Numerical research on internal solitary waves in stratified fluid

Internal solitary waves are a common nonlinear large amplitude internal wave in nature, which can cause strong divergence and sudden strong currents in seawater during propagation, thus having great destructive effects on marine structures. The paper focuses on numerical wave generation based on three types of internal solitary wave theories. The waves are numerical simulated by RANS method in the paper. The numerical results of internal solitary wave theories are compared and analyzed. It shows that under the same numerical water tank scale and calculation conditions, the mKdV theory can obtain the largest wave amplitude, followed by the eKdV theory and the KdV theory. In addition, background currents will accelerate the propagation speed of internal solitary waves and increase their amplitude slightly.


Introduction
The uneven heating of the sea surface by solar radiation, as well as the atmospheric dynamics and thermodynamics of different climate zones, result in significant differences in the temperature, salinity, and density of seawater.The temperature and salinity differences in seawater cause density stratification, which is also a common phenomenon in global oceans [1], [2].The sea areas in China have a wide continental slope and complex terrain, spanning three climate zones: temperate, subtropical, and tropical.The four seasons alternate significantly, and the vertical stratification of density is significant.In an ocean with stable stratification of density, internal waves are triggered when seawater currents flow through rapidly changing underwater ridges in sea [3].The northern part of the South China Sea is one of the regions with the most frequent occurrence of internal solitary waves in the global ocean.As shown in figure 1, the largest amplitude internal solitary wave in the world was detected in this sea area.The extreme ISW captured on 4 December 2013 with a maximum amplitude of 240m and a peak westward current velocity of 2.55 m/s [4].Due to the fact that internal solitary waves are not as turbulent and surging as waves on the sea surface, they are hidden within the ocean, often making it difficult for people to prevent them.The wave height of internal solitary waves is generally much higher than that of surface waves on the sea surface, and can even reach several hundred meters.The characteristic liquid length can also reach several hundred to several thousand meters.Even if the sea surface is very calm, it is inevitable that there are large internal waves below the surface.The internal solitary waves we often refer to belong to tidal internal waves, which contain enormous energy.Internal solitary waves have characteristics such as large amplitude (up to hundreds of meters) and strong shear currents, and can propagate over long distances of hundreds of kilometres.They are typical medium and small scale dynamic processes in the ocean.Internal solitary waves cause strong isodensity surface fluctuations, leading to reciprocating horizontal flows with strong vertical shear.So internal solitary waves can cause serious damage to offshore engineering facilities, especially oil and gas platforms [5]- [8].
People's interest in studying solitary waves in the ocean began approximately between the 1960s and 1970s.The significant development of marine instruments and equipment during this period, as well as research achievements in applied mathematics and remote sensing science, led to a series of advancements in the observation and theory of solitary waves in the ocean.The theory of internal solitary waves includes models such as KdV, eKdV, and mKdV.Theoretical research has been conducted internationally as early as the late 19th century, and the most classic and earliest internal solitary wave model is the Korteweg de Vries (KdV equation) mathematical model, which is a theoretical model of internal solitary waves derived based on the shallow water small amplitude assumption [9].In subsequent research, to broaden the applicability of KV theory, scholars introduced a cubic nonlinear term into the KdV equation based on the dispersion and nonlinear conditions of the marine environment, and obtained the mKdV equation.Numerical wave generation research is the foundation for conducting research on the influence of internal solitary waves.
The paper focuses on the numerical wave generation research of various internal solitary wave theories.The results and analysis are presented in the main text.

MCC Theory
Definition ε=a/h and μ=h/L, which are referred to as the nonlinear and dispersive parameters of internal solitary waves respectively.The applicable condition of MCC theory is μ<<1, which requires the internal solitary wave to be weakly dispersive, but there is no restrictive requirement for nonlinear parameters.Therefore, this is a type of weakly dispersive and strongly nonlinear theory regarding internal standing waves in two-layer fluids.
Assuming ζ is the displacement of the internal solitary wave interface, a is the amplitude, and c is the wave velocity, the MCC theoretical solution of the stationary internal solitary wave in two layers of fluid is: (1)

