Rotation-induced nonlinear wavepackets in internal waves

The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary wave propagation. An ocean eddy is produced by a quasi-geostrophic model in f-plane, and the one-dimensional nonlinear variable-coefficient extended Korteweg-de Vries (eKdV) equation is used to simulate an internal solitary wave passing through the mesoscale eddy field. The results suggest that the mode structures of the linear internal wave are modified due to the presence of the mesoscale eddy field. A cyclonic eddy and an anticyclonic eddy have different influences on the background environment of the internal solitary wave propagation. The existence of a mesoscale eddy field has almost no prominent impact on the propagation of a smallamplitude internal solitary wave only based on the first mode vertical structure, but the mesoscale eddy background field exerts a considerable influence on the solitary wave propagation if considering high-mode vertical structures. Furthermore, whether an internal solitary wave first passes through anticyclonic eddy or cyclonic eddy, the deformation of wave profiles is different. Many observations of solitary internal waves in the real oceans suggest the formation of the waves. Apart from topography effect, it is shown that the mesoscale eddy background field is also a considerable factor which influences the internal solitary wave propagation and deformation.

The numerical solutions for the coexisting fields of surface and internal solitary waves have been obtained, where the set of nonlinear equations based on the variational principle for steady waves are solved using the Newton- Raphson method. The relative phase velocity of surface-mode solitary waves is smaller in the coexisting fields of surface and internal solitary waves than in the cases without the coexistence of internal waves. The relative phase velocity of internal-mode solitary waves is also smaller in the coexisting fields of surface and internal solitary waves than in the cases without surface waves. The interfacial position of an internal mode internal solitary wave in a coexisting field of surface and internal waves can exceed the critical level determined in the corresponding case without a surface wave. The wave height ratio between internal-mode surface and internal solitary waves is smaller than the corresponding linear shallow water wave solution, and the difference increases, as the relative wave height of internal-mode internal solitary waves is increased.

Internal solitary waves exist widely in the oceans, and their generations, propagation evolutions, and dissipations have profound effects on the ocean environment, topography, and marine structures. Typically, two basic theoretical models are now being developed to govern the evolutions of internal solitary waves at the interface of two immiscible inviscid fluids. One is a unidirectional wave propagation model described by the KdV (Korteweg-de Vries) equation, and the other is a bidirectional wave propagation model depicted by the Miyata-Choi-Camassa (MCC) equation. Neither of them, however, can effectively characterize the course of the evolution of large-amplitude internal solitary wave. In this paper, a modified unidirectional internal solitary wave model is established by adjusting the coefficients of the original unidirectional model. The adjusted coefficients are determined through asymptotic analysis by matching with the MCC model. The efficacy of the modified coefficients is investigated by comparing the modified model with the original model. The experiments on the generation of internal solitary waves with varying amplitudes are carried out by comparing the internal solitary wave solution of the modified equation. It is shown that the modified model is suitable for describing the waveform of internal solitary waves with small, medium, and large amplitudes within the limiting amplitude of the MCC model. By quantitatively analyzing the agreement of the effective wavelength, wave speed, and waveform of steady-state internal solitary waves between the unidirectional model and the MCC model, the applicability of the modified model in characterization of the properties of steady-state internal solitary waves is further investigated. In addition, the stability of unidirectional theoretical model is analyzed for simulating the propagation of large-amplitude internal solitary wave under flat bottom condition. It is found that the unidirectional model is suitable for initiating its own internal solitary solution provided that the numerical scheme is stable. It is shown that the modified unidirectional model can be used to characterize large-amplitude internal solitary waves, and is also expected to be applied to the study of marine structure hydrodynamics.

