Internal Wave Model Research Articles

The effect of statistical wind corrections on global wave forecasts

The skill of modern wave models is such that the quality of their forecasts is, to a large degree, determined by errors in the forcing wind field. This work explores the extent to which large-scale systematic biases in modelled waves from a third generation wave model can be attributed to the forcing winds. Three different sets of winds with known global bias characteristics are used to force the WAVEWATCH III model. These winds are based on the Australian Bureau of Meteorology’s ACCESS model output, with different statistical corrections applied. Wave forecasts are verified using satellite altimeter data. It is found that a negative bias in modelled Significant Wave Height (Hs) has its origins primarily in the forcing, however, the reduction of systematic wind biases does not result in universal improvement in modelled Hs. A positive bias is present in the Southern Hemisphere due primarily to an overestimation of high Hs values in the Southern Ocean storm tracks. A positive bias is also present in the east Pacific and East Indian Ocean. This is due both to the over-prediction of waves in the Southern Ocean and lack of swell attenuation in the wave model source terms used. Smaller scale features are apparent, such as a positive bias off the Cape of Good Hope, and a negative bias off Cape Horn. In some situations, internal wave model error has been compensated for by error in the forcing winds.

Numerical Simulation of a Weakly Nonlinear Model for Internal Waves

Abstract Internal waves arise in a wide array of oceanographic problems of both theoretical and engineering interest. In this contribution we present a new model, valid in the weakly nonlinear regime, for the propagation of disturbances along the interface between two ideal fluid layers of infinite extent and different densities. Additionally, we present a novel high-order/spectral algorithm for its accurate and stable simulation. Numerical validation results and simulations of wave-packet evolution are provided.

The problem of sound propagation through the fluctuating ocean and the interplay between ocean acoustics and physical oceanography

While the ocean provides a particularly amenable environment for acoustic remote sensing, navigation, and communication, the confounding effects of fluctuations have bedeviled ocean acousticians for decades. To attack this problem the communities of ocean acoustics and physical oceanography became new bedfellows, with the development of ocean acoustic tomography and path-integral/moment-transport theories driven by internal-wave models being major accomplishments for describing low and high-frequency fluctuations. With the advent of relatively inexpensive oceanographic mooring equipment, experiments in the last decades have deployed nearly as many oceanographic instruments as acoustic instruments, thus leading to fundamental discoveries concerning internal tides, solitons, random internal waves and spicy thermohaline structure. These measurements are in turn providing strong constraints on acoustic fluctuation theories and Monte Carlo simulations used to interpret acoustic observations. This presentation will give a review of what has been learned about ocean sound-speed structure and how this information can be better integrated into acoustic fluctuations calculations. It will be shown that acoustic fluctuation theory has developed to the point in which reasonable inversions for internal-wave parameters are now possible.

Transport theory applied to shallow water acoustics: The relative importance of surface scattering and linear internal waves

Acoustic fields in shallow water have a statistical nature due to complex, time-evolving sound speed fields and scattering from rough boundaries. A coupled-mode transport theory [Creamer (1996), Colosi and Morozov (2009)] allows for the prediction of acoustic field second moments like mean intensity and coherence. This was previously applied to study low frequency acoustic fluctuations in an environment typical of that of the Shallow Water 2006 (SW06) experiment on the New Jersey Continental shelf. Here the propagation was found to be strongly adiabatic and random sound speed fluctuations from internal waves radically altered acoustic interactions with intense nonlinear internal wave packets. Here, we extend the SW06 study to examine the ability of transport equations to describe high frequency (>1 kHz) sound in shallow water. Mode coupling rates from internal waves are expected to be larger, and scattering effects from rough surfaces need treatment. The aforementioned transport theory is merged with the rough surface scattering transport theory of Thorsos et al (2009). Oceanographic and sea surface measurements are used to constrain the internal wave and sea surface models. The relative importance of linear internal waves and surface scattering effects are studied using transport theory and Monte Carlo simulations.

A Model for Large-Amplitude Internal Waves with Finite-Thickness Pycnocline

We derive a new model of internal solitary wave propagation governed by the Euler equations for a continuously stratified, incompressible fluid in the Boussinesq approximation. The model is derived by a systematic joint asymptotic expansion for long wavelengths and thin pycnoclines. The resulting solutions extend those of the strongly nonlinear two-layer system (Choi and Camassa in J. Fluid Mech. 396:1–36, 1999) to continuous stratifications, and capture non-trivial features of the internal pycnocline structure. These solutions can be used, among other applications, to perform shear instability calculations in analytic form.

