Inertial wave singularities from two-dimensional tips and wedges
Abstract
This study examines the structure of the inertial wave field generated by small harmonic oscillations of a two-dimensional wedge in an unbounded, uniformly rotating fluid. We demonstrate that inviscid singularities generically emerge in the form of singular eigenmodes satisfying homogeneous boundary conditions and outgoing wave conditions. These modes produce velocity fields that behaves as |x ⊥ | α-1 along the rays emanating from the wedge apex, where x ⊥ denotes the distance from the critical line. The complex exponents α are determined by a simple discrete relation depending only on wedge geometry and wave propagation angle. For inviscid forcing, the exponent is selected by finite-energy constraints ( e(α) ≥ 1/2). A viscous approximation is then constructed by extending classical self-similar solutions to complex exponents, capturing the internal structure and evolution of the wave beam. Higher-order corrections enforcing no-slip boundary conditions are also derived. For viscous forcing, the boundary layer flow induces a tangential mismatch that is balanced by radial flux in the bulk, thereby fully prescribing the amplitude of the only singular eigenmode with radial flux (the Dirac eigenmode with α = 0). The theory is validated through direct numerical simulations for both inviscid and viscous forcing, showing excellent agreement in both unbounded and bounded configurations. Finally, the framework suggests broader implications, including wave scattering by sharp edges, extensions to three-dimensional geometries, and potential explanations for edge-emitted sheet-like structures observed in librating or precessing titled cubes.
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BibTeX
@misc{dizes2026inertial,
title = {{Inertial wave singularities from two-dimensional tips and wedges}},
author = {Stéphane Le Dizès and Benjamin Favier},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://amu.hal.science/hal-05669299v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Inertial wave singularities from two-dimensional tips and wedges
AU - Stéphane Le Dizès
AU - Benjamin Favier
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://amu.hal.science/hal-05669299v2
AB - This study examines the structure of the inertial wave field generated by small harmonic oscillations of a two-dimensional wedge in an unbounded, uniformly rotating fluid. We demonstrate that inviscid singularities generically emerge in the form of singular eigenmodes satisfying homogeneous boundary conditions and outgoing wave conditions. These modes produce velocity fields that behaves as |x ⊥ | α-1 along the rays emanating from the wedge apex, where x ⊥ denotes the distance from the critical line. The complex exponents α are determined by a simple discrete relation depending only on wedge geometry and wave propagation angle. For inviscid forcing, the exponent is selected by finite-energy constraints ( e(α) ≥ 1/2). A viscous approximation is then constructed by extending classical self-similar solutions to complex exponents, capturing the internal structure and evolution of the wave beam. Higher-order corrections enforcing no-slip boundary conditions are also derived. For viscous forcing, the boundary layer flow induces a tangential mismatch that is balanced by radial flux in the bulk, thereby fully prescribing the amplitude of the only singular eigenmode with radial flux (the Dirac eigenmode with α = 0). The theory is validated through direct numerical simulations for both inviscid and viscous forcing, showing excellent agreement in both unbounded and bounded configurations. Finally, the framework suggests broader implications, including wave scattering by sharp edges, extensions to three-dimensional geometries, and potential explanations for edge-emitted sheet-like structures observed in librating or precessing titled cubes.
ER -