Dynamics Of Internal Waves Research Articles

A novel variable-coefficient extended Davey–Stewartson system for internal waves in the presence of background flows

We propose a novel variable-coefficient Davey–Stewartson type system for studying internal wave phenomena in finite-depth stratified fluids with background flows, where the upper- and lower-layer fluids possess distinct velocity potentials, and the variable-coefficient terms are primarily controlled by the background flows. This realizes the first application of variable-coefficient DS-type equations in the field of internal waves. Compared to commonly used internal wave models, this system not only describes multiple types of internal waves, such as internal solitary waves, internal breathers, and internal rogue waves, but also aids in analyzing the impact of background flows on internal waves. We provide the influence of different background flow patterns on the dynamic behavior and spatial position of internal waves, which contribute to a deeper understanding of the mechanisms through which background flows influence internal waves. Furthermore, the system is capable of capturing variations in the velocity potentials of the upper and lower layers. We discover a connection between internal waves under the influence of background flows and velocity potentials. Through the variations in velocity potentials within the flow field, the dynamic behaviors of internal waves can be indirectly inferred, their amplitude positions located, and different types of internal waves distinguished. This result may help address the current shortcomings in satellite detection of internal wave dynamics and internal rogue waves.

The Atlantic Niño Mode: A Thermodynamic or a Dynamic Phenomenon?

AbstractThe Atlantic Niño is an important source of the year‐to‐year variability of the tropical Atlantic, consisting in an irregular oscillation of the Sea Surface Temperature (SST) in the eastern tropical Atlantic. The physical mechanism underlying this oscillation is topic of debate. Some theories, known as dynamical, suggest that the Atlantic Niño is driven by internal wave dynamics. Other theories, termed thermodynamic, propose that the eastern tropical Atlantic SST variability is caused by thermodynamic processes associated with heat fluxes at the ocean surface and/or heat transport by oceanic currents. Here, we examine the SST variability from 1940 to 2022 and find that it can be described in terms of two spatial modes: one in the central and the other in eastern tropical Atlantic. However, focusing on the period from 1993 to 2022, these patterns behave differently. In 1993–2009 they concur in determining the eastern tropical Atlantic SST variability, while in 2010–2022 they act in opposite direction. Therefore, we use Sea Surface Height data and heat fluxes advected by oceanic currents to explore the relative contribution of wave dynamics and heat advection to the eastern tropical Atlantic SST fluctuation. We show that, between 1993 and 2010, the eastern tropical Atlantic SST variability is mainly determined by heat advected from the south by horizontal currents, while between 2010 and 2022 it can be explained in terms of wave dynamics. This finding suggests that the two spatial patterns determining the eastern tropical Atlantic SST variability, are two distinct physical phenomena forced by different physical mechanisms.

The modified cubic Benjamin–Ono equation describing internal solitary waves in the deep ocean and its related properties

In this article, we investigate the propagation of internal solitary waves in deep ocean. Based on the principles of nonlinear theory, perturbation expansion, and multi-scale analysis, a time-dependent modified cubic Benjamin–Ono (mCBO) equation is derived to describe internal solitary waves in the deep ocean with stronger nonlinearity. When the dispersive term ∂3f∂X3 vanishes, the mCBO equation transforms into the cubic BO equation. Similarly, when the dispersive term ∂3f∂X3 becomes zero and the nonlinear term ∂f3∂X degenerates into ∂f2∂X, the mCBO equation reduces to the BO equation. Furthermore, if the integral term ∂2∂X2ℵ(f) disappears, it simplifies to the mKdV equation. To gain deeper insight into the characteristics of solitary waves, conservation of mass and momentum associated with them are discussed. By employing Hirota's bilinear method, we obtain soliton solutions for the mCBO equation and subsequently investigate interactions between two solitary waves with different directions, leading to the occurrence of important events such as rogue waves and Mach reflections. Additionally, we explore how certain parameters influence Mach stem while drawing meaningful conclusions. Our discoveries reveal the complex dynamics of internal solitary waves within the deep ocean and contribute to a broader understanding of nonlinear wave phenomena.