Trapped Core Formation within a Shoaling Nonlinear Internal Wave

Abstract Large-amplitude (100–200 m) nonlinear internal waves (NLIWs) were observed on the continental slope in the northern South China Sea nearly diurnally during the spring tide. The evolution of one NLIW as it propagated up the continental slope is described. The NLIW arrived at the slope as a nearly steady-state solitary depression wave. As it propagated up the slope, the wave propagation speed C decreased dramatically from 2 to 1.3 m s−1, while the maximum along-wave current speed Umax remained constant at 2 m s−1. As Umax exceeded C, the NLIW reached its breaking limit and formed a subsurface trapped core with closed streamlines in the coordinate frame of the propagating wave. The trapped core consisted of two counter-rotating vortices feeding a jet within the core. It was highly turbulent with 10–50-m density overturnings caused by the vortices acting on the background stratification, with an estimated turbulent kinetic energy dissipation rate of O(10−4) W kg−1 and an eddy diffusivity of O(10−1) m2 s−1. The core mixed continually with the surrounding water and created a wake of mixed water, observed as an isopycnal salinity anomaly. As the trapped core formed, the NLIW became unsteady and dissipative and broke into a large primary wave and a smaller wave. Although shoaling alone can lead to wave fission, the authors hypothesize that the wave breaking and the trapped core evolution may further trigger the fission process. These processes of wave fission and dissipation continued so that the NLIW evolved from a single deep-water solitary wave as it approached the continental slope into a train of smaller waves on the Dongsha Plateau. Observed properties, including wave width, amplitude, and propagation speed, are reasonably predicted by a fully nonlinear steady-state internal wave model, with better agreement in the deeper water. The agreement of observed and modeled propagation speed is improved when a reasonable vertical profile of background current is included in the model.

PHYSICAL PARAMETERS OF THE RELATIVISTIC SHOCK WAVES IN GRBs: THE CASE OF 30 GRBs

Using the modified internal shock wave model, we fit the gamma ray burst (GRB) light and spectral curves of 30 GRBs observed with BATSE. From the best fitting, we obtain basic parameters of the relativistic shells which are in good agreement with predictions given earlier. We compare measured GRB parameters with those obtained from the model and discuss connections between them in the frame of the physical processes laying behind GRB events.

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

Modeling and Prediction of Internal Waves in the South China Sea

Abstract : Nonlinear internal solitary waves generated within Luzon Strait move westward across the northern South China Sea, refract around Dongsha Atoll, and dissipate on the Chinese continental shelf after a journey of over 500 km lasting more than four days. In the last 10 years a great deal of observational, theoretical, and modeling effort has been directed toward understanding and predicting these solitary waves and their effects on the oceanography of the northern South China Sea. This paper reviews a variety of modeling approaches (two- and three-dimensional, kinematic, hydrostatic, and nonhydrostatic) that have been employed to gain insight into the generation mechanisms and physics of the South China's Sea's nonlinear solitary waves with the goal of predicting wave characteristics such as phase speed, amplitude, and arrival time.

Propagation speeds of strongly nonlinear near-surface internal waves in the Strait of Georgia

[1] A novel aerial observational method for studying internal features in the coastal ocean is developed and tested in a study of large nonlinear internal solitary-like waves. Photogrammetrically rectified oblique photo images from a circling aircraft are used to track a number of internal wave packets for periods of up to one hour in the Strait of Georgia, British Columbia, Canada. Combining these sequences with coincident water column data allows us to obtain a more complete view of the spatial structure of internal waves. Highly accurate measurements of wave propagation speeds and directions are possible. The applicability of various weakly nonlinear theories in modeling propagation of the observed large-amplitude waves is tested. The measured wave speeds enable us to differentiate between classic internal wave models. The linear, KdV (Korteweg-de Vries), and BO (Benjamin-Ono) models are applied with and without background shear. After background shear effects are included, it is found that a continuously stratified BO equation can predict propagation speeds within observational error, and that this is not true for other theories. The technique may be useful in future studies of oblique internal wave interactions.

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.

