width of the right minor water compartment (perturbation region) is 0.3 m, the same value as in the laboratory experiment configuration. Three numerical experiments are performed, as shown in Table 1, initialized with different interface displacements. The differential equations for the momentum and salinity are discretized and implemented with the finite volume method within Open Field Operation and Manipulation (OpenFOAM) framework, the initial velocity field is zero in the domain, no flux and no slip conditions are applied along all the boundaries, thus the initial static state is driven by the initial baroclinic pressure gradient force caused by the inhomogeneous horizontal salinity distribution. All numerical experiments are integrated for 50 s, though our concern here is mainly on the internal wave generation, propagation and transformation processes, some reflection features are also briefly analyzed, as shown in the next section.

3. The analysis of results

3.1 General model results

Fig.7 ISW generation processes for Case 2 from 0 s to 7 s, the time interval is 1 s

The temporal evolution of the density interfaces (defined as the isopycnals of the averaged density of the upper and lower layers) in the whole domain are shown in Fig.3 (Case 1), Fig.4 (Case 2) and Fig.5 (Case 3), respectively. From Figs.3 and 4 it can be seen that the general internal wave generation, the propagation and reflection characteristics of Case 1 and Case 2 are similar. At first, the ISWs of depression are generated at the right side of the domain, with a leading ISW and small scale trailing interface oscillations, the ISW propagates to the left and reflects at the left boundary. It can be seen that the ISW propagation speed is well predicted by the linear theoretical model in the current scenarios. However, a distinct result shows in Fig.5 for Case 3, where it can be seen that firstly a wave of elevation is generated, as it propagates to the left, the amplitude of the leading elevation wave diminishes gradually and a train of depression ISWs emerge behind, the wave propagation speed is reduced as compared with that predicted by the linear model. The above numerical observations conform to the findings in the laboratory experiments of Chen et al.[16]. It may be more understandable if their conclusions are rephrased as follows: the initial type of the ISW is determined by the relative height of the perturbation density interface and the environmental density interface (a wave of depression is generated if the position of the perturbation density interface is lower than that of the environmental density interface, otherwise, a wave of elevation is generated.), while the final type of wave is determined by the position of the environmental density interface, namely, the relative thickness of the upper and lower layers (if the thickness of the upper layer is smaller than that of the lower layer, the wave of depression emerges finally, otherwise, the wave of elevation type is generated).

3.2 Internal wave generation

Fig.8 ISW generation processes for Case 3 from 0 s to 7 s, the time interval is 1 s

Chen et al.[16]show snapshots of video images to exemplify the internal wave generation via the gravity collapse, the density interface is indicated by the dye injected beforehand. This method is commonly used in laboratory experiments to visualize the density interface evolution, however, as can be discerned from Chen et al.’s demonstration, two issues of deficiencies are evident, firstly, the dye injection is not continuous along the density interface, e.g., at the step location, which renders it impossible to know the temporal evolution of the interface; and secondly, the dye is extremely diffusive, thus it is hard for us to discern the interface after the initialization. These two issues can be easily addressed by the high-resolution numerical results, as shown in Fig.6 (Case 1), Fig.7 (Case 2) and Fig.8 (Case 3). In these three figures, the temporal evolutions of the density interface during the first seven seconds are presented to see the difference of the ISW generation processes, and the temporal interface evolutions predicted by the theoretical model are also presented for comparison. From these figures it can be seen that the ISW generation mechanisms are similar among cases, which resemble the lock-exchange phenomenon, while some distinctions are also evident. For example, the vortex structures shown in Figs.6(d) and 6(f) are obvious, while these phenomena are less evident at the corresponding time in Fig.7 and even less in Fig.8. These vortex structures might be caused by the Kelvin-Helmholtz instability at the stratified fluid interface. For all our numerical simulations, the stratifications are identical, the differences among these cases lie in the relative height between the environmental density interface and the perturbation density interface. The larger the relative height (as Case 1), the larger the shear across the density interface, the more prone to form the vortex structures it will be. It should be stressed that the accurate description of the ISW generation process via this mechanism is beyond the capability of any theoretical two-layer stratified fluid model, because the temporal evolution of the density interface, is not a function of the horizontal coordinate, as shown in Figs.6(d), 6(f), 7(c) and 7(d).