Symmetry-based closed-form solutions and conserved quantities of the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in fluid mechanics

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

Modeling sound speed profile based on ocean normal mode

IntroductionStatistical methods such as empirical orthogonal functions (EOFs) are often used to model the sound speed profile (SSP). However, their statistical nature often leads to the sample dependence and physical fuzziness.MethodThis study proposes a technique for modeling the SSP from the perspective of ocean dynamics. It employs the ocean normal mode, which is the mode of fluid particles motion, to deduce perturbations in the SSP, which is called the ocean mode basis (OMB).ResultThe results of SSP reconstruction of in-situ samples showed that a few leading orders of the OMB can provide a compact representation of the SSP. Oscillations of the contours and gradient of the sound speed in thermocline were analyzed by using the first two orders of the projection coefficients of the relationship between the OMB and the baroclinic mode. As a physical model, this technique can also be used to characterize the dynamics of internal solitary waves. Furthermore, the OMB derived from archival date was used for SSP inversion. The results showed that the OMB can reconstruct SSP of a reasonable resolution without requiring in-situ samples.DiscussionCompared with statistical models, the OMB can better explain the ocean dynamics underlying variations in the SSP while requiring fewer samples.

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.

Exploring two-dimensional internal waves: A new three-coupled Davey–Stewartson system and physics-informed neural networks with weight assignment methods

In order to investigate various two-dimensional internal waves in a two-layer finite-water depth fluid with different velocity potentials in each layer, we derive a new (2+1)-dimensional system of three coupled, two-component Davey–Stewartson type equations under the long wave assumption involving complex amplitude and real velocity potentials in the upper and lower layers. It is found that exact solutions of this system can describe internal solitary waves, internal rogue waves, and internal breathers, and more interestingly, it is proposed for the first time that the patterns of changes in the velocity potentials of the upper and lower layers can be used as an indicator for recognizing the occurrence, disappearance, and amplitudes of these internal waves. In view of the application of deep learning techniques in the simulation, prediction and monitoring of internal waves, an improved physical information neural network (PINN) method is introduced to successfully simulate dynamics of internal solitary waves, internal rogue waves, and internal breathers, as well as the corresponding velocity potentials. The effects of no weight assignment and two different weight assignment methods in the loss function on the training results are analyzed in detail, which reveals that better simulating results can be achieved via a fixed-weight loss function for small-amplitude internal waves, while the introduction of weights has little effect on the simulations for larger-amplitude internal waves. In addition, the loss function with adaptive weights performs well in the fast simulation of internal rogue waves and poorly in the learning effect of the velocity potentials, which is contrary for internal solitary waves, but is not applicable in the simulation of internal breathers.

A tri-point correlation method for ocean current estimation and its applications in near-bottom turbulent mixing on the shelf of the Northern South China Sea

We propose a tri-point correlation method to estimate ocean currents using moored temperature observation data. The transit time for two time series of temperature is determined with the cross correlation technique, enabling the formulations of an equation system to determine the current speed and angle. To validate our method, we utilize a high-resolution temperature dataset collected at the shelf bottom of the northern South China Sea and compare the inferred currents with direct measurement results with a current meter. The results demonstrate a high level of agreement between the inferred currents and the directly measured ones. Using this temperature dataset and the synchronously inferred currents, we further estimate near-bottom turbulent mixing, including diapycnal mixing and vertical heat flux. During the observation period, the mean values of the turbulent kinetic energy dissipation rate, turbulent diffusivity, and vertical heat flux are 3.8×10−8 m2/s3, 2.6×10−4 m2/s, and 10.0 W/m2, respectively. These inferred turbulent mixing quantities exhibit a positive correlation with the dynamics of internal waves.

Mathematical Modeling of Dynamics of Internal Gravity Waves in the Ocean with Arbitrary Distribution of Buoyancy Frequency

Mathematical Modeling of Dynamics of Internal Gravity Waves in the Ocean with Arbitrary Distribution of Buoyancy Frequency

Manifestation of internal waves in the Southern Caspian Sea using satellite imagery