The exact within potential flow integral equation approach of Evans and Ford [Proc. R. Soc. London Ser. A 452, 373 (1996)] for the normal solitary wave, is here generalized to 2-layer, ‘‘internal’’ solitary waves. This differs in its mathematical form from other exact integral equation methods based on the complex velocity potential. For both ‘‘rigid lid’’ (i.e., flat toplayer surface) and ‘‘free-surface’’ boundary conditions, a set of coupled non-linear integral equations are derived by an application of Green’s theorem. For each point on the layer interface(s), these describe functional constraints on the profiles and interface fluid velocity moduli; the exact profiles and velocities being those forms that satisfy these constraints at all such interface points. Using suitable parametric representations of the profiles and interface velocity moduli as functions of horizontal distance, x, and utilizing tailored quadrature methods [Int. J. Comput. Math. B 6, 219 (1977)], numerical solutions were obtained by the Newton–Raphson method that are highly accurate even at large amplitudes. For ‘‘rigid lid’’ boundary conditions, internal wave solutions are presented for layer density and depth ratios typical of oceanic internal wave phenomena as found in the Earth’s marginal seas. Their various properties, i.e., mass, momentum, energy, circulation, phase and fluid velocities, streamline profiles, internal pressures, etc., are evaluated and compared, where possible, with observed properties of such phenomena as reported, for example, from the Andaman Sea. The nature of the limiting (or ‘‘maximum’’) internal wave is investigated asymptotically and argued to be consistent with two ‘‘surge’’ regions separating the outskirts flow from a wide mid-section region of uniform ‘‘conjugate flow’’ as advocated by Turner and Vanden-Broeck [Phys. Fluids 31, 286 (1988)].

Internal waves play a significant role in the resuspension and transport of fine sediment adjacent to the sea bottom in the coastal region. Inall (2009) found the horizontal current and diffusion of mass due to the breaking of internal waves in the Fjord by using a fluorescent tracer. Internal waves have been shown to cause crucial mass transport and affect the ecological system. In particular, an internal solitary wave that progresses without changing the profile contributes to mass transport on a sloping bottom. Therefore, it is essential to clarify how an internal solitary wave breaks over a slope and transports mass. When pycnocline thickness is negligible in a twolayer fluid, an internal solitary wave breaking over a slope can be categorized into four breaker types by applying the latest classification based on wave slope, bottom slope gradient and an internal Reynolds number. Aghsaee et al. (2010) demonstrated that an internal solitary wave breaking over a slope can be categorized into four breaker types: surging, collapsing, plunging, and fission breakers using three-dimensional numerical simulations. They used wave slope and bottom slope gradient. However, some plunging and collapsing breakers were not appropriately categorized. Nakayama et al. (2019) solved the classification problem by introducing an internal Reynolds number based on the Korteweg–De Vries equation. However, it remains unsolved if this classification can categorize the breaking of an internal solitary wave under thick pycnocline conditions. This study uses numerical simulations to investigate the applicability of the classification under changing pycnocline thickness (Nakayama et al., 2021).

A numerical study of rotation effect on the propagation of nonlinear internal solitary waves in the northern South China Sea

In the weakly nonlinear long-wave regime, internal solitary waves are often modelled by the Korteweg– de Vries equation, which is well known to support an exact solitary wave solution. However, when the effect of background rotation is taken into account, an additional term is needed and the outcome is the Ostrovsky equation. Although the additional term would appear to be relatively mild, being a linear long-wave perturbation, it has the drastic effect of destroying the solitary wave solution. Instead an initial solitary-like disturbance decays into radiating oscillatory waves, with the eventual formation of a nonlinear envelope wave packet, whose carrier wavenumber is determined by an extremum in the group velocity. In this paper, we will use a combination of theoretical analyses, numerical simulations and laboratory experiments to describe this process.