Variational principles for two kinds of extended Korteweg—de Vries equations

Variational principles are constructed using the semi-inverse method for two kinds of extended Korteweg—de Vries (KdV) equations, which can be regarded as simple models of the nonlinear oceanic internal waves and atmospheric long waves, respectively. The obtained variational principles have also been proved to be correct.

Three-dimensional numerical modeling of tsunami-related internal gravity waves in the Hawaiian atmosphere

The tremendous tsunami following the 2011 Tohoku Earthquake produced internal gravity waves (IGWs) in the neutral atmosphere and large disturbances in the overlying ionospheric plasma while propagating through the Pacific ocean. To corroborate the tsunamigenic hypothesis of these perturbations, we use a 3D numerical modeling of the ocean-atmosphere coupling, to reproduce the tsunami signature observed in the airglow by the imager located in Hawaii and clearly showing the shape of the modeled IGW. The agreement between data and synthetics not only supports the interpretation of the tsunami-related-IGW behavior, but strongly shows that atmospheric and ionospheric remote sensing can provide new tools for oceanic monitoring and tsunami detection.

The Generation and Evolution of Nonlinear Internal Waves in the Deep Basin of the South China Sea

Abstract Time series observations of nonlinear internal waves in the deep basin of the South China Sea are used to evaluate mechanisms for their generation and evolution. Internal tides are generated by tidal currents over ridges in Luzon Strait and steepen as they travel west, subsequently generating high-frequency nonlinear waves. Although nonlinear internal waves appear repeatedly on the western slopes of the South China Sea, their appearance in the deep basin is intermittent and more closely related to the amplitude of the semidiurnal than the predominant diurnal tidal current in Luzon Strait. As the internal tide propagates westward, it evolves under the influence of nonlinearity, rotation, and nonhydrostatic dispersion. The interaction between nonlinearity and rotation transforms the internal tide into a parabolic or corner shape. A fully nonlinear two-layer internal wave model explains the observed characteristics of internal tide evolution in the deep basin for different representative forcing conditions and allows assessment of differences between the fully and weakly nonlinear descriptions. Matching this model to a wave generation solution for representative topography in Luzon Strait leads to predictions in the deep basin consistent with observations. Separation of the eastern and western ridges is close to the internal semidiurnal tidal wavelength, contributing to intensification of the westward propagating semidiurnal component. Doppler effects of internal tide generation, when combined with a steady background flow, suggest an explanation for the apparent suppression of nonlinear wave generation during periods of westward intrusion of the Kuroshio.

Study of interaction between intense internal wave solitons in the context of the Choi-Camassa model

Based on the approximated theory developed by the authors to describe the interaction of complex solitons that are formed by kinks, i.e., drops of fields of different polarity, the features of the dynamics of intense internal wave solitons are studied in the context of nonintegrable Choi-Camassa equations. These equations describe the evolution of long internal waves with arbitrary amplitudes in a two-layer fluid. Making allowance for the complex character of the limiting amplitude solitons made it possible to study the dynamics of their interaction as longitudinal (not pointlike) formations for the first time. A series of the features of soliton interaction caused by the complexity of their structure is discovered and explained. These include the appearance of solitary waves with opposite polarity in the collision process, anomalously large phase shifts, etc. The existence of these features is verified by numerical experiments in the context of internal wave models permitting the existence of complex solitons.

INFLUENCE OF STOKES DRIFT ON SALT WEDGE INTRUSION EVALUATED USING FULLY-NONLINEAR AND STRONGLY-DISPERSIVE WAVE EQUATIONS

This paper describes the influence of surface waves on salt-wedge intrusion in terms of radiation stress. Radiation stress may increase salt-wedge intrusion when surface waves propagate up a river. This study thus aims to reveal the effect of radiation stress on the distance of salt-wedge intrusion by using fully nonlinear strongly dispersive internal wave equations and three-dimensional numerical computation model, Fantom3D. Fully nonlinear strongly dispersive internal wave model reveals the possibility that large radiation stress is induced near the river mouth and increases the distance of salt-wedge intrusion. Three-dimensional numerical model also demonstrates that there is a significant difference in the intrusion distance by taking into account radiation stress.