Iran's South Caspian coast is an important agricultural hotspot and densely populated area that attracts about 20 million domestic tourists in summer. Therefore, not only the coastal area ecosystem of the South Caspian Sea (SCS) but also their marine ecosystem is exposed to different environmental pollution. The ecosystem of the SCS especially in the onshore area strongly affected by Internal Waves (IWs) dynamics. IWs can capture, concentrate and transport, floating plastic debris, larvae, contaminants, and toxic pollutants in this basin. Therefore, detecting and zoning the SCS for IWs is critical to the management and protection of this enclosed basin. Measured data obtained from CTD and satellite imagery archives from Landsat 7, 8, and Sentinel-2 have been used to analyze and manifest IWs. Analysis and comparison of different satellite images reveal that the maximum wavelength of the IWs packet was about 400 m with a crest length of 18,000 m. However, the minimum packet had a wavelength of about 20 m with a crest length of 360 m. Based on the occurrence of IWs extracted by edge-finding algorithms on satellite images, the SCS can be divided into six original zones. Four zones of them located near the coastal line where freshwater rivers discharge to the sea in the rainy season and two other zones are in the deep water, which form under the dominant sea surface wind regime and sharp changing of bathymetry.

Fiber‐Optic Observations of Internal Waves and Tides

AbstractAlthough typically used to measure dynamic strain from seismic and acoustic waves, Rayleigh‐based distributed acoustic sensing (DAS) is also sensitive to temperature, offering longer range and higher sensitivity to small temperature perturbations than conventional Raman‐based distributed temperature sensing. Here, we demonstrate that ocean‐bottom DAS can be employed to study internal wave and tide dynamics in the bottom boundary layer, a region of enhanced ocean mixing but scarce observations. First, we show temperature transients up to about 4 K from a power cable in the Strait of Gibraltar south of Spain, associated with passing trains of internal solitary waves in water depth <200 m. Second, we show the propagation of thermal fronts associated with the nonlinear internal tide on the near‐critical slope of the island of Gran Canaria, off the coast of West Africa, with perturbations up to about 2 K at 1‐km depth and 0.2 K at 2.5‐km depth. With spatial averaging, we also recover a signal proportional to the barotropic tidal pressure, including the lunar fortnightly variation. In addition to applications in observational physical oceanography, our results suggest that contemporary chirped‐pulse DAS possesses sufficient long‐period sensitivity for seafloor geodesy and tsunami monitoring if ocean temperature variations can be separated.

3D hydrodynamic simulations of massive main-sequence stars – I. Dynamics and mixing of convection and internal gravity waves

ABSTRACT We performed 3D hydrodynamic simulations of the inner $\approx 50{{\ \rm per\ cent}}$ radial extent of a $25\,\,\mathrm{\mathrm{M}_\odot }$ star in the early phase of the main sequence and investigate core convection and internal gravity waves in the core-envelope boundary region. Simulations for different grid resolutions and driving luminosities establish scaling relations to constrain models of mixing for 1D applications. As in previous works, the turbulent mass entrainment rate extrapolated to nominal heating is unrealistically high ($1.58\times 10^{-4}\,\,\mathrm{\mathrm{M}_\odot \, {\mathrm{yr}}^{-1}}$), which is discussed in terms of the non-equilibrium response of the simulations to the initial stratification. We measure quantitatively the effect of mixing due to internal gravity waves excited by core convection interacting with the boundary in our simulations. The wave power spectral density as a function of frequency and wavelength agrees well with the GYRE eigenmode predictions based on the 1D spherically averaged radial profile. A diffusion coefficient profile that reproduces the spherically averaged abundance distribution evolution is determined for each simulation. Through a combination of eigenmode analysis and scaling relations it is shown that in the N2-peak region, mixing is due to internal gravity waves and follows the scaling relation DIGW-hydro ∝ L4/3 over a $\gtrapprox 2\,\,\mathrm{\mathrm{dex}}$ range of heating factors. Different extrapolations of the mixing efficiency down to nominal heating are discussed. If internal gravity wave mixing is due to thermally enhanced shear mixing, an upper limit is $D_\mathrm{IGW}\lessapprox 2$ to $3\times 10^{4}\,\,\mathrm{cm^2\, s^{-1}}$ at nominal heating in the N2-peak region above the convective core.

