The oblique parabolic equation model for linear and nonlinear wave shoaling

Yaron Toledo*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Abstract Common simplified models for surface gravity waves result in parabolic-type equations. These equations mostly assume a negligible reflection from the bottom variations but account for both refraction and diffraction effects. A common deficiency of these equations is an inherent assumption of normally incident waves, which cause an increasing error, as the incident waves propagate from an increasing attack angle. In the nonlinear formulation of this type of equation, previous research added the assumption of small crossing angles between interacting waves, which also limits the applicability of these nonlinear models to narrow directional spectra. The current paper presents a parabolic approximation for oblique incident waves that overcomes these limitations. This is done by introducing a perturbation solution for the wave's phase function, which in its lowest order corresponds to oblique incident waves on a bottom with no lateral changes. The resulting curved wave ray structure replaces the simplified straight one used in the derivation process of various former models. In the nonlinear model, the nonlinear interactions are calculated between the frequency modes while taking into account the different propagating directions. Numerical results of the oblique parabolic equation show great advantages compared to other formulations that use a straight wave ray structure and the small crossing angle assumption. The nonlinear model was used to investigate the nonlinear shoaling of two similar monochromatic waves approaching from different angles. This fundamental nonlinear interaction problem shows that there is an energy transfer to waves of the primary harmonic that approach at larger attack angles than the two incident waves. This is counter-intuitive. Unlike in the case of linear wave shoaling, where the waves reduce their attack angle in the refraction process, in the nonlinear shoaling process the attack angle can increase. Fundamental numerical simulations of the linear and nonlinear oblique parabolic models are shown to be in excellent agreement with the initial mild-slope equations. This confirms the benefits of applying these models to various shoaling scenarios for both linear and nonlinear waves.

Original languageEnglish
Pages (from-to)103-133
Number of pages31
JournalJournal of Fluid Mechanics
StatePublished - 25 Jan 2013
Externally publishedYes


  • surface gravity waves
  • waves/free-surface flows


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