Solitary Wave Solution of Flat Surface Internal Geophysical Waves with Vorticity

A fluid system bounded by a flat bottom and a flat surface with an internal wave and depth-dependent current is considered. The Hamiltonian of the system is presented and the dynamics of the system are discussed. A long-wave regime is then considered and extended to produce a KdV approximation. Finally, a solitary wave solution is obtained

The Dynamics of Flat Surface Internal Geophysical Waves with Currents

A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.Comment: LaTeX, 21 pages, 1 figure, available online in J. Math. Fluid Mech. (2016

Hamiltonian Approach to Internal Wave-Current Interactions in a Two-Media Fluid with a Rigid Lid

We examine a two-media 2-dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface with wind generated surface waves but considered bounded above by a lid by an assumption that surface waves have negligible amplitude. An internal wave driven by gravity which propagates in the positive $x$-direction acts as a free common interface between the media. The current is such that it is zero at the flatbed but a negative constant, due to an assumption that surface winds blow in the negative $x$-direction, at the lid. We are concerned with the layers adjacent to the internal wave in which there exists a depth dependent current for which there is a greater underlying than overlying current. Both media are considered incompressible and having non-zero constant vorticities. The governing equations are written in canonical Hamiltonian form in terms of the variables, associated to the wave (in a presence of a constant current). The resultant equations of motion show that wave-current interaction is influenced only by the current profile in the 'strip' adjacent to the internal wave.Comment: 13 pages, 1 figur

Fluid-Dynamic Models of Geophysical Waves

Geophysical waves are waves that are found naturally in the Earth\u27s atmosphere and oceans. Internal waves, that is waves that act as an interface between uids of dierent density, are examples of geophysical waves. A uid system with a at bottom, at surface and internal wave is initially considered. The system has a depth-dependent current which mimics a typical ocean set-up and, as it is based on the surface of the rotating Earth, incorporates Coriolis forces. Using well established uid dynamic techniques, the total energy is calculated and used to determine the dynamics of the system using a procedure called the Hamiltonian approach. By tuning a variable several special cases, such as a current-free system, are easily recovered. The system is then considered with a non- at bottom. Approximate models, including the small amplitude, long-wave, Boussinesq, Kaup-Boussinesq, Korteweg-de Vries (KdV) and Johnson models, are then generated using perturbation expansion techniques, that is using small arbitrary parameters. Solutions are obtained that model waves that move without change of form called solitary waves. These waves can be referred to as solitons when their particle-like behaviour is considered. The Johnson model is used to model the `birth\u27 of new solitons when a single soliton hits an underwater ramp. The presented models have applications for climatologists, meteorologists, oceanographers, marine engineers, marine biologists and applied mathematicians

Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom

A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth.Comment: 18 pages, 6 figures, 1 tabl

Models of Internal Waves in the Presence of Currents

A fluid system consisting of two domains is examined. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. An internal wave propagating in one direction, driven by gravity, acts as a free common interface between the fluids. Various current profiles are considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are formulated. The presented models provide potential applications to modelling of internal geophysical waves

Benjamin-Ono Model of an Internal Wave Under a Flat Surface

A two-layer uid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is infinitely deep, with a higher density than the upper layer which is bounded above by a at surface. The uids are incompressible and inviscid. A Hamiltonian formulation for the dynamics in the presence of a depth-varying current is presented and it is shown that an appropriate scaling leads to the integrable Benjamin-Ono equation

Surface profile prediction from bottom pressure measurements with application to marine current generators

A knowledge of wave kinematics is requisite for most aspects of marine engineering, yet still relatively little is known in the context of wave-current interactions. There is a requirement for wavepower applications to predict the wavefield at a Wave Energy Converter (WEC), particularly for the application of control algorithms. For marine current generators a similar requirement arises. The modelling of wave-current interactions possesses a rich history, yet the presence of vorticity immediately introduces major mathematical complications into modelling considerations. Improving our understanding of the relationship between the dynamic pressure function and the underlying fluid kinematics for rotational ocean waves has implications for both wave and tidal resource characterisation. This paper considers the pressure-streamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. Using a novel pressure-streamfunction reformulation of the governing equations, an explicit formula was recently derived by two of the authors in terms of series solutions detailing the relationship between the pressure, streamfunction and the vorticity distributions. In particular, for linear waves, a description is provided of the role which the pressure function on the sea-bed plays in determining the free- surface profile elevation. These are the first results in this direction for water waves with vorticity, and it is shown that this approach provides a good approximation for a range of current conditions

Surface waves over currents and uneven bottom

The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth. Emergence of new solitons is observed as a result of the wave interaction with the uneven bottom