Strongly non-linear Boussinesq-type model of the dynamics of internal solitary waves propagating in a multilayer stratified fluid

We propose a system of first-order balance laws that describe the propagation of internal solitary waves in a multilayer stratified shallow water with non-hydrostatic pressure in the upper and lower layers. The construction of this model is based on the use of additional variables, which make it possible to approximate the Green–Naghdi-type dispersive equations by a first-order system. In the Boussinesq approximation, the governing equations allow one to simulate the propagation of non-linear internal waves, taking into account fine density stratification, a weak velocity shear in the layers, and uneven topography. We obtain smooth steady-state soliton-like solutions of the proposed model in the form of symmetric and non-symmetric waves of mode-2 adjoining to a given multilayer constant flow. Numerical calculations of the generation and propagation of large-amplitude internal waves are carried out using both the proposed first-order system and Green–Naghdi-type equations. It is established that the solutions of these models practically coincide. The advantage of the first-order equations is the simplicity of numerical implementation and a significant reduction in the calculation time. We show that the results of numerical simulation are in good agreement with the experimental data on the evolution of mode-2 solitary waves in tanks of constant and variable height.

Multiscale Wave-Turbulence Dynamics in the Atmosphere and Ocean

The atmosphere and oceans present an ongoing first-rate challenge to science and mathematics because they operate on an extremely broad ranges of scales, from molecular to planetary in length and from below seconds to millennia in time. This is the reason why climate simulations still suffer from leading-order uncertainties. Conceptual simplifications, such as scale-separation assumptions and the neglect of many physical processes, have enabled past progress in understanding the interactions of the basic dynamic constituents, i.e. large-scale mean flows, medium-scale waves and vortices, and small-scale turbulence. But present-day research is stretching the validity of this framework. For example, it is recognized that intermediate-scale waves and vortices are key elements linking all relevant players, and are often characterized by nonlinear interactions on comparable scales and also by additional physical nonlinearities due to effects such as air moisture. Motivated by recent advances in mathematical wave–vortex and wave–wave interaction theory, turbulence theory, and the study of internal wave dynamics as well as their numerical parametrization, the workshop gathered leading experts in these fields to foster a synthesis of new approaches and thereby a new level of understanding and numerical treatment of climate dynamics.

Internal solitary waves generated by a moving bottom disturbance

The strongly nonlinear Miyata–Choi–Camassa model under the rigid lid approximation (MCC-RL model) can describe accurately the dynamics of large-amplitude internal waves in a two-layer fluid system for shallow configurations. In this paper, we apply the MCC-RL model to study the internal waves generated by a moving body on the bottom. For the case of the moving body speed $U=1.1c_{0}$ , where ${c_0}$ is the linear long-wave speed, the accuracy of the MCC-RL results is assessed by comparing with Euler's solutions, and very good agreement is observed. It is found that when the moving body speed increases from $U=0.8c_{0}$ to $U=1.241c_{0}$ , the amplitudes of the generated internal solitary waves in front of the moving body become larger. However, a critical moving body speed is found between $U=1.241c_{0}$ and $U=1.242c_{0}$ . After exceeding this critical speed, only one internal wave right above the body is generated. When the moving body speed increases from $U=1.242c_{0}$ to $U=1.5c_{0}$ , the amplitudes of the internal waves become smaller.

Sensitivity of Seawater Temperature and Salinity to their Restoring Times, Appending in the Boundary Conditions for these Variables at the Free Surface of the Laptev Sea in the No-Ice Period

In order to study sensitivity of seawater temperature and salinity in the no-ice Laptev Sea to their restoring times, appending in the boundary conditions for these variables at the free surface, the 3D finite-element hydrostatic model QUODDY-4 and the indirect means of describing internal tidal waves have been applied. The latter is based on the use of the corrected eddy diffusivity representing the sum of the same uncorrected diffusivity and the diapycnal diffusivity. The first of them characterizes the influence of non-tidal factors, computed using 2.5-level eddy closure scheme, the second, determined by the ratio of internal tidal waves induced baroclinic tidal energy dissipation to the buoyancy frequency in square, the influence of purely tidal factor. The estimate of this dissipation obtained from a solution of auxiliary problem on the internal tidal waves dynamics and energetics is attached. Seawater temperature and salinity in the subsurface and near-bottom layers of the Sea and also their vertical distributions along the 120E meridional transection obtained for the strong, moderate, and mixed restoring times are discussed. As a result, it is clarified that seawater temperature and salinity are weakly sensitive to changes in the restoring time. The saying follows from a comparison of model averaged (over a chosen period and by the Sea area) values of seawater temperature and salinity, and from their local vertical profiles, computed with regard to accepted estimates of the restoring time.

