Analytical solutions of long nonlinear internal waves: Part I

Hamdi, Samir; Morse, Brian; Halphen, Bernard; Schiesser, William E.

doi:10.1007/s11069-011-9757-0

Cited by 28 publications

(16 citation statements)

References 30 publications

(30 reference statements)

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“…This section describes a model of nonlinear internal waves following [8], [7]. As Figure 1 shows, an internal wave is a wave that travels beneath the surface of the ocean, along a pycnocline, which is a surface of constant water density.…”

Section: A Modeling Of Nonlinear Internal Wavesmentioning

confidence: 99%

“…However, nonlinear waves are needed to account for the advection required for larval transport [1]. Many models exist for nonlinear waves [7], [8] to account for the wide variety of conditions and bathymetries found in the ocean. Scientists widely use drogues drifting passively as monitoring platforms to gather relevant ocean data [9], [10], [11].…”

Section: Introductionmentioning

confidence: 99%

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Robust estimation and aggregation of ocean internal wave parameters using Lagrangian drifters

Ouimet

Cortés

2014

2014 American Control Conference

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This paper considers a group of drogues whose objective is to estimate the physical parameters that determine the dynamics of ocean nonlinear internal waves. While underwater, individual drogues do not have access to absolute position information and only rely on inter-drogue measurements. Building on this data and the structure of the drogue dynamics under the flow induced by an internal wave, this paper improves in three different ways upon our previous strategy, termed PARAMETER DETERMINATION STRATEGY, which determines all wave parameters. The first is by showing that with sufficiently fast sampling, the extended algorithm determines the wave parameters. The second is by showing its applicability to situations where two internal waves are present simultaneously. With the extended algorithm, multiple estimates are calculated for each parameter. Thus, with the presence of noisy measurements, the third contribution is a method to aggregate parameter estimates to reduce the error. Simulations illustrate the algorithm performance under noisy measurements, the effect that the initial drogue locations has on robustness, and the effectiveness of parameter aggregation.

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Section: A Modeling Of Nonlinear Internal Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust estimation and aggregation of ocean internal wave parameters using Lagrangian drifters

Ouimet

Cortés

2014

2014 American Control Conference

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“…At the same time, we are unaware of investigations of the integrability and/or stability of solitons for an equation of this form. Hamdi et al [7,8] obtained two partial invariants and single soliton solutions for a more general class of equations including Eq. (2).…”

Section: Dynamics Of Solitons In a Nonintegrablementioning

confidence: 99%

“…(2) for solitons with amplitudes of (a) 1 and 0.5, (b) 1 and -0.5, (c) 1.118 and 0.5, and (d) 1.118 and -0.5. The plots are given in dimensionless coordinates(7).…”

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confidence: 99%

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

et al. 2012

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Localized perturbations of internal gravity wave fields constitute an integral and important component of nonlinear wave motions of a stratified fluid. It is important to analyze these perturbations because these waves often carry significant amounts of energy and play the leading role in processes in the environ ment. In particular, they affect the transport of bottom sediments in the shelf regions of the oceans, propaga tion of pollutants and impurities in the bulk of the fluid and on its surface, mixing of water, energy dissipation, transfer of acoustic signals, etc. It is impossible to understand the mechanisms of this action without detailed studies of the properties of isolated long lived pulses, accurate determination of their characteristics and features of the dynamics under various conditions in the medium, and the existence of localized nonra diating waveforms depending on the external factors of the medium.The modified Korteweg-de Vries equation with cubic nonlinearity is applicable to many physical problems with symmetry. Anisotropy for waves in a fluid is associated with the direction of the gravita tional force. For this reason, the classical Kortewegde Vries equation, which was first derived for waves on water, is derived in the first approximation of perturba tion theory for weakly nonlinear and weakly dispersive waves in a fluid. At the same time, the structure of the wave field in a stratified fluid is determined by the ver tical distribution of the density field, more precisely, by the distribution of the buoyancy (Brunt-Väisälä) frequency. This distribution can be symmetric; in par ticular, it can include two identical pycnoclines (regions with a strong change in the density). This case is described by the modified Korteweg-de Vries equa tion [1]. Moreover, cubic nonlinearity for certain posi tions of pycnoclines becomes anomalously small and even vanishes [1]. In this case, terms of the next per turbation order should be taken into account. Such an extended modified Korteweg-de Vries equation was recently derived in [2]. This equation is not integrable and the dynamics of solitons of this equation is ana lyzed in this work.First, we determine the situation described by this equation. We consider a three layer (in density) water flow in the gravitational field. The flow is bounded in the vertical direction by the bottom and free surface. The latter can be replaced by a rigid surface (this approximation in oceanology is called the rigid lid condition and is valid at small variations in the density, which are typical of incompressible stratified fluids). Let h be the thickness of the lower and upper layers; H be the total depth of the fluid; ρ 1 = ρ, ρ 2 = ρ + Δρ, and ρ 3 = ρ 2 + Δρ = ρ + 2Δρ be the densities of the lower, middle, and upper layers, respectively; and η(x, t) and ζ(x, t) be the lower and upper interfaces, respectively. We assume that the relative density changes at the Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the co...

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“…In a previous paper (Hamdi et al 2011), the authors derived solitary wave solutions for the more general Gardner equation that includes nonlinear terms of any order:…”

Section: Introductionmentioning

confidence: 99%

Conservation laws and invariants of motion for nonlinear internal waves: part II

et al. 2011

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In this paper, we derive three conservation laws and three invariants of motion for the generalized Gardner equation. These conserved quantities for internal waves are the momentum, energy, and Hamiltonian. The approach used for the derivation of these conservation laws and their associated invariants of motion is direct and does not involve the use of variational principles. It can be easily applied for finding similar invariants of motion for other general types of KdV, Gardner, and Boussinesq equations. The stability and conservation properties of discrete schemes for the simulations of internal waves propagation can be assessed and monitored using the analytical expressions of the constants of motion that are derived in this work.

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Analytical solutions of long nonlinear internal waves: Part I

Cited by 28 publications

References 30 publications

Robust estimation and aggregation of ocean internal wave parameters using Lagrangian drifters

Robust estimation and aggregation of ocean internal wave parameters using Lagrangian drifters

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Conservation laws and invariants of motion for nonlinear internal waves: part II

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