Mode-2 Internal Solitary Waves Research Articles

Observations of anomalously strong mode-2 internal solitary waves in the central Andaman sea by a mooring system

During the period from July 2016 to July 2018, a mooring system deployed in the central Andaman Sea recorded a significant number of highly intense mode-2 internal solitary waves (ISWs), with wave-induced current strengths comparable to those of local mode-1 ISWs. Distinct propagation characteristics and seasonal variations are uncovered, with mode-2 ISWs being identified as primarily propagating eastward and exhibiting a notable frequency peak from November to March. A significant correlation is established between the occurrence of mode-2 ISWs and higher-mode internal tides (ITs), particularly characterized by a high correlation with the third Empirical Orthogonal Function (EOF) mode. Combined with satellite remote sensing images, it is further confirmed that mode-2 ISWs are generated by the nonlinear steepening of ITs and propagate over long distances through resonance with higher-mode ITs. In contrast, our findings suggest that mode-1 ISWs are predominantly generated by the Lee-wave mechanism, especially south of the Ten Degree Channel, and typically propagate northeastward. This study underscores the complex interplay of ocean stratification and seabed topography in the genesis and propagation of ISWs.

Dynamics of multi-mode nonlinear internal waves in fluids of great depth

This paper is mainly concerned with the dynamics of strongly nonlinear internal waves in two-dimensional fluids of great depth. A fully nonlinear model for a three-layer fluid of great depth containing topography, a Miyata–Choi–Camassa (MCC)-type system, is developed based on the generalization of the Ablowitz–Fokas–Musslimani global-relation formulation for free-surface water waves. Mode-1 internal solitary waves are numerically found in the MCC-type equation using the Petviashvili iteration technique and in the full Euler equations based on the boundary integral equation method. The comparison between the results validates the broad applicability of the MCC-type system. Mode-2 internal solitary waves are found in the MCC-type equations and are shown to be a type of gap solitary wave. Using the multi-mode internal solitary-wave solutions in the MCC-type equations, we apply the conformal mapping technique to calculate streamlines and particle trajectories. For unsteady simulations, we propose a pseudo-spectral algorithm to handle the time-coupled equations and investigate the generation mechanism of multi-mode internal solitary waves due to the resonance effect of local protrusion on the rigid wall, as well as their collisions when the wall is flat. For flows past protrusion on the rigid wall, the flow speed can be divided into subcritical, transcritical, and supercritical. The shedding of multi-mode internal solitary waves can occur only in the transcritical region of flow speed. We implement the carving of bifurcation curves with inflection to achieve the regional delimitation and, based on this, the interpretation of the generation mechanism from a physical point of view.

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.

Spatial and temporal variability in mode-1 and mode-2 internal solitary waves from MODIS-Terra sun glint off the Amazon shelf

