The square root depth wave equations

Cotter, Colin J.; Holm, Darryl D.; Percival, James

doi:10.1098/rspa.2010.0124

Cited by 17 publications

(14 citation statements)

References 10 publications

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“…Their idea was to use a technique commonly used in the one fluid case (water waves) to improve the dispersion relation of asymptotic models; it consists in working with a velocity variable that differs from the averaged velocity v; this can typically be the velocity at some fixed depth or a given level line, of the fluid domain (see [39,5,7] and more generally Section 5.2 of [33]). This approach has also been used in the context of interfacial waves [8,38]; the idea in [13] (see also [2], and [16] for a related approach) was to use it to remove the Kelvin-Helmholtz instabilities from the standard GN/GN model. The authors rewrote the GN/GN equations (35) in the variables (ζ ,v ± r ) instead of (ζ , v ± ), wherev ± r is the horizontal velocity at the fixed heightẑ ±…”

Section: A First Class Of Regularized Modelsmentioning

confidence: 99%

The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models

Lannes

Mei

2015

Fields Institute Communications

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The goal of this paper is to describe the formation of Kelvin-Helmholtz instabilities at the interface of two fluids of different densities and the ability of various shallow water models to reproduce correctly the formation of these instabilities. Working first in the so called rigid lid case, we derive by a simple linear analysis an explicit condition for the stability of the low frequency modes of the interface perturbation, an expression for the critical wave number above which Kelvin-Helmholtz instabilities appear, and a condition for the stability of all modes when surface tension is present. Similar conditions are derived for several shallow water asymptotic models and compared with the values obtained for the full Euler equations. Noting the inability of these models to reproduce correctly the scenario of formation of Kelvin-Helmholtz instabilities, we derive new models that provide a perfect matching. A comparisons with experimental data is also provided. Moreover, we briefly discuss the more complex case where the rigid lid is replaced by a free surface. In this configuration, it appears that some frequency modes are stable when the velocity jump at the interface is large enough; we explain why such stable modes do not appear in the rigid lid case.

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Section: A First Class Of Regularized Modelsmentioning

confidence: 99%

The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models

Lannes

Mei

2015

Fields Institute Communications

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“…The result has been extended to models with finite depth and flat bottom [2,3], for internal waves between layers of different density [4] as well as waves with added shear for constant vorticity [5][6][7][8][9]. A multi-layer model based on the Green-Naghdi approximation has been proposed in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Compelli

Ivanov

Todorov

2017

Phil. Trans. R. Soc. A.

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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.Mathematics Subject Classification (2010): 35Q53, 35Q35, 37K05, 37K40

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“…For some general facts concerning the description of waves interacting with currents we refer to the following reviews and monographs [10,49,37,52] and the references therein. The present study draws from previous single medium irrotational [54], [3], [46], [47], [48] and rotational [9], [11], [10], [12], [50], [17], [53], [42] studies as well as from studies of two-media systems such as [1], [2], [22], [21], [18], [19], [16], [15], [4], [5], [6], [7], [20], [27], [28], [29], [41], [44], [45].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian model for coupled surface and internal waves in the presence of currents

Ivanov¹

2017

Nonlinear Analysis: Real World Applications

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We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of 'wave' variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave-current interaction is influenced only by the current profile in the 'strips' adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.

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The square root depth wave equations

Cited by 17 publications

References 10 publications

The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models

The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models

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

Hamiltonian model for coupled surface and internal waves in the presence of currents

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