Interaction between small-scale surface waves and large-scale internal waves

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.

The various mechanisms producing internal waves in estuaries are reviewed and a short discussion of lee waves is given. Elementary analytical methods are applied to demonstrate the interaction between large-scale internal waves and turbulence in shallow estuaries. The production of turbulent kinetic energy owing to long internal waves is calculated for horizontal and undulating beds. The relevance to the numerical modelling of estuarine flows is briefly outlined. Two-layer models are discussed in some detail, and it is shown that including friction introduces an instability beyond a certain internal Froude number.

Dynamics of surface and internal long waves generated by atmospheric pressure disturbances

<p style='text-indent:20px;'>We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.</p>

Two-dimensional weakly nonlinear surface gravity–capillary waves in an ideal fluid of finite water depth are considered and nonlinear evolution equations which are correct up to the third order of wave steepness are derived including the applied pressure on the free surface. Since no assumptions are made on the length scales, the equations can be applied to a fluid of arbitrary depth and to disturbances with arbitrary wavelength. For one-dimensional gravity waves, these evolution equations are reduced to those derived by Matsuno (1992). Most of the known equations for surface waves are recovered from the new set of equations as special cases. It is shown that one set of equations has a Hamiltonian formulation and conserves mass, momentum and energy. The analysis for irrotational flow is extended to two-dimensional uniform shear flow.

This paper focuses on the study of linear and non-linear surface and internal waves, in a complete setting, using a two-layer model of a stratified fluid. The respective Korteweg-de Vries evolutionary equations have been obtained, analysed, and compared with the ‘rigid lid’ model data. Boussinesq-type equations have been derived for the interacting modes pertaining to one type and to different types. It is shown that in addition to the known mechanisms of interaction between internal and surface waves, interaction between long non-linear baroclinic modes and barotropic modes, propagating in the same direction, is likely in such a system.

Instability observations associated with wave breaking in the stable-stratified deep-ocean

Numerical solutions of surface and internal solitary waves are obtained through a new method, where advection equations on physical quantities including surface/interface displacements and velocity potential are solved to find convergent solutions by applying the Newton-Raphson method. The nonlinear wave equations derived using a variational principle are adopted as the fundamental equations in the present study. Surface and internal solitary waves obtained through the proposed method are compared with the corresponding theoretical solutions, as well as numerical solutions of the Euler equations, to verify the accuracy of solutions through the proposed method especially for internal solitary waves of large amplitude progressing at a large celerity with a flattened wave profile. Properties of surface and internal solitary waves are discussed considering vertical distribution of horizontal and vertical velocity, as well as kinetic and potential energy. Numerical simulation using a time-dependent model has also been performed to represent propagation of surface and internal solitary waves with permanent waveforms.

The nonhydrostatic pressure effects on the generation and propagation of wind-forced internal waves are studied with a two-dimensional numerical ocean model. A one-way directed wind pulse over a stratified ocean initiates surface and internal waves in a closed basin. The studies are performed with horizontal grid sizes in the range from 1 km to 62.5 m. The experiments are performed with both a hydrostatic and a nonhydrostatic model, facilitating systematic studies of the sensitivity of the numerical model results to the grid size and to the nonhydrostatic pressure adjustments. The results show that the nonhydrostatic pressure effects are highly dependent on the grid size and grow with increased resolution. In the internal depression wave, the horizontal nonhydrostatic pressure gradients reach the same order of magnitude as the hydrostatic gradients in the high-resolution nonhydrostatic studies. In these studies, the nonhydrostatic pressure gradients approximately balance the corresponding hydrostatic pressure gradients in the internal depression wave, and the wave degenerates into a train of soliton waves. The time for the soliton form to develop agrees with the steepening timescale calculated from Korteweg-de Vries theory. In the high-resolution hydrostatic model, the internal depression wave takes the form of a single wave front. When the internal waves are generated in the boundary layers, the nonhydrostatic pressure gradients are much smaller than the hydrostatic gradients and the generation processes are not effected by the nonhydrostatic pressure with the present range of grid sizes.

Acoustic fields in shallow water have a statistical nature due to complex, time-evolving sound speed fields, and scattering from rough boundaries. Previously, coupled-mode transport theory [Raghukumar and Colosi (2012)] was applied to high frequency acoustic fluctuations in an environment typical of the Shallow Water 2006 (SW06) experiment on the New Jersey continental shelf. As a consequence of the strong adiabatic component in SW06 propagation, a hybrid approach was used to calculate mode coherences where mode energies from the Dozier-Tappert approach were combined with adiabatic phase terms. Mode energies, coherences and acoustic intensities were examined, and it was found that internal and surface waves preferentially couple low and high modes respectively. Here, we extend that study to include higher moments such as scintillation index and shift focus to modes that are coupled by both internal and surface waves. 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.