Abstract. The Amazon shelf is a key region for intense internal tides (ITs) and nonlinear internal solitary wave (ISWs) generation associated with them. The region shows well-marked seasonal variability (from March to July, MAMJJ, and from August to December, ASOND) of the circulation and stratification, which can both induce changes in the ISW physical characteristics. The description of the seasonal and neap–spring tidal variability in the ISWs off the Amazon shelf is performed for the first time using a meaningful data set composed of 140 MODIS-Terra imagery from 2005 to 2021, where about 500 ISW signatures were identified in the sun glint region. Previous studies have documented the existence of mode-1 ISWs, but the region appears as a newly described hotspot for mode-2 ISWs. ISW packets separated by typical mode-1 (95–170 km; 2.1–3.8 m s−1) and mode-2 (46–85 km; 1.0–1.9 m s−1) IT wavelengths have been identified and mapped coming from different IT generation sites. For each ISW, a group of waves (3 to 10) is generally follows the largest crest. The intra-packet distance between each wave in the group is about 10 to 20 km. Regions of higher occurrence of ISWs are spaced by a IT mode-1 wavelength. We make the assumption that it might correspond to the IT reflection beams at the surface, which may generate newer ISWs. The mean mode-1 and mode-2 inter-packet distances do not show significant differences according to their IT generation sites. The ISW activity is higher (more than 60 % of signatures) during spring tides than neap tides. In the region under the influence of the North Equatorial Counter Current (NECC), ISWs are separated by a mean mode-1 IT wavelength which is 14.3 % higher during ASOND than during MAMJJ due to a deeper thermocline and the reinforcement of the NECC. These ISWs are also characterized by a wider inter-packet distance distribution (higher standard deviation) that may be related to the stronger eddy kinetic energy (EKE) during ASOND compared to MAMJJ. The mean inter-packet distance of mode-2 ISWs remains almost unchanged during the two seasons, but the inter-packet distance distribution is wider in ASOND than in MAMJJ as for mode 1. Note that these results need to be treated with caution, as only few occurrences of mode-2 waves were found during MAMJJ. In the region of the NECC, the direction of propagation for all modes is very similar in MAMJJ (about 30∘ clockwise from the north), whereas, for ASOND, the ISWs propagate in a wider pathway (from 0 to 60∘ clockwise from the north), due to a much larger eddy activity. During ASOND, as the background flux goes further east, the inter-packet distances become larger (4 % for mode 1 and 7.8 % for mode 2). These results show that the reinforcement of the NECC in ASOND appears to play a role in diverting the waves towards the east, increasing their phase velocities and their eastern traveling direction component when compared to MAMJJ. Calculations of the IT velocities using the Taylor–Goldstein equation supported our results regarding the presence of ISWs associated with mode-2 ITs and additionally the IT seasonal variability.

Experimental investigation on the interaction between mode-2 internal solitary wave and horizontal transverse cylinder

The experimental study on the interaction between a mode-2 internal solitary wave (ISW) and a horizontal transverse cylinder has been carried out in a large gravity-stratified fluid flume. One kind of making-wave method of “rotating-blade-gate” is proposed to achieve the experimental simulation of the standard mode-2 ISW in the laboratory flume. The conductivity probe array and particle image velocimetry are used to measure the mode-2 ISW and its wave-flow structure of the interaction with the horizontal transverse cylinder, and the micro-amplitude force sensor is used to measure the forces of the mode-2 ISW on the cylinder model. It is shown that the mode-2 ISW geometric structure is characterized by a convex and concave oval wave envelope shape and the counterclockwise and clockwise circulations are formed in its convex and concave domains, respectively. The resulting pair of upper and lower antisymmetric circulations moves forward together, which causes a horizontal strong flow at the symmetric center of the wave envelope consistent with the wave propagation direction. The horizontal transverse cylinder is subjected to the combined action of horizontal flow, vertical flow, and density change induced by the mode-2 ISW, in which the horizontal strong flow, induced flow separation, and vortex structure located at the symmetric center of the wave envelope are the signature features that are different from other positions. The theoretical analysis model of the forces exerted by the mode-2 ISW on a horizontal transverse cylinder is established. The spatial distribution characteristic of the force exerted by the mode-2 ISW on a horizontal cylinder is obtained, and the essential reason for the maximum horizontal and vertical loads on the cylinder is revealed. The horizontal maximum load corresponds to the inhomogeneous change of the horizontal velocity along the depth, and the vertical maximum load depends on the maximum density change along the depth, of which variation trend corresponds to the linear change of wave amplitudes.