Subsurface imbalance stimulated in a mesoscale eddy. Part I: Observations

This study presents observational evidence of a stimulated imbalance below the mixed layer from a fast-tow cruise survey with approximately 4 km resolution that traverses a cyclonic mesoscale eddy in the central South China Sea. The observed physical and biochemical properties reveal both geostrophic and ageostrophic physics in the mesoscale eddy. In addition to the well-known eddy-induced features, banded structures of temperature and chlorophyll anomaly across the section indicate unbalanced motions. These features are notably active near the eddy's edge and show larger amplitudes of variance than those induced by the mesoscale eddy. The results also show remarkable unbalanced variances of temperature and salinity, as well as lateral temperature-salinity compensation on isopycnals, likely due to stirring associated with internal waves. Spectral analysis suggests that there exist significant unbalanced variances of temperature and chlorophyll at ∼13 km. Our further analysis suggests that symmetric instability may have been stimulated due to strong lateral buoyancy gradients and vertical shear of velocity, which may account for the imbalance signature in the pycnocline. The observed imbalances point to the instability due to the interaction between mesoscale, submesoscale, and internal wave dynamics, which may play an important role in catalyzing the forward energy cascade in the ocean.

Impact of Vertical Mixing Parameterizations on Internal Gravity Wave Spectra in Regional Ocean Models

AbstractWe present improvements in the modeling of the vertical wavenumber spectrum of the internal gravity wave continuum in high‐resolution regional ocean simulations. We focus on model sensitivities to mixing parameters and comparisons to McLane moored profiler observations in a Pacific region near the Hawaiian Ridge, which features strong semidiurnal tidal beams. In these simulations, the modeled continuum exhibits high sensitivity to the background mixing components of the K‐Profile Parameterization (KPP) vertical mixing scheme. Without the KPP background mixing, stronger vertical gradients in velocity are sustained in the simulations and the modeled kinetic energy and shear spectral slopes are significantly closer to the observations. The improved representation of internal wave dynamics in these simulations makes them suitable for improving ocean mixing estimates and for the interpretation of satellite missions such as the Surface Water and Ocean Topography mission.

Ocean circulation over the Saya de Malha Bank in the South West Indian Ocean

The Saya de Malha Bank is one of the major banks of the Mascarene Plateau in the South West Indian Ocean. It is known for its unique ecosystem, remoteness (nearest island is about 300 km away) and complex oceanographic conditions. This study presents the results of a survey conducted in May 2018 on-board the R/V Dr Fridtjof Nansen, and aims to provide a descriptive overview of the interaction of the large-scale South Equatorial Current (SEC) over the shallow Saya de Malha Bank. The analysis of the current pattern revealed a two-layered structure of the current over the shallow topography of the bank compared to the vertically rigid-structure of the SEC throughflow in its deeper region. This two-layered flow consists of a surface layer and a sub-thermocline layer. The top layer flow, carrying the lower salinity mass (Tropical Surface Water) is driven by the Ekman dynamics observed in the southern hemisphere whereas the sub-thermocline current layer is most likely governed by the tidal and internal wave dynamics generated by the topographic relief of the bank.

Diurnal and semidiurnal internal waves on the southern slope of the Yermak Plateau

The Yermak Plateau (YP) is located across the Arctic–Atlantic gateway in the northwest of the Svalbard archipelago. In this region, internal waves are believed to cause intense turbulent mixing and hence influence the heat budget in the Arctic Ocean. Based on year-long observations from three moorings, the characteristics and energetics of diurnal and semidiurnal internal waves on the southern slope of the YP are investigated in this study. Diurnal internal waves induce large isothermal displacements exceeding 100 m, which are nearly one order of magnitude greater than those of semidiurnal internal waves. In addition, diurnal internal waves are strong in winter but weak in summer, while the semidiurnal internal waves exhibit complicated temporal variation. For the diurnal internal waves, their available potential energy is greater than the horizontal kinetic energy; whereas the situation is opposite for the semidiurnal ones. This feature is further clarified with two-dimensional numerical simulations. Due to the larger tidal excursion, diurnal tidal forcing yields the generation of stronger higher harmonics, i.e., the semidiurnal internal waves. In contrast, higher harmonics are rather weak under the semidiurnal forcing. Moreover, a large proportion of energy for both diurnal and semidiurnal internal waves is dissipated locally. Results of this study can provide useful insight on the dynamics of internal waves in the Arctic Ocean.