Nonlinear dynamics of surface and internal waves in a stratified ocean under the influence of the Earth's rotation is discussed. Attention is focussed upon guided waves long compared to the ocean depth. The effect of rotation on linear processes is reviewed in detail as well as the existing nonlinear models describing weakly and strongly nonlinear dynamics of long waves. The influence of rotation on small-scale waves and two-dimensional effects are also briefly considered. Some estimates of the influence of the Earth's rotation on the parameters of real oceanic waves are presented and related to observational and numerical data.

<p>The Angolan and Peruvian shelves are located in upwelling regions along the eastern boundaries of the tropical Atlantic and Pacific Oceans. They are sites of important fisheries supported by high productivity which is driven by fluxes of nutrients from deep to near surface water along the coast. Mixing associated with internal waves is believed to play a role in this process. Recent field observations have shown the presence of an active internal wave field that includes internal solitary waves. In this talk results of high-resolution two-dimensional simulations of internal wave generation by tide-topography interactions on the Angolan and Peruvian shelves are presented. The simulations show the generation of internal wave beams at near-critical slopes and the generation of high-frequency internal solitary waves. The high-frequency IW spectrum is enhanced when small scale bathymetric ripples are included. Wave generation during winter and summer stratifications will be compared.</p>

The velocity potentials of a point source moving at a constant velocity in the upper layer of a two-layer fluid are obtained in a form amenable to numerical integration. Each fluid layer is of finite depth, and the density difference between the two layers is not necessarily small. The far-field asymptotic behavior of the surface waves and internal waves are also derived using the method of stationary phase. They show that the wave system at the free surface or at the interface each contains contributions from two different modes: a surface-wave mode and an internal-wave mode. When the density difference between the two layers is small or the depth of the upper layer is large, the surface-wave mode mainly affects the surface waves while the internal-wave mode mainly affects the internal waves. However, for large density difference, both modes contribute to the surface wave or internal wave system. For each mode, both divergent and transverse waves are present if the total depth Froude number is less than a certain critical Froude number which is mode-dependent. For depth Froude number greater than the critical Froude number, only divergent waves exist for that mode. This classification is similar to that of a uniform fluid of finite depth, where the critical Froude number is simply unity. The surface waves and internal waves are also calculated using the full expressions of the source potentials. They further confirm and illustrate the features observed in the asymptotic analysis.

In this review we consider the propagation of nonlinear waves in magnetically structured plasmas. We start from studying nonlinear surface waves on magnetic interfaces and give a derivation of the equation governing propagation of these waves in incompressible plasmas. We present the results of a numerical solution of this equation. Then we discuss various generalizations of this equation.We proceed to study slow nonlinear waves in magnetic slabs. We present the Benjamin-Ono equation describing nonlinear slow sausage surface waves in magnetic slabs, and discuss its solutions in the form of algebraic solitons. Then we consider nonlinear slow body waves and write down the governing equation for these waves. We discuss its numerical solution which shows that the evolution of an initial perturbation results in a gradient catastrophe.After that we consider nonlinear slow sausage surface and slow body waves in magnetic tubes, which are similar to the corresponding counterparts in magnetic slabs. We present a system of two equations describing both types of waves. We briefly discuss the derivation of the Leibovich-Roberts equation for slow sausage surface waves from this system of equations, and present the results of its numerical study. Then we give a brief description of the derivation of the governing equation for slow body waves, which is very similar to its counterpart for slow body waves in magnetic slabs and has very similar properties. We complete this part of the review with presenting the governing equation for non-axisymmetric surface waves in a thin magnetic tube in an incompressible plasma.In the last part of the paper we consider the damping of nonlinear waves in weakly dissipative magnetically structured plasmas. We briefly discuss the application of the obtained results to the problem of heating of the solar corona.KeywordsSurface WaveSolitary WaveSolar PhysMagnetic TubeKink ModeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Kolmogorov’s theory of turbulence in an incompressible fluid of constant density at high Reynolds numbers has provided a cornerstone for the interpretation of oceanic spectral measurements of turbulence. The most convincing verification of the theory came from observations by Grant, Stewart and Moilliet under conditions that clearly satisfy the basic premises of the theory, but subsequent measurements have explored the influence of ambient stratification and shear on both the energy and temperature spectra. As the turbulence decays, the larger scales of motion interact more weakly as do internal waves, so that Kolmogorov’s cascade becomes disrupted. It has long been known that the existence of a k -5/3 region in the spectrum does not require local isotropy and it is indicated that the success of Kolmogorov scaling in collapsing measured spectra in the dissipation range, does not require a continuing energy cascade from larger scales. Several questions remain unresolved, particularly the reasons for the shape of temperature spectra that have been measured in turbulence generated by large-scale internal waves in a tidal channel.