Optical remote sensing image characteristics of large amplitude convex mode-2 internal solitary waves: an experimental study

Optical remote sensing image characteristics of large amplitude convex mode-2 internal solitary waves: an experimental study

Assessment of KdV and EKdV theories for simulating internal solitary waves in the continental slope of the South China Sea

Korteweg-de Vries (KdV) and Extended KdV (EKdV) are two important nonlinear theoretical models that are most commonly used for the description of internal solitary waves (ISWs). However, most of the previous studies based on the applications of these two models focus only on laboratory-based simulations. Moreover, the modeling data in those studies were rarely compared with the in-situ measurements. In this study, three moorings were deployed to the west of Dongsha Islands of the northern South China Sea (SCS), based on the KdV and EKdV theoretical approaches, to simulate and statistically analyze 854 mode-1 ISWs and 37 mode-2 ISWs captured during the period of observation. The results from this study indicated that both theories can be used to well describe the mode-1 ISWs with η0/H<0.12 (η0 and H are the amplitude of ISW and depth of seawater, respectively). Generally, when the depth of pycnocline was not close to H/2, the EKdV theory was more accurate than the KdV theory in simulating the waveforms and vertical velocities of mode-1 ISWs. However, for mode-1 ISWs with larger amplitudes, the horizontal velocities of the upper layer calculated by the EKdV theory were extremely large. The EKdV theory was no longer applicable when the depth of pycnocline was close to H/2 and η0 exceeded its limiting value of the EKdV theory. For mode-2 ISWs, the waveforms simulated by the EKdV theory were in close agreement with the in-situ measurements. However, neither of the two theories can accurately simulate the velocities. Further, the statistical results of propagation speeds of 49 mode-1 ISWs highlighted that the EKdV theory performed slightly better than the KdV theory in calculating the propagation speed of ISWs. Finally, the MCC (Miyata-Choi-Camassa) theory was introduced as a supplement to simulate larger-amplitude ISWs. The MCC theory was found to show better performance than the above two theories in simulating the ISWs with η0/H>0.15 but the trend was opposite when η0/H<0.15.

Experimental Study on Optical Imaging of Convex Mode-2 Internal Solitary Waves in Calm Water

The optical imaging mechanism of convex mode-2 Internal Solitary Waves (ISWs) in calm water without wind is investigated based on experimental and simulation calculations. By constructing an optical imaging system in the laboratory, the optical images of ISWs, the matching in-situ data, and the free surface displacement caused by ISWs are acquired simultaneously. The experimental outcomes show that the convex mode-2 ISWs exhibit bright-dark stripes in the optical image and produce a depressed free surface displacement on the water surface. The physical model of free surface displacement is constructed in Unigraphics NX (UG) and imported into Light Tools (LT) for ray tracing. The consistency between simulation calculations and experimental results shows that the free surface displacement generated by the mode-2 ISWs will change the reflection of light, leading to bright-dark stripes in the optical image. A series of experiments were designed to explore the influence of amplitude on the optical imaging of mode-2 ISWs when the thickness of the intermediate layer is different or the same. The results show that the amplitude of the wave will affect the imaging effect of ISWs in optical image, and the large amplitude will make the stripes clearer in the image.

Large mode-2 internal solitary waves in three-layer flows

In this paper, we investigate mode-2 solitary waves in a three-layer stratified flow model. Localised travelling wave solutions to both the fully nonlinear problem (Euler equations), and the three-layer Miyata–Choi–Camassa equations are found numerically and compared. Mode-2 solitary waves with speeds slower than the linear mode-1 long-wave speed are typically generalised solitary waves with infinite tails consisting of a resonant mode-1 periodic wave train. Herein, we evidence the existence of mode-2 embedded solitary waves, that is, we show that for specific values of the parameters, the amplitude of the oscillations in the tail are zero. For sufficiently thick middle layers, we also find branches of mode-2 solitary waves with speeds that extend beyond the mode-1 linear waves and are no longer embedded. In addition, we show how large amplitude embedded solitary waves are intimately linked to the conjugate states of the problem.

