Experimental Investigation on a convex Mode-2 internal solitary wave and its induced flow field

The second mode internal solitary wave (Mode-2 ISW) and its wave-induced flow field structure are investigated by conducted physical experiments in a large stratified fluid flume. The Mode-2 ISW is generated by the method of the gravity collapse of fluid, and the measurements of its wave-flow structure are obtained by the combined method of conductivity probe array and particle image velocimetry (PIV). The flow field structure and its influencing factors were analysed quantitatively. The results reveal the characteristic of the Mode-2 ISW waveform and density field and the effect of the wave amplitude on the flow and shear characteristics of induced flow field.


Introduction
Internal solitary waves (ISWs) are widely distributed in the stratified ocean with the form of multi-mode [1][2][3], in which more than 90% of the kinetic energy is contained in the first and second modes [4].The Mode-2 ISWs are usually in the shape of "double-humped" packets with convex upper waveform and concave lower waveform observed [5][6][7], as shown in Figure 1.At present, there is not enough knowledge about this kind of distinctive ISW with complex wave-flow structure.

Figure 1.
The observed convex Mode-2 ISW with wave amplitude of 30m [8] With the enrichment of the internal wave data obtained from the field observation and satellite remote sensing observation, more and more attention has been paid to the study of the multiple-mode ISW including the generation and dissipation.Yang et al first reported that Marine measured data of multiple solitary waves within the second baroclinic mode on the continental slope of the northern South China Sea (SCS) [9].Liang studied the generation mechanism of the field data of the mode -2 ISWs reported by Yang et al., the results shows that the Mode-2I SW is generated by the half-day internal tide (IT) [10].Magalhaes et al observed five continuous and regularly spaced envelope of Mode-2 ISWs simultaneously in Andaman Sea by using remote sensing technology, results show that Mode-2 ISW may have been overlooked due to their small size in previous remote sensing observations.In fact, the presence of Mode-2 ISW is more extensive than before studies [11].Belogortsev et al. numerically simulated the propagation characteristics of the mode-2 convex and concave solitary waves on the inclined bottom boundary, and proved the possibility of their mode transition from the mode-1 to mode-2 during the propagating process from the deep sea to the shelf [12].
Due to the complex interaction of the actual ocean phenomena and the uncertain excitation mechanism of the large amplitude second-order mode internal solitary wave, it is very difficult to obtain the all-essential characteristics of its generation and evolution by in-situ observation method.The understanding of the characteristics of ISW within the second baroclinic mode is still very limited.The development of modern stratified fluid experiment and measurement technology provides an effective way to understand related problems.
In view of the above, the convex Mode-2 ISW generated by the gravity collapse method are studied and analyzed in a large stratified flume, and the wave-flow field structure of the convex Mode-2 ISW is measured by PIV, the evolution characteristics of the Mode-2 ISW and its wave-induced flow field can be obtained.The paper structure is arranged as follows: Section 1 is Introduction, Section 2 is Experimental Method, Section 3 is Results and Analysis, and Section 4 is Conclusion.

