Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves

Ibragimov, Ranis N.

doi:10.1016/j.cnsns.2006.06.011

Cited by 8 publications

(4 citation statements)

References 23 publications

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“…For example, in the particular case when the Earth rotation is ignored, f ¼ 0, these equations have been studied by Tabaei [30,31] (colliding and reflecting internal wave beams), Lombard and Riley [23] (linearized stability of interacting exact solutions) and Dauxois and Young [7] (reflection of internal waves off a near-critical slope). The case of nonzero earth rotation, f -0, has been considered in our previous studies [13] to model weakly non-linear wave interactions.…”

Section: Model Equationsmentioning

confidence: 99%

“…The characteristic equation kdx þ mdz ¼ 0 of the first equation in (13) yields that the operator mX 2 À kX 3 has, along with t; v; q; and w, the following invariant:…”

Section: Symmetries and An Invariant Solutionmentioning

confidence: 99%

“…In the simplest case when explicit viscosity and diffusion terms are ignored and the Boussinesq approximation is employed, the governing equations of uniformly stratified incompressible fluid are (Ibragimov [13,14]) oũ ot…”

Section: Model Equationsmentioning

confidence: 99%

See 2 more Smart Citations

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Ibragimov¹,

Ibragimov²

2010

Communications in Nonlinear Science and Numerical Simulation

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Section: Model Equationsmentioning

confidence: 99%

“…The characteristic equation kdx þ mdz ¼ 0 of the first equation in (13) yields that the operator mX 2 À kX 3 has, along with t; v; q; and w, the following invariant:…”

Section: Symmetries and An Invariant Solutionmentioning

confidence: 99%

See 1 more Smart Citation

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Ibragimov¹,

Ibragimov²

2010

Communications in Nonlinear Science and Numerical Simulation

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“…Resonant triads are the basic building blocks for understanding mode coupling in systems of weakly interacting dispersive waves in which the leading order nonlinearity of the underlying wave equation is quadratic in the wave amplitude, ψ. Physical examples of such systems include capillary waves on fluid interfaces [1], Rossby waves in geophysical fluid dynamics [2,3], mode coupling in nonlinear optics [4], drift waves in magnetized plasmas [5] and internal waves in stratified fluids [6,7]. For simplicity, let us suppose we can represent such wave fields in terms of Fourier harmonics.…”

Section: Introductionmentioning

confidence: 99%

Externally forced triads of resonantly interacting waves: Boundedness and integrability properties

Harris

Bustamante

Connaughton

2012

Communications in Nonlinear Science and Numerical Simulation

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We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, H, is zero we reduce this dynamical system to the motion of a particle in a one-dimensional time-independent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when H = 0 the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When H = 0 we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincaré sections of this system provide strong evidence that the motion remains bounded when H = 0 and is typically quasi-periodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics.

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Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System

Hu¹,

Xiao²,

Deng³

2023

Geometric Mechanics and Its Applications

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Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves

Cited by 8 publications

References 23 publications

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Externally forced triads of resonantly interacting waves: Boundedness and integrability properties

Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System

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