Numerical modeling of energy dissipation of internal solitary waves encountering step topography

In this study, we conducted simulation experiments using a large-eddy simulation method to examine the interaction of internal solitary waves (ISWs) with step topography. We investigated the propagation, transmission, reflection, and breaking of depression ISWs. The features of ISW depend on the geometrical parameters of the step topography that define the initial setting. The relationships between the ISWs geometric and energy loss features as well as the initial setting parameters were analyzed; related empirical relations were developed. Following the analysis of the vortex changes during the process of wave-topography interaction, the different breaking processes and their development rules were discussed. The results showed that, as the obstacle height increased, the degree of energy conversion during the interaction increased. There is no explicit relationship between the loss of energy and energy conversion during the interaction of an ISW with an obstacle. The maximum wave energy loss obtained by eliminating the effect of the tailed wave generated during wave generation and considering the energy conversion is approximately 5%–10% lower than that obtained with traditional methods. Additionally, the mode-2 ISW was detected during the wave-topography interaction, and its velocity profiles in the vertical direction were analyzed, which may be one of the reasons for the accelerated energy dissipation in the system.

Regional study of mode-2 internal solitary waves at the Pacific coast of Central America using marine seismic survey data

Abstract. In this paper, a regional study of mode-2 internal solitary waves (ISWs) at the Pacific coast of Central America is carried out using the seismic reflection method. The observed relationship between the dimensionless propagation speed and the dimensionless amplitude (DA) of the mode-2 ISW is analyzed. When DA &lt; 1.18, the dimensionless propagation speed seems to increase with increasing DA, divided into two parts with different growth rates. When DA &gt; 1.18, the dimensionless propagation speed increases with increasing DA at a relatively small growth rate. We suggest that the influences of seawater depth (submarine topography), pycnocline depth, and pycnocline thickness on the propagation speed of the mode-2 ISW in the study area cause the relationship between dimensionless propagation speed and DA to diversify. The observed relationship between the dimensionless wavelength and the DA of the mode-2 ISW is also analyzed. When DA &lt; 1, the nondimensional wavelengths seem to change from 2.5 to 7 for a fixed nondimensional amplitude. When DA &gt; 1.87, the dimensionless wavelength increases with increasing DA. Additionally, the seawater depth has a great influence on the wavelength of the mode-2 ISW in the study area. Overall, the wavelength increases with increasing seawater depth. As for the vertical structure of the amplitude of the mode-2 ISW in the study area, we find that it is affected by the nonlinearity of the ISW and the pycnocline deviation (especially the downward pycnocline deviation).

The shoaling mode-2 internal solitary waves in the Pacific coast of Central America investigated by marine seismic survey data

In this paper, the shoaling mode-2 internal solitary waves (ISWs) in the Pacific coast of Central America are studied for the first time using seismic oceanography method. It is found that there is an asymmetry in the vertical amplitude variation of the mode-2 ISWs front and rear during the shoaling process. The amplitude may weaken the effect of the pycnocline deviation on the stability of the mode-2 ISWs. For large amplitudes, the pycnocline needs to deviate greatly, so that the mode-2 ISWs become unstable; for small amplitude, the pycnocline deviates very little, and the mode-2 ISWs become unstable. The pre-stack migration sections of the common offset gathers are used to calculate the apparent phase velocities of the mode-2 ISWs. It is found that in deep water, the seafloor depths and amplitudes simultaneously control the magnitude of the apparent phase velocities of the mode-2 ISWs; in shallow water, the amplitudes mainly control the magnitude of the apparent phase velocities of the mode-2 ISWs. For the mode-2 ISWs with small amplitudes (2a/h2≤2) in shallow water, the apparent phase velocities increase with the amplitudes, and increase rapidly; for the mode-2 ISWs with large amplitudes (2a/h2>2) in shallow water, the apparent phase velocities increase with the amplitudes, but increase slowly. By calculating the relative seismic spectrum energy, it is found that the seismic spectrum energy in the submesoscale wavenumber range corresponds well to the mode-2 ISWs waveform. The mode-2 ISWs in the shoaling process have submesoscale characteristics.

