An unsteady wave driver for narrowbanded waves: modeling nearshore circulation driven by wave groups

In this paper, we derive an unsteady refraction-diffraction model for narrowbanded water waves for use in computing coupled wave-current motion in the nearshore. The end result is a variable coefficient, nonlinear Schrodinger-type wave driver (describing the envelope of narrow-banded incident waves) coupled to forced nonlinear shallow water equations (describing steady or unsteady mean flows driven by the short-wave field). Comparisons with experimental data show that good accuracy can be obtained for cases of nonbreaking wave transformation. Numerical simulations show that the interaction of wave groups with longshore topographic nonuniformities generates strong edge wave resonances, providing a generating mechanism for low-order edge waves.

[1]  P. Liu,et al.  Derivation of the third-order evolution equations for weakly nonlinear water waves propagating over uneven bottoms , 1989 .

[2]  Ge Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4 , 2000, Journal of Fluid Mechanics.

[3]  M. Longuet-Higgins.  Longshore currents generated by obliquely incident sea waves: 1 , 1970 .

[4]  James T. Kirby,et al.  Higher‐order approximations in the parabolic equation method for water waves , 1986 .

[5]  Chiang C. Mei,et al.  A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation , 1985, Journal of Fluid Mechanics.

[6]  D. H. Peregrine,et al.  Surf Zone Currents , 1998 .

[7]  Miguel Onorato,et al.  The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains , 2000 .

[8]  J. Kirby,et al.  Boussinesq modeling of a rip current system , 1999 .

[9]  R. Dalrymple,et al.  Rip current instabilities , 2001, Journal of Fluid Mechanics.

[10]  A. Bowen,et al.  Simulations of Dissipative, Shore-Oblique Infragravity Waves , 2003 .

[11]  James T. Kirby,et al.  Nonlinear Evolution of Shear Instabilities of the Longshore Current , 1997 .

[12]  I. A. Svendsen.  Mass flux and undertow in a surf zone , 1984 .

[13]  Robert A. Dalrymple,et al.  An approximate model for nonlinear dispersion in monochromatic wave propagation models , 1986 .

[14]  Chiang C. Mei,et al.  On slowly-varying Stokes waves , 1970, Journal of Fluid Mechanics.

[15]  Robert A. Dalrymple,et al.  Verification of a parabolic equation for propagation of weakly-nonlinear waves , 1984 .

[16]  M. Longuet-Higgins,et al.  Radiation stresses in water waves; a physical discussion, with applications , 1964 .

[17]  Robert A. Dalrymple,et al.  Wave-current interaction models for rip currents , 1978 .

[18]  M. Nicholas,et al.  Coastal engineering. , 1969, Science.

[19]  M. Losada,et al.  Standing edge waves on a pocket beach , 2001 .

[20]  Hemming A. Schäffer,et al.  Infragravity waves induced by short-wave groups , 1993, Journal of Fluid Mechanics.

[21]  J. Berkhoff,et al.  Computation of Combined Refraction — Diffraction , 1972 .

[22]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[23]  C. Mei,et al.  Nonlinear excitation of long-trapped waves by a group of short swells , 1981, Journal of Fluid Mechanics.

[24]  R. Davis,et al.  Excitation of edge waves by waves incident on a beach , 1974 .

[25]  Anthony J. Bowen,et al.  Rip currents: 1. Theoretical investigations , 1969 .

[26]  K. Stewartson,et al.  On three-dimensional packets of surface waves , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[27]  Carlos J. Lozano,et al.  Refraction–diffraction model for linear surface water waves , 1980, Journal of Fluid Mechanics.

[28]  A. C. Radder,et al.  On the parabolic equation method for water-wave propagation , 1978, Journal of Fluid Mechanics.

[29]  Chiang C. Mei,et al.  Effects of slowly varying depth and current on the evolution of a Stokes wavepacket , 1983 .

[30]  M. S. Longuet-Higgins,et al.  Longshore currents generated by obliquely incident sea waves: 2 , 1970 .

[31]  R. Guza,et al.  Infragravity Edge Wave Observations on Two California Beaches , 1987 .

[32]  A. C. Radder,et al.  Verification of numerical wave propagation models for simple harmonic linear water waves , 1982 .

[33]  H. Schäffer.  Edge waves forced by short-wave groups , 1994, Journal of Fluid Mechanics.

[34]  Robert A. Holman,et al.  Generation of edge waves in shallow water , 1997 .

[35]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[36]  Karsten Trulsen,et al.  A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water , 1996 .

[37]  R. Holman,et al.  Nonlinear shear instabilities of alongshore currents on plane beaches , 1996, Journal of Fluid Mechanics.

[38]  G. Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves , 1995, Journal of Fluid Mechanics.

[39]  Robert A. Dalrymple,et al.  A parabolic equation for the combined refraction–diffraction of Stokes waves by mildly varying topography , 1983, Journal of Fluid Mechanics.

[40]  R. Holman,et al.  Shear instabilities of the mean longshore current: 1. Theory , 1989 .

[41]  B. Gallagher.  Generation of surf beat by non-linear wave interactions , 1971, Journal of Fluid Mechanics.