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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.25435 |
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| _version_ | 1866915929192923136 |
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| author | Kirkeby, Adrian Halsne, Trygve |
| author_facet | Kirkeby, Adrian Halsne, Trygve |
| contents | Starting from the free surface Euler equations, we derive a leading-order system in terms of surface variables, depending on the surface current and on the bathymetry through the depth-dependent Dirichlet-to-Neumann (DN) operator. The resulting system is shown to be well-posed using the theory of hyperbolic systems of pseudo-differential operators. We then consider wave propagation in slowly varying environments. As an explicit approximation to the DN operator, the semiclassical Weyl quantization of the symbol $g_b(X,ξ)=|ξ|\tanh(b(X)|ξ|)$ is shown to be both asymptotically accurate and consistent with the self-adjoint structure of the true operator, and to provide the natural framework for asymptotic analysis of the wave system. A central consequence of the resulting framework is that classical asymptotic models - including the wave action equation, the mild-slope equation, the Schrödinger equation, and the action balance equation - emerge systematically from a single formulation. By deriving these equations, we show how the simple leading order system with the Weyl quantization of the DN operator provides a unified and mathematically consistent framework for the asymptotic linear theory of wave-current-bathymetry interaction, hence providing a transparent, rigorous and accessible route from the primitive Euler equations to the mentioned asymptotic models. Throughout, numerical experiments are included to illustrate the analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25435 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wave-Current-Bathymetry Interaction Revisited: Modeling, Analysis and Asymptotics Kirkeby, Adrian Halsne, Trygve Analysis of PDEs 76B15, 35Q35, 35B40, 35S10 Starting from the free surface Euler equations, we derive a leading-order system in terms of surface variables, depending on the surface current and on the bathymetry through the depth-dependent Dirichlet-to-Neumann (DN) operator. The resulting system is shown to be well-posed using the theory of hyperbolic systems of pseudo-differential operators. We then consider wave propagation in slowly varying environments. As an explicit approximation to the DN operator, the semiclassical Weyl quantization of the symbol $g_b(X,ξ)=|ξ|\tanh(b(X)|ξ|)$ is shown to be both asymptotically accurate and consistent with the self-adjoint structure of the true operator, and to provide the natural framework for asymptotic analysis of the wave system. A central consequence of the resulting framework is that classical asymptotic models - including the wave action equation, the mild-slope equation, the Schrödinger equation, and the action balance equation - emerge systematically from a single formulation. By deriving these equations, we show how the simple leading order system with the Weyl quantization of the DN operator provides a unified and mathematically consistent framework for the asymptotic linear theory of wave-current-bathymetry interaction, hence providing a transparent, rigorous and accessible route from the primitive Euler equations to the mentioned asymptotic models. Throughout, numerical experiments are included to illustrate the analysis. |
| title | Wave-Current-Bathymetry Interaction Revisited: Modeling, Analysis and Asymptotics |
| topic | Analysis of PDEs 76B15, 35Q35, 35B40, 35S10 |
| url | https://arxiv.org/abs/2603.25435 |