Resonant coupling of mode-1 and mode-2 internal waves by topography

Abstract

Mooring observed mode-2 internal solitary waves in the northern South China Sea

The mode-2 internal solitary waves (ISWs) generated by mode-2 internal tide (IT) are identified by mooring observations in the northern South China Sea (SCS) from 2016 to 2017. Two mode-2 ISWs with a re-appearance period of 24.9 h observed on 29 and 30 July 2016 are characterized by type-b ISWs. They occurred when the isotherms compressed obviously in the vertical direction. Modal decomposition of IT horizontal currents shows that the vertical compression of the isotherms is mainly caused by diurnal mode-2 IT. The analysis of the role of the density stratification reveals that a deeper and thinner pycnocline is favorable for generation of mode-2 ISWs rather than pycnocline intensity. By comparing the mode-2 nonlinear, dispersion coefficients and the Ursell numbers calculated based on the stratification associated with different kinds of ITs with the observation results, it is shown that the diurnal mode-2 IT plays a crucial role in the generation of the mode-2 ISWs. When the diurnal mode-2 IT interacts with the semidiurnal IT and causes a deeper and thinner pycnocline, the mode-2 ISWs are easily excited.

Numerical Investigation of Bidirectional Mode-1 and Mode-2 Internal Solitary Wave Generation from North and South of Batti Malv Island, Nicobar Islands, India

AbstractStrong bidirectional internal solitary waves (ISWs) generate from a shallow channel between Car Nicobar and Chowra Islands of Nicobar Islands, India, and propagate toward the Andaman Sea (eastward) and Bay of Bengal (westward). Batti Malv Island separates this shallow channel into two ridges, north of Batti Malv (NBM) and south of Batti Malv (SBM). First, this study identifies the prominent mode-1 and mode-2 ISWs emerging from NBM and SBM using synthetic aperture radar images and then explores their generation mechanism(s) using a nonlinear, unstructured, and nonhydrostatic model, SUNTANS. During spring tide, flow over NBM is supercritical with respect to mode-1 internal wave. Model simulations reveal that mode-1 ISWs are generated at NBM by a “lee wave mechanism” and propagate both in the east and west directions depending on the tidal phases. However, the flow over SBM is subcritical with respect to mode-1 internal wave. The bidirectional propagating mode-1 ISWs evolve from a long-wave disturbance induced by “upstream influence.” But, during spring tide, with an increased tidal flow over SBM, it is observed that the westward propagating ISWs are formed by a dispersed hydraulic jump observed over the ridge. Moreover, the bidirectional mode-2 waves from SBM are generated by a lee wave mechanism. An energy budget comparison reveals that the region surrounding NBM is efficient in radiating low-mode baroclinic energy (0.98 GW), while SBM is highly efficient in converting barotropic to baroclinic energy (4.1 GW).

Long lived second mode internal solitary waves in the Andaman Sea

Internal waves are density oscillations propagating along the ocean’s inner stratification, which are now acknowledged as a key constituent of the ocean’s dynamics. They usually result from barotropic tides, which flow over bottom topography, causing density oscillations to propagate along the pycnocline with a tidal frequency (i.e. internal tides). These large-scale waves propagate away from their forcing bathymetry and frequently disintegrate into nonlinear short-scale (higher-frequency) internal wave packets. Typically, short-scale internal wave observations in the ocean are associated with vertical structures (in the water column) of the lowest fundamental mode. Higher vertical modes have recently been documented as well, but these are commonly short-lived (up to a few hours). However, unprecedented satellite images showing long-lived short-scale mode-2 internal waves have now been documented in the Andaman Sea, and we report here the companion results of a non-hydrostatic and fully nonlinear numerical model. The simulations reproduce the waves’ main characteristics as observed in satellite imagery, and the results suggest a resonant coupling with a larger-scale mode-4 internal tide as an explanation for their long-lived character.