It can be seen from Figs.6 and 7 that, although the theoretical model fails to predict the vortex structures during the initialization processes because of its inherent linear feature, the final wave position and the polarity feature are generally in good agreement with numerical results, as shown in Figs.6(h) and 7(h). This is, however, not true for Case 3, as shown in Fig.8(h), where the wave generated is of elevation type, while the environmental stratification favors the wave type of depression, which slows the corresponding propagation speed.

3.3 Temporal evolution of interface

The temporal evolution of the density interface at the middle of the computational domain is shown in Fig.9, where one sees the perfect ISW of depression and the accompanied trailing oscillations for Cases 1 and 2, and an elevation-type internal wave followed by the trailing internal wave of depression type for Case 3, in keeping with the recorded wave profiles in Chen et al.’s laboratory experiments qualitatively, which further validates our numerical results.

3.4 The shape of ISW

The propagation of ISWs can be described to the first order of magnitude in the amplitude via the weakly nonlinear Korteweg-de Vries (KdV) equation[20],

A particular solution for Eqs.(13) is the solitary wave of the following form,

where the phase speed c and the length scale are given by

The classic KdV equation is derived under a strong restriction of weak nonlinearity and is likely not to give right results in certain circumstances when the solitons are extremely nonlinear[21]. The Extended Korteweg-de Vries (EKdV) equation accounts for a higher degree of nonlinearity,

A solitary wave solution of Eqs.(16) is of the following form,

where ν is a nonlinearity parameter that falls in the range from 0 to 1.

The numerical results provide an opportunity for us to evaluate the performance of the KdV and EKdV equations in predicting the shape of the ISWs generated. Here the wave profiles at instants 15 s and 30 s are chosen for the first two cases. The reason to choose these two specific instants is that they are not affected by the ISW generation and reflection processes, and could be representative as the “young” and“mature” ISWs for their respective case scenario. The comparisons are shown in Fig.10.

From the figure, one can see some common features among the two cases, e.g., the shape of the wave obtained by the KdV theory is steeper than that obtained by the EKdV theory for all comparisons. Distinct differences are also evident. We will make the comparisons for Case 2 first, as shown in Figs.10(b) and 10(d). Though the ISW amplitude decreases a little with the time, the differences from the model results, obtained by the KdV and EKdV theories, are very small. This can be explained by the fact that the initial forcing for Case 2 is weak, thus produces small amplitude ISWs. In that case, the effect of nonlinearity does not dominate, thus the wave shapes inferred from both KdV and EKdV equations conform to that produced from the numerical output. However, this does not hold for Case 1, as shown in Figs.10(a) and 10(c). In Fig.10(a), none of the KdV and EKdV theories predicts the ISW shape well, while for a later instant, the wave shape obtained by the EKdV theory agrees well with that obtained by the model, as shown in Fig.10(c). The reason for the distinct difference of the ISW shape obtained by the KdV and EKdV theories is that the amplitude is larger in Case 1 than that in Case 2, the nonlinearity effect is more important in such circumstance. Since both the classic KdV and EKdV theories are valid for steady state ISWs, when the effect of nonlinearity is equal to the effect of dispersion. Theinconsistency between the model output and the theoretical predictions in Fig.10(a) might be due to the fact that the initially generated ISW still adjusts itself to the environmental stratification, finally the wave shape predicted by the EKdV theory agrees better with the numerical result in Case 1 as the amplitude is larger. As a short conclusion, the applicability of the EKdV theory is wider than the KdV theory from the results shown in this study, as it incorporates a higher order nonlinear term.

Fig.10 Comparison of model predicted wave shape and those predicted by the KdV and EKdV models

3.5 Energy analysis

One undisputable merit of the numerical simulation is that it could provide quantitative results for all quantities, such as the velocity, the temperature, the salinity, the turbulent kinetic energy and its dissipate rate in the whole computational domain, which are inaccessible for laboratory measurements as a whole. The availability of these quantitative data helps us better understand the related phenomena in depth, e.g., the energy evolution process, which will be discussed in this subsection.

Fig.11 The temporal evolutions of total kinetic, potential energy and total mechanical energy (E) in Case 1, normalized by its initial available potential energy

Fig.12 The temporal evolutions of the kinetic energy in Cases 1, 2 and 3, normalized by their initial available potential energy, respectively

The temporal evolutions of the kinetic energy and the potential energy as well as the total mechanical energy of the system in Case 1 are shown in Fig.11. The overall tendency shown in this figure indicates that this is mainly a laminar process, where the turbulent mixing is not important and the total mechanical energy is approximately conserved. From the initial static state, the potential energy drops sharply and is converted to the kinetic energy. As the ISW forms and propagates leftward from the right boundary, the amplitude of the ISW decreases slightly, accompanied with a conversion from the kinetic energy to the potential energy. When the internal wave reaches the left boundary, the kinetic energy diminishes swiftly and is converted into the potential energy, an extrema of theinterface displacement is reached. The kinetic energy of the system increases again when the reflection process initiates.

