Nonlinear dispersion in wave-current interactions

Holm, Darryl D.; Ruiao, Hu,

doi:10.3934/jgm.2022004

Cited by 1 publication

(1 citation statement)

References 39 publications

(105 reference statements)

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“…Hence, in this regime the wavefronts of the nonlinear ring waves at the outer front of the wave structure eventually become significantly different from the wavefronts of the linear waves in its inner tail. We note that fission of ring waves has also manifested itself, in different settings, in numerical experiments in [23,27].…”

Section: Asymmetric Configuration With Dmentioning

confidence: 76%

Internal ring waves in a three-layer fluid on a current with a constant vertical shear

et al. 2023

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Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustment equation—a nonlinear first-order ordinary differential equation describing the dependence of the linear long-wave speed on its angle to the direction of the current. The compact ring waves, well studied in the absence of a current, correspond to the singular solution (envelope of the general solution) of this equation, and they can exist only under certain conditions. The constructed solutions reveal qualitative differences in the shapes of the wavefronts of the two baroclinic modes: the wavefront of the faster mode is elongated in the direction of the current, while the wavefront of the slower mode is squeezed. Moreover, depending on the vorticity strength, several different regimes have been identified. When the vertical shear is weak, part of the wavefront is able to propagate upstream, while when the shear is strong enough, the whole wavefront propagates downstream. A richer pattern of behaviour is observed for the slower mode. As the shear increases, singularities of the swallowtail-type may arise and, eventually, solutions with compact wavefronts crossing the downstream axis cease to exist. We show that the latter is related to the long-wave instability of the base flow. We obtain the cKdV-type amplitude equation and examine analytical expressions for its coefficients. Using this cKdV-type equation we numerically model the evolution of the waves for both modes. The initial stage of the evolution is in agreement with the leading-order predictions for the deformations of the wavefronts. Then, as the wavefronts expand, strong dispersive effects occur in the upstream direction. Moreover, when nonlinearity is enhanced, fission of waves is observed in the upstream part of the ring waves.

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Section: Asymmetric Configuration With Dmentioning

confidence: 76%