Experimental Method
The experiments of Mode-2 ISW and its induced flow field are simulated in a large-scale stratified fluid flume with main scale of 1200cm(length)×120cm(width)×100cm(height).As shown in Figure 2, the Cartesian coordinate system is established.The density stratified fluid environment was prepared by using the method of "double-tank" [13], and the preparation process was summarized as follows: firstly, the fluid with thickness of 40cm and density of 998 kg/m 3 was injected into the flume to prepare the upper layer, and the fluid with thickness of 40cm and density of 1029 kg/m 3 is then injected into the flume to prepare the lower layer.The pycnocline with approximately linear density change is formed due to the mixing near the interface between salt water and fresh water.The density distribution of the fluid environment is obtained by a conductive probe ( ) , as shown in Figure 3. ,which can record the full waveform of ISW and its initial amplitude [14].Define the density isopycnals of the maximum convex displacement and concave displacement of the Mode-2 ISW packet as the characteristic convex and concave waveforms, respectively, and their corresponding maximum displacements are the upper convex wave amplitude 1 a and the lower concave wave amplitude 2 a .The two-dimensional particle image velocimetry (PIV) system can be used to measure the velocity vector distribution.
Dimensionless parameters are defined as following.Dimensionless environment parameter 13 / hh , dimensionless density difference parameter , the density isopycnals of the Mode-2 ISW is measured by using a conductivity probe array.Figure .4shows the wave-induced density field and characteristic waveform of the heading Mode-2 ISW.The waveform is characterized by obvious "double peak" soliton shape with a convex upper interface and a concave lower interface.The scale of Mode-2 ISW along the horizontal direction is greater than that of the one along the vertical direction.The propagating of Mode-2 ISW causes obvious density disturbance near the ISW envelope, while no significant density disturbance can be observed at the center of the pycnocline.There is a strong induced horizontal flow along the propagating Mode-2 ISW direction near the depth of density pycnocline center.The induced vertical flow is characterized by the convex wave envelope accompanied by the wave-front upwelling and the wave-backward sinking flow, while the flow field near the downward concave waveform is the opposite, and the vertical velocity near the depth of density pycnocline center is tend to be zero., the Mode-2 ISW amplitude effect on its induced flow field is investigated based on the measurements of flow field.Figure 6.(a) and Figure 6.(b) show the variation of peak values of the dimensionless velocity with dimensionless Mode-2 ISW amplitudes, respectively.The peak value of the positive horizontal velocity u is always greater than that of the negative peak value for various ISW amplitudes, and both the peak value of positive horizontal velocity and that of the negative one obviously increases with increasing amplitude.The peak value of the horizontal velocity is approximately twice of the vertical one.and peak values of positive and negative vertical velocity are close and increase with increasing amplitude.Due to the significant feature of strong horizontal positive currents caused by the Mode-2solitary waves caused near the pycnocline, the relationship between the vertical influence range of horizontal positive currents and wave amplitude is analysed.As shown in Figure 8, the vertical influence range of horizontal currents increases with the increase of wave amplitude, and the growth rate for larger amplitude is greater than that for smaller amplitude.

Conclusion
The experimental study of the internal solitary wave with second mode and its wave-induced flow field structure is conducted in a large physical flume.The conclusions are summarized as follows.
(1) The waveform of a convex Mode-2 ISW is characterized with the shape with convex isopycnals and concave isopycnals propagating along with continuous density pycnocline.The induced flow field is characterized with a pair of antisymmetric vortices propagating along with pycnocline.
(2) The influence of initial wave amplitude on induced flow field is analysed.The horizontal and vertical velocity of the Mode-2 ISW increase with increasing dimensionless characteristic amplitude, and the increase of the amplitude of Mode-2 ISW will also enhance its horizontal and vertical shear of the velocity field, additionally, the vertical influence range of horizontal currents increases with the increase of wave amplitude.

Figure 4 .
Figure 4. Waveform and density field induced by the Mode-2 ISW

Figure 5 (
a) shows the overall characteristic of the Mode-2 ISW flow field, which is characterized by two antisymmetric vortices propagating along the pycnocline.

Figure 5 (
b) shows the distribution of horizontal velocity flow field, the horizontal flow is characterized by the shear distribution near the waveforms in the upper and lower layer.

Figure 6 .
Figure 6.The peak values of the dimensionless velocity vary with the dimensionless Mode-2 ISW amplitude.(a) Horizontal velocity, (b) Vertical velocity.

Figure 7 .
Figure 7.The peak values of the dimensionless velocity shear strength vary with the dimensionless amplitude.(a) Horizontal shear.(b) Vertical shear

Figure 8 .
Figure 8.The vertical influence range of horizontal positive currents vary with the dimensionless amplitude.