KdV theoretical model
When ε＜＜1，μ＜＜1，ε=O(μ), i.e. the internal solitary wave is weakly dispersive, weakly nonlinear, and the two are balanced, the MCC theory will degenerate into the well-known KdV theory as follows: (2) The theoretical solution of KdV is:

mKdV theory
The critical value of the thickness ratio between the upper and lower layers of the fluid is： Under the conditions of weak non rotational and weak dispersion, when the thickness ratio of the upper and lower fluids approaches the critical value, the term c1 in KdV theory is 0. At this point, the KdV theoretical model is no longer applicable.Therefore, the mKdV theoretical model is developed, and the theoretical solution is:

eKdV theory
By adding a cubic nonlinear term to the KdV equation, an equation suitable for moderate nonlinearity and weak dispersion conditions can be obtained, namely eKdV: ( ) The theoretical solution of the eKdV equation is: cosh

Numerical simulation results
Water is assumed to be incompressible and isothermal Newtonian fluids.Flows are three dimensional, turbulent.The continuity equation and RANS equations can be written in Cartesian tensor notation as follows: Where __ i u is the mean velocity component, ' i u is the fluctuating velocity component (i=l, 2, 3).For simulation of viscous flow, the equations ( 8) and ( 9) are solved.
In this paper, we adopt Volume of Fluid (VOF) method in simulating free surface flow.VOF algorithm was proposed by B. D. Nichols and C. W. Hirt in 1973 and designed to capture discontinuous layer [10].It introduces a volume fraction of fluid as an additional dependent variable.In the method, αi denotes the ith fluid volume fraction in the cell.It is used to determine the position and shape of interface.For two-phase flow in the paper, the continuity equation includes both phases (lighter water and heavier water).Mass for each phase will be conservative.The tracking of the interface between the two phases is accomplished by the solution of a continuity equation for the volume fraction.For the heavier water, this equation has the following form: Control volume will be filled with heavier and lighter water.The volume fraction will be summed to one.Written as: Where subscript h denotes the heavier water and l denotes the lighter water.The main parameters of the water tank used in the numerical calculation are shown in Table 1, with a length of L=15000 and a height of H=1600 meters.The depth of the upper layer is h1=400m, and the depth of the lower layer fluid is h2=1200m.The density ratio of the upper and lower layers of fluid is 0.986, and the kinematic viscosity coefficients of both the upper and lower layers of fluid are 1.003 × 10 -6 m 2 /s.Table 1 shows the main parameters of the numerical water tank.Figure 2 shows the wave curves with the theoretical numerical wave generation of KdV, eKdV, and mKdV.The background current velocity is 0. Obviously, there is no phase difference in the wave curves under the three theories, and the difference in wave length is relatively small.The main difference is reflected in the wave amplitude.The mKV theory has the largest amplitude, followed by the eKdV theory, and the KdV theory has the smallest.
Figure 3 shows the situation where the background current velocity is 0.1m/s.It is similar to Figure 4.The difference is mainly reflected in the wave amplitude, and the mKdV theory has the largest wave amplitude, followed by the eKdV theory, and the KdV theory has the smallest.The amplitude statistics of numerical wave generation under different internal solitary wave theories and ocean currents are shown in Table 2.It can be observed that the current increases the amplitude of the wave under the same theory.As shown in figures 4~6, there is a phase difference in the liquid shape curve with and without ocean currents under the same theoretical model, which means that the internal isolated liquid propagates faster under the influence of currents.This phenomenon should be caused by the faster propagation of internal standing waves and lower computational dissipation under the current influence.

Conclusion
In the paper, the phenomenon of internal waves in the ocean are introduced as well as the theory and development history of internal solitary waves.Then the internal waves are simulated based on the RANS method with three internal solitary wave theories.The three theories are KdV, eKdV and mKdV.Finally, the numerical results of internal solitary wave theories are compared and analyzed.It indicates that under the same numerical water tank and calculation conditions, the mKdV theory can obtain the largest wave amplitude, followed by the eKdV theory, and the KdV theory is the smallest.In addition, background currents will accelerate the propagation speed of internal solitary waves and increase their amplitude slightly.

Figure 1 .
Figure 1.The extreme ISW event captured at M10 on 4 December 2013 [4].(a) Shadings indicate the temperatures ( ℃ ) measured by the thermistor chains, and the lines represent the isothermal displacements at every 100 m between 200 and 1000 m.(b) Shadings indicate the zonal velocity anomalies (m/s) measured by the upward-looking ADCP, and the white line shows the shape of velocity anomalies with magnitude exceeding 2m/s.(c) Shadings indicate vertical velocity (m/s) measured by the upward and downward-looking ADCPs.

Figure 4 .Figure 5 .Figure 6 .
Figure 4. Comparison of waveform curves of the KdV theoretical model

Table 1 .
Main parameters of the numerical water tank.

Table 2 .
Comparison of Wave Amplitudes