Article Type:
Research Article

NOTES AND CORRESPONDENCE Constituent Boussinesq Equations for Waves and Currents

Colin Y. Shen Naval Research Laboratory, Washington, D.C.

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Print Publication:
01 Mar 2001
DOI:
https://doi.org/10.1175/1520-0485(2001)031<0850:NACCBE>2.0.CO;2
Page(s):
850–859
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Abstract

The Boussinesq long-wave equations in constituent form are generalized to encompass both the irrotational long-wave dynamics and the rotational current dynamics. For irrotational long waves, the generalized equations are shown to represent both the weakly nonlinear and fully nonlinear Boussinesq-type wave equations given previously in the literature. The generalized equations are of additional interest in that they are effectively model equations for a weakly nonhydrostatic wave/current system, in which the conventional hydrostatic ocean current/wave model is a limiting case. The derived equations are related to the surface flow variables that are accessible to synoptic surface wave/current measurements such as those available from radar remote sensing.

Corresponding author address: Dr. Colin Y. Shen, Naval Research Laboratory, Code 7250, Washington, DC 20375.

Abstract

The Boussinesq long-wave equations in constituent form are generalized to encompass both the irrotational long-wave dynamics and the rotational current dynamics. For irrotational long waves, the generalized equations are shown to represent both the weakly nonlinear and fully nonlinear Boussinesq-type wave equations given previously in the literature. The generalized equations are of additional interest in that they are effectively model equations for a weakly nonhydrostatic wave/current system, in which the conventional hydrostatic ocean current/wave model is a limiting case. The derived equations are related to the surface flow variables that are accessible to synoptic surface wave/current measurements such as those available from radar remote sensing.

Corresponding author address: Dr. Colin Y. Shen, Naval Research Laboratory, Code 7250, Washington, DC 20375.

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  • Ainsworth, T. L., S. R. Chubb, R. A. Fusina, R. M. Goldstein, R. W. Jansen, J. S. Lee, and G. R. Valenzuela, 1995: INSAR imagery of surface currents, wave fields, and fronts. IEEE Trans. Geosci. Remote Sens.,33, 1117–1123.

  • Bauer, E., S. Hasselmann, K. Hasselmann, and H. C. Graber, 1992: Validation and assimilation of SEASAT altimeter wave heights using the WAM wave model. J. Geophys. Res.,97, 12 671–12 682.

  • Benjamin, T. B., J. L. Bona, and J. J. Mahony, 1972: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London,272A, 47–78.

  • Chen, Q., P. A. Madsen, H. A. Sch;auaffer, and D. R. Basco, 1998: Wave-current interaction based on an enhanced Boussinesq approach. Coastal Eng.,33, 11–39.

  • De Las Heras, M. M., G. Burgers, and P. A. E. M. Janssen, 1994: Variational wave data assimilation in a third-generation wave model. J. Atmos. Oceanic Technol.,11, 1350–1369.

  • Jonsson, I. G., O. Brink-Kj;ahr, and G. P. Thomas, 1978: Wave action and set-down for waves on a shear current. J. Fluid Mech.,87, 401–416.

  • Kaihatu, J. M., 1999: Data assimilation in phase resolving ocean wave models. Abstracts, Fifth SIAM Conf. on Mathematical and Computational Issues in the Geosciences, San Antonio, TX, Soc. Ind. Appl. Math. p 94.

  • Madsen, P. A., and H. A. Sch;auaffer, 1998: Higher order Boussinesq-type equations for surface gravity waves=mDerivation and analysis. Philos. Trans. Roy. Soc. London,356A, 3123–3184.

  • ——, R. Murray, and O. R. S;torensen, 1991: A new form of Boussinesq equations with improved linear dispersion characteristics. Coastal Eng.,15, 371–388.

  • Mahadevan, A., J. Oliger, and R. Street, 1996: A nonhydrostatic mesoscale ocean model. Part I: Well-posedness and scaling. J. Phys. Oceanogr.,26, 1868–1880.

  • Marom, M., R. M. Goldstein, E. B. Thornton, and L. Shemer, 1990: Remote sensing of ocean waves by interferometric synthetic aperture radar. Nature,345, 793–795.

  • Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophy. Res.,102, 5733–5752.

  • Mei, C. C., 1983: The Applied Dynamics of Ocean Surface Waves. J. Wiley and Sons, 740 pp.

  • Murray, R. J., 1989: Short wave modelling using new equations of Boussinesq type. Proc. Ninth Australasian Conf. on Coastal and Ocean Engineering Adelaide, Australia, Institution of Engineers, 331–336.

  • Nwogu, O., 1993: An alternative form of the Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coast. Ocean Eng.,119, 618–638.

  • Peregrine, D. H., 1967: Long waves on a beach. J. Fluid Mech.,27, 815–827.

  • Rego, V. S., and C. F. Neves, 1997: A Boussinesq-type wave model with vertical shear. Proc. Waves '97: The Third Int. Symp. on Ocean Wave Measurement and Analysis, Reston, VA, ASCE, 446–460.

  • Wei, G., J. T. Kirby, S. T. Grilli, and R. Subramanya, 1995: A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech.,294, 71–92.

  • Whitham, G. B., 1974: Linear and Nonlinear Waves. J. Wiley and Sons, 636 pp.

  • Witting, J. M., 1984: A unified model for the evolution of nonlinear water waves. J. Comput. Phys.,56, 203–236.

  • Yoon, S. B., and P. L. F. Liu, 1989: Interaction of currents and weakly nonlinear water waves in shallow water. J. Fluid Mech.,205, 397–419.

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