Interfacial waves with free-surface boundary conditions: an approach via a model equation

Dias, Frédéric; Ильичев, А. Т.

doi:10.1016/s0167-2789(01)00149-x

Cited by 22 publications

(47 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Choi and Camassa (1999) studied large amplitude interfacial waves, again with rigid lid upper boundary conditions. Dias and Ilichev (2001) modeled both classical and generalized interfacial solitary waves beneath a linear free surface, and Parau and Dias (2001) numerically computed such solutions. Finally, Craig et al (2004Craig et al ( , 2005a consider both the cases of rigid lid and free surface upper boundary conditions in the long-wave scaling regime.…”

Section: Introductionmentioning

confidence: 99%

Coupling between internal and surface waves

2010

View full text Add to dashboard Cite

In a fluid system in which two immiscible layers are separated by a sharp free interface, there can be strong coupling between large amplitude nonlinear waves on the interface and waves in the overlying free surface. We study the regime where long waves propagate in the interfacial mode, which are coupled to a modulational regime for the freesurface mode. This is a system of Boussinesq equations for the internal mode, coupled to the linear Schrödinger equations for wave propagation on the free surface, and respectively a version of the Korteweg-de Vries equation for the internal mode in case of unidirectional motions. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler's equations, as described in (Benjamin and Bridges, J the perturbation theory for the modulational regime that is given in (Craig et al. to appear). We focus in particular on the situation in which the internal wave gives rise to localized bound states for the Schrödinger equation, which are interpreted as surface wave patterns that give a characteristic signature of the presence of an internal wave soliton. We also comment on the discrepancies between the free interface-free surface cases and the approximation of the upper boundary condition by a rigid lid.

show abstract

Section: Introductionmentioning

confidence: 99%

Coupling between internal and surface waves

2010

View full text Add to dashboard Cite

show abstract

“…The plateaus can become infinitely long; therefore, the flow in the far field and the flow in the middle can be referred to as parallel conjugate flows [30,29]. From this point of view, we can calculate the limiting value of F by solving the following algebraic equation:

false(8 + R false) F^{6} - 12 false(1 + H false) F^{4} + 2 false(3 - R false) {false(1 + H false)}^{2} F^{2} - false(1 - R false) {false(1 + H false)}^{3} = 0.

This relation was first given by Dias & Il'ichev, and readers are referred to [16] for the detailed derivation (they derived (3.11) in the absence of an elastic cover, but their result remains valid for our problem, because the elastic sheet does not affect the result when the free surface and the interface are flat). The theoretic limiting value given by (3.11) is 0.1849, which is in agreement with the numerical result (≈0.1836) as shown in figure 7 a .…”

Section: Numerical Resultsmentioning

confidence: 99%

Numerical study of interfacial solitary waves propagating under an elastic sheet

et al. 2014

View full text Add to dashboard Cite

Steady solitary and generalized solitary waves of a two-fluid problem where the upper layer is under a flexible elastic sheet are considered as a model for internal waves under an ice-covered ocean. The fluid consists of two layers of constant densities, separated by an interface. The elastic sheet resists bending forces and is mathematically described by a fully nonlinear thin shell model. Fully localized solitary waves are computed via a boundary integral method. Progression along the various branches of solutions shows that barotropic (i.e. surface modes) wave-packet solitary wave branches end with the free surface approaching the interface. On the other hand, the limiting configurations of long baroclinic (i.e. internal) solitary waves are characterized by an infinite broadening in the horizontal direction. Baroclinic wave-packet modes also exist for a large range of amplitudes and generalized solitary waves are computed in a case of a long internal mode in resonance with surface modes. In contrast to the pure gravity case (i.e without an elastic cover), these generalized solitary waves exhibit new Wilton-ripple-like periodic trains in the far field.

show abstract

“…As a result, we can reduce the set of conditions in Theorem 1 to 0. The case of = 0 separating the two scenarios (two or four real solutions) corresponds to the special case for which the straight line described by (7) becomes tangent to the curve (5) (see [9]).…”

Section: A Geometrical Formulation Of the Problemmentioning

confidence: 99%

“…On the other hand, the presence of a top free surface might have some important effects that cannot be captured by the rigid-lid model. One example is the generalized solitary waves that can only exist for the free-surface case (see [6,7]). In this case, internal solitary waves with multi-humped profiles have been observed by Barros and Gavrilyuk [8].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

Barros

Choi

2009

Stud Appl Math

View full text Add to dashboard Cite

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin-Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.

show abstract

Interfacial waves with free-surface boundary conditions: an approach via a model equation

Cited by 22 publications

References 22 publications

Coupling between internal and surface waves

Coupling between internal and surface waves

Numerical study of interfacial solitary waves propagating under an elastic sheet

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

Contact Info

Product

Resources

About