Fig.13 The spatial distribution of kinetic energy in Case 1 (upper panel), Case 2 (middle panel) and Case 3 (lower panel). The bin size along the x axis is 0.01 m

The temporal evolution of the kinetic energy in different cases are shown in Fig.12, with the general trends of development being similar, as analyzed above, however, three distinct features are evident. Firstly, the kinetic energy booms up to more than 50% of its initial APE during the generation phase, and then drops a little to around 50%, which could probably attribute to the inertial effect during the gravity collapse phase and the later self adjustment. Secondly, the system’s kinetic energy is approximately one half of its initial available potential energy during the ISW propagation phase, accompanied with a slight conversion from the kinetic energy to the potential energy as the ISW adjusts to the environmental stratification. Thirdly, it is worth noting that the occurrence of the maximum total kinetic energy is not at the ISW generation phase, but at the ISW reflection stage.

The spatial distributions of the kinetic energy in the three cases at instants 10 s, 20 s and 30 s are presented in Fig.13, from which it can be seen that in Cases 1 and 2 the kinetic energy packets are approximately conserved and propagate with the ISW speed, though their spatial distributions flatten slightly with time. This indicates that the kinetic energy of the system comes mainly from the ISW induced currents. The kinetic energies contained in the ISWs in Case 1 (Case 2), defined as the integration of the kinetic energy from the left end to the extrema location just behind leading wave, are approximately 95.09%, 90.49%, 89.13% (94.98%, 90.31%, 88.31%) of their total kinetic energies at instants 10s, 20s, and 30s, respectively. Though the temporal evolution of the total kinetic energy in case 3 resembles that in Cases 1 and 2, as shown in Fig.12, its spatial distribution diverges dramatically as shown in the lower panel of Fig.13. The kinetic energy distribution collapses spatially as the time elapses.

4. Conclusions

In this paper, systematic theoretical and numerical studies are carried out for the ISW generation and propagation characteristics by the gravity collapse mechanism. The numerical results could reproduce all the phenomena observed in laboratory experiments performed by Chen et al.[16]and the numerical model is used to overcome the deficiencies inherent in their laboratory experiments, permitting a quantitative understanding of this ISW generation mechanism and evolution features.

The theoretical model indicates that a rectangletype wave is generated, which propagates back and forth in the domain. The numerical results produce more realistic outputs, from which it can be seen that the initial ISW type is determined by the relative height of the perturbation density interface and the environmental density interface, while the final ISW type depends on the relative height of the upper layer and the lower layer of the environmental fluid. One notable phenomenon is that the propagation speed of the ISW inferred from the theoretical model conforms to that of the numerical results considerably well if no ISW polarity conversion process occurs. The difference of the wave shapes predicted by the KdV and EKdV theories is obvious when the amplitude of the ISW is relatively large, the prediction based on the EKdV theory is more appropriate under this circumstance as it incorporates a higher-order term of nonlinearity. The result shows that the total mechanical energy is generally conserved, a rapid conversion from the potential energy to the kinetic energy occurs during the ISW generation phase. As the ISW forms and propagates away from its source region, the total kinetic energy is approximately 50% of its initial APE, though a slight conversion to the potential energy is accompanied. The spatial distributions of the kinetic energy tend to flatten slowly as the ISW adjusts itself to the environmental stratification if no polarity conversion occurs, on the other hand, the spatial distribution of the kinetic energy collapses very quickly and covers a larger area for the polarity conversion case.

The present study demonstrates that the laboratory scale numerical model is an indispensable tool that can further extend our understanding of the interested process.

Acknowledgement

This work was supported by the State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences (Grant No. LTO1104).

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Appendix

Transformation of (,)xtη

where

10.1016/S1001-6058(11)60276-X

* Project supported by the National Natural Science Foundation of China (Grant Nos. 61072145, 41176016), the Fund for Creative Research Groups by National Natural Science Foundation of China (Grant No. 41121064).

Biography: LIN Zhen-hua (1981-), Male, Ph. D.