An iterative method to solve a regularized model for strongly nonlinear long internal waves

We present a simple iterative scheme to solve numerically a regularized internal wave model describing the large amplitude motion of the interface between two layers of different densities. Compared with the original strongly nonlinear internal wave model of Miyata [10] and Choi and Camassa [2], the regularized model adopted here suppresses shear instability associated with a velocity jump across the interface, but the coupling between the upper and lower layers is more complicated so that an additional system of coupled linear equations must be solved at every time step after a set of nonlinear evolution equations are integrated in time. Therefore, an efficient numerical scheme is desirable. In our iterative scheme, the linear system is decoupled and simple linear operators with constant coefficients are required to be inverted. Through linear analysis, it is shown that the scheme converges fast with an optimum choice of iteration parameters. After demonstrating its effectiveness for a model problem, the iterative scheme is applied to solve the regularized internal wave model using a pseudo-spectral method for the propagation of a single internal solitary wave and the head-on collision between two solitary waves of different wave amplitudes.

A unified linear theory of wavelike perturbations under general ocean conditions

A linearized instability analysis model with five unknowns was proposed to describe disturbance motions under general oceanic background conditions, including large-scale current shear, density stratification, frontal zone, and arbitrary topography. A unified linear theory of wavelike perturbations for surface gravity waves, internal gravity waves and inertial gravity waves was derived for the adiabatic case, and the solution was then found using Fourier integrals. In this theory, we discarded the assumptions widely accepted in the literature concerning derivations of wave motions such as the irrotationality assumption for surface gravity waves, the rigid-lid approximation for internal gravity waves, and the long-wave approximation for inertial gravity waves. Analytical solutions based on this theory indicate that the complex dispersion relationships between frequency and wave-number describing the propagation and development of the three types of wavelike perturbation motions include three components: complex dispersion relationships at the sea surface; vertical invariance of the complex frequency; and expressions of the vertical wave-number (phase). Classical results of both surface waves and internal waves were reproduced from the unified theory under idealized conditions. The unified wave theory can be applied in the dynamical explanation of the generation and propagation properties of internal waves that are visible in the satellite SAR images in the southern part of the China Seas. It can also serve as the theoretical basis for both a numerical internal-wave model and analytical estimation of the ocean fluxes transported by wavelike perturbations.

Experimental demonstration of the utility of pressure sensitivity kernels in time-reversal

Pressure sensitivity kernels were recently applied to time-reversal acoustics in an attempt to explain the enhanced stability of the time-reversal focal spot [Raghukumar et al., J. Acoust. Soc. Am. 124, 98-112 (2008)]. The theoretical framework developed was also used to derive optimized source functions, closely related to the inverse filter. The use of these optimized source functions results in an inverse filter-like focal spot which is more robust to medium sound speed fluctuations than both time-reversal and the inverse filter. In this paper the theory is applied to experimental data gathered during the Focused Acoustic Fields experiment, conducted in 2005, north of Elba Island in Italy. Sensitivity kernels are calculated using a range-independent sound-speed profile, for a geometry identical to that used in the experiment, and path sensitivities are identified with observed arrivals. The validity of the kernels in tracking time-evolving Green's functions is studied, along with limitations that result from a linearized analysis. An internal wave model is used to generate an ensemble of sound speed profiles, which are then used along with the calculated sensitivity kernels to derive optimized source functions. Focal spots obtained using the observed Green's functions with these optimized source functions are then compared to those obtained using time-reversal and the inverse-filter. It is shown that these functions are able to provide a focal spot superior to time-reversal while being more robust to sound speed fluctuations than the inverse filter or time-reversal.

A continuously stratified nonlinear model for internal solitary waves in the northern South China Sea

A continuously stratified nonlinear model is set up to study the impact of topographical character on the generation of internal solitary waves over a sill by tidal flow. One of the reasons why almost all of the generated internal solitary waves propagate westward in the northern South China Sea is explained. The model simulations describe the generation and propagation of internal waves well. When the strength of imposed barotropic tides and the water stratification stay unchanged, the steepness of the sill slope can control both (a) whether or not the waves induced over a sill by tidal flow are linear internal waves or nonlinear internal solitary waves, and (b) the amplitude of the internal solitary waves generated. If the steepness of the sill is asymmetric, the nonlinear internal solitary waves may be induced on the steeper side of the sill. These conclusions are supported by a numerical experiment with a monthly-mean stratification and an actual seafloor topography from the Luzon Strait.