Convergence of Multilayer Nonhydrostatic Models in Relation to Boussinesq-Type Equations
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 144, Issue 2
Abstract
Multilayer nonhydrostatic models have gained popularity in the computation of ocean wave processes, but a recent attempt to make a connection to the Boussinesq-type approach raises some fundamental issues that require a rigorous assessment beyond the conventional framework of approximation. Such an endeavor is readily feasible for depth-integrated nonhydrostatic models because of the existence of an equivalent Boussinesq form. An examination of the governing equations shows that defining the nonhydrostatic pressure at layer interfaces for depth integration gives rise to a leading-order approximation of the dispersion relation distinct from the Taylor-series expansion