Internal waves are waves that travel within the interior of the water column. Its existence is owned to the stratified density structure between two or continuous layers of fluids (Apel et al., 1987). Internal solitary waves (ISWs) are nonlinear internal waves, which are frequently observed all over the world oceans, where strong tides and stratification occur over varying topography features (Apel et al., 1985; Colosi et al., 2001; Osborne & Burch, 1980). They typically occur in packets at tidal intervals, suggesting that they mainly originate from the tide-topography interactions over variable topography (Gerkema et al., 1995). Depending on the different environmental conditions, there are two main mechanisms for the generation of ISWs: lee-wave mechanism (Maxworthy, 1979) and nonlinear internal tide mechanism (Lee & Beardsley, 1974). The lee-wave mechanism states that the lee-wave is formed by the ebb tide and released when the tide changes from ebb tide to flood tide, and evolves into a rank-ordered internal solitary wave (ISW) packet. By the nonlinear internal tide mechanism, internal tides spawn ISWs in three steps: initial generation of a front due to topographic blocking, nonlinear steepening of the front, and formation of a rank-order ISW packet under effects of nonlinearity and dispersion (Helfrich & Melville, 2006; Zhao & Alford, 2006). Internal solitary waves are important for many practical reasons. As they are commonly observed wherever strong tides and stratification occur next to the irregular topography, thus they are often prominent features seen in optical and radar satellite imagery of coastal waters. They can propagate over several hundred kilometers and transport both mass and momentum. They can also induce considerable velocity shears that can impose unexpectedly large stresses on offshore oil-drilling rigs and lead to turbulence and mixing. In addition, the mixing often introduces bottom nutrients into the water column, thereby fertilizing the local region and modifying the biology system therein (Jackson, 2004). ISWs in the South China Sea (SCS) have been observed at a variety of locations from Luzon Strait to the continental shelf (Cai et al., 2002; Lien et al., 2005; Liu et al., 2004; Ramp et al., 2004; Xu et al., 2010; Zhao et al., 2004). Until recently, considerable effort has been focused on the study of ISWs in the northeastern SCS (Farmer et al., 2009; Moore et al., 2007; Ramp et al., 2004). In contrast, Due to the shortage of high-quality data sets, studies of the ISWs in the northwestern SCS are quite limited and nonlinear internal waves in this region have been

High-resolution, non-hydrostatic simulation of internal tides and solitary waves in the southern East China Sea

Internal solitary waves (IWs) in the northwest Bay of Bengal were studied using ENVISAT ASAR imageries. Simultaneous in situ observations were also carried out to study the characteristics of the internal waves. Internal wave features have often been observed over the ocean area between 18.5°–19.5° N and 84.5°–85.5° E. The direction of propagation of IWs is generally toward the coast with wavelengths ranging from 0.4 to 1.1 km. The phase speeds of the IWs vary from 0.12 to 1.3 m/s. Internal wave amplitudes computed using Korteweg and de Vries model show good agreement with in situ data. Based on their direction of propagation and time series analysis, IWs generated were found to be of tidal forcing origin over local topographic features. IWs of different types like wave packets or wave trains and single solitons of weak manifestations were observed depending on the phase and the tidal amplitude as well as the stratification of the ocean, which in turn vary due to the dynamic and complex nature of Bay of Bengal, as it experiences strong monsoon winds and low pressure systems with calm weather in between them. The internal waves are often observed during non-monsoon period over this region. During monsoon period, continuous vertical mixing takes place due to strong winds causing unfavorable conditions for IW formation and so the observation of IWs during monsoons is rather rare.

