2009
DOI: 10.1111/j.1467-9590.2009.00433.x
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A Higher‐Order Internal Wave Model Accounting for Large Bathymetric Variations

Abstract: A higher-order strongly nonlinear model is derived to describe the evolution of large amplitude internal waves over arbitrary bathymetric variations in a two-layer system where the upper layer is shallow while the lower layer is comparable to the characteristic wavelength. The new system of nonlinear evolution equations with variable coefficients is a generalization of the deep configuration model proposed by Choi and Camassa [1] and accounts for both a higher-order approximation to pressure coupling between t… Show more

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Cited by 18 publications
(19 citation statements)
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“…We can not use directly the Banach fixed point principle to study existence of solutions of this system, because the operator A does not map the ball X(T ) ⊂ H s in X(T ). This is also the case of previous models derived in [8] and [18] where a kinematic equation in the form of equation (4.3) was employed. Furthermore, the new formulation (2.31)-(2.32) was stated in terms of the fluid velocity at a fixed depth Z 0 , which is easier of measuring than the commonly used depth-averaged velocity employed for instance in [8,9].…”
Section: Existence and Uniqueness Of Solutionsmentioning
confidence: 99%
“…We can not use directly the Banach fixed point principle to study existence of solutions of this system, because the operator A does not map the ball X(T ) ⊂ H s in X(T ). This is also the case of previous models derived in [8] and [18] where a kinematic equation in the form of equation (4.3) was employed. Furthermore, the new formulation (2.31)-(2.32) was stated in terms of the fluid velocity at a fixed depth Z 0 , which is easier of measuring than the commonly used depth-averaged velocity employed for instance in [8,9].…”
Section: Existence and Uniqueness Of Solutionsmentioning
confidence: 99%
“…One of the authors also revisited the two-dimensional potential theory case in the presence of highly irregular bottom variations: a weakly dispersive, weakly nonlinear Boussinesq system was given in Nachbin (2003), while a fully dispersive Boussinesq-type system was presented in Artiles & Nachbin (2004), where the conformal mapping framework was combined with expansions of the Dirichlet to Neumann map; iterates of the linear Dirichlet to Neumann map were used in the equation. This weakly nonlinear, fully dispersive system was then extended to the case of nonlinear internal waves in Ruiz de Zárate et al (2009). In all these cases a conformal mapping was used to generate a curvilinear coordinate system, which is also boundary fitted, but that has no restrictions regarding a vanishing Jacobian.…”
Section: Introductionmentioning
confidence: 99%
“…In the simplest variant, the rigid walls are both horizontal, and the flow is potential within each layer, but additional ingredients can be incorporated into the model, such as a background shear current and large bathymetric variations (see Refs. [13][14][15][16][17], and references therein). In the general case, an analytical study of this system requires quite lengthy calculations, especially when fully nonlinear dispersive approximations of higher orders are considered [16][17][18][19][20][21][22][23][24].…”
mentioning
confidence: 99%
“…[13][14][15][16][17], and references therein). In the general case, an analytical study of this system requires quite lengthy calculations, especially when fully nonlinear dispersive approximations of higher orders are considered [16][17][18][19][20][21][22][23][24]. In the present work, it will be shown that in the case ǫ ≪ 1 (the so called Boussinesq limit), there exists an elegant and remarkably short way how to derive fully nonlinear dispersive models of high orders.…”
mentioning
confidence: 99%
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