Evolution of internal solitary waves on the slope-shelf topography in the northern South China Sea

Based on a non-hydrostatic two-dimensional and high-resolution model, evolution of internal solitary waves (ISWs) on the typical slope-shelf topography in the northern South China Sea is investigated numerically, and the influences of the initial amplitude, seasonal stratification and topographic characteristics are analyzed with a series of sensitivity runs. The results indicate that the initial amplitude affects the fission of ISW, resulting in three wave groups for large ISW and two wave groups for small ISW. In addition, the generation of mode-2 waves is influenced since energetic beams are engendered by large initial ISW, which impact the pycnocline and generate the mode-2 ISWs. Seasonal stratification has significant impacts on the evolution of the ISW. In winter, the changing sign of the nonlinearity coefficient at the bump near the shelf break implies the inversion of polarity of the ISW. Therefore, the transmitted and fissioned waves behave differently from those in summer and annual stratifications. Furthermore, the speed and energy of the leading wave are minimal in winter but maximal in summer. The bump near the continental shelf has two impacts: promoting the fission of the incident ISW and generating mode-2 ISWs by increasing the Ursell number (the ratio of nonlinear coefficient to dispersion coefficient). However, the formation of the trailing nonlinear wave packet is not affected by these factors, despite of some differences in waveforms.

Wave generation through the interaction of a mode-2 internal solitary wave and a broad, isolated ridge

An investigation of a mode-2 internal solitary wave passing over a broad ridge finds that it is gradually adjusted into multiple waves of different form. The leading wave remains intact but is reduced in amplitude and energy. Incident amplitude, speed, and ridge height determine the strength of the resultant waves.

The interaction of a mode-1 internal solitary wave with a step and the generation of mode-2 waves

ABSTRACTIn this study, we examine the transformation of a mode-1 internal solitary wave incident on a bottom step, and the consequent generation of mode-2 internal solitary waves. A linear long-wave theory of mode coupling in the vicinity of the step is used to estimate the mode-1 and mode-2 wave reflection and transmission coefficients, and hence the energy fluxes. Away from the step, the wave evolution of the transmitted and reflected waves is simulated by the Korteweg–de Vries equation. Specific calculations are made using a three-layer fluid model. Three different regimes based on the layer thicknesses are examined and discussed in detail for either depression or elevation mode-1 incident waves. The common features found are that the transmitted waves (mainly mode-1) are the dominant part; most of the incident energy is transmitted and only a small part is reflected. The amplitudes of the generated mode-2 waves and the reflected mode-1 waves increase when either the upper- or middle-layer thickness increases. When the lower layer is thin enough, the amplitude of the transmitted mode-2 wave can be larger than the mode-1 waves, and the reflected energy can increase considerably which we infer may be due to a blocking effect of the step on the lower layer. The evolution away from the step is either fission into several solitary waves, or the development of a rarefaction wave followed by an undular bore, depending on the relative signs of the wave amplitudes and the nonlinear coefficient in the Korteweg–de Vries equation.

The evolution of mode-2 internal solitary waves modulated by background shear currents

Abstract. The evolution of mode-2 internal solitary waves (ISWs) modulated by background shear currents was investigated numerically. The mode-2 ISW was generated by the “lock-release” method, and the background shear current was initialized after the mode-2 ISW became stable. Five sets of experiments were conducted to assess the sensitivity of the modulation process to the direction, polarity, magnitude, shear layer thickness and offset extent of the background shear current. Three distinctly different shear-induced waves were identified as a forward-propagating long wave, oscillating tail and amplitude-modulated wave packet in the presence of a shear current. The amplitudes of the forward-propagating long wave and the amplitude-modulated wave packet are proportional to the magnitude of the shear but inversely proportional to the thickness of the shear layer, as well as the energy loss of the mode-2 ISW during modulation. The oscillating tail and amplitude-modulated wave packet show symmetric variation when the background shear current is offset upward or downward, while the forward-propagating long wave was insensitive to it. For comparison, one control experiment was configured according to the observations of Shroyer et al. (2010); in the first 30 periods, ∼ 36 % of total energy was lost at an average rate of 9 W m−1 in the presence of the shear current; it would deplete the energy of initial mode-2 ISWs in ∼ 4.5 h, corresponding to a propagation distance of ∼ 5 km, which is consistent with in situ data.