This study presents the first comprehensive characterization of internal solitary waves (ISWs) in the Alor Strait, Indonesia, utilizing multi-modal observations from the Jala Citra Expedition 3-2023 “Flores.” High-resolution water column imaging via echosounder, coupled with satellite synthetic aperture radar (SAR) imagery, revealed exceptional ISWs with amplitudes frequently exceeding 100 m, with scattering volume (SV) value ranges from −73 to −60 dB. The Korteweg-de Vries (KdV) equation was employed to analyze the subsurface wave structure, correlating with SAR-derived bright-dark patterns. In situ measurements confirmed bi-directional wave propagation towards the Flores and Sawu Seas. This research demonstrates that the Alor Strait, not just the Ombai Strait, is a significant generator of internal waves in the southern Indonesian archipelago. Time-series analysis indicated that intra-seasonal events (ENSO) significantly modulate ISW characteristics. Notably, the study uncovered a potential link between ISWs, bolus activity, and sand wave evolution, with observations suggesting ISW-induced formation of sand wave crests up to 20 m high, ∼4 km long, and SV value ranges from −18.38 to −17.96 dB. These findings expand our understanding of internal wave generation in narrow Indonesian straits bordering the Indian Ocean and highlight the critical role of ISWs in shaping benthic morphodynamics, with implications for sediment transport and marine geohazards.

Oblique reflection of large internal solitary waves in a two-layer fluid

To address the difficulty of achieving rapid motion prediction for underwater vehicles under internal solitary waves (ISWs) disturbances using conventional experimental and high-fidelity numerical approaches, this study proposes a fast prediction method by coupling the vertical plane motion equations of an underwater vehicle with an internal wave force prediction model. The goal is to achieve rapid prediction of the motion response of the navigation vehicle in an internal wave disturbance environment. Based on this method, motion responses under small-amplitude and large-amplitude ISWs were simulated and validated against high-precision CFD (computational fluid dynamics) results. For small-amplitude ISWs, the linear motion equations yield superior prediction accuracy in trajectory, attitude, and velocity. For large-amplitude ISWs, linear equations perform better in trajectory and early-stage pitch prediction, whereas nonlinear equations provide improved accuracy in late-stage pitch motion and vertical velocity prediction. The results demonstrate that the proposed method is particularly suitable for real-time prediction under small-amplitude ISW conditions, while further improvement is required for large-amplitude and highly nonlinear motion scenarios. This work provides an efficient tool for rapid motion prediction of underwater vehicles in ISW environments and supports operational safety in internal wave-affected regions.

It is well known that internal waves (IWs) of tidal frequency (i.e., internal tides) are successfully detected in sea surface height (SSH) by satellite altimetry. Shorter period internal solitary waves (ISWs), whose periods (and spatial scales) are an order of magnitude smaller than tidal internal waves, have been generally assumed too small to be detected with conventional altimeters. This is because conventional (pulse-limited) radar altimeter footprints are somewhat larger than or of similar size, at best, as the typical wavelengths of the ISWs. Here we demonstrate that the synthetic aperture radar altimeter (SRAL) on board the Sentinel-3A can detect short-period ISWs. A variety of signatures owing to the surface manifestations of the ISWs are apparent in the SRAL Level-2 products over the ocean. These signatures are identified in several geophysical parameters, such as radar backscatter (sigma0), sea level anomaly (SLA), and significant wave height (SWH). Radar backscatter is the primary parameter in which ISWs can be identified owing to the measurable sea surface roughness perturbations in the along-track sharpened SRAL footprint. The SRAL footprint is sufficiently small to capture radar power fluctuations over successive wave crests and troughs, which produce rough and slick surface patterns arrayed in parallel bands with scales of a few kilometers. The ISW signatures are unambiguously identified in the SRAL because of the exact synergy with OLCI (Ocean Land Colour Imager) images, which in cloud-free conditions allow clear identification of the ISWs in the sunglint OLCI images. We show that both sigma0 and SLA yield realistic estimates for routine observation of ISWs with the SRAL, which is a significant improvement from previous observations recently reported for conventional pulse-limited altimeters (Jason-2). Several case studies of ISW signatures are interpreted in light of our knowledge of radar backscatter in the internal wave field. An analysis is presented for the tropical Atlantic Ocean off the Amazon shelf to infer the frequency of the phenomena, being consistent with previous satellite observations in the study region.

Modelling and observations of oceanic nonlinear internal wave packets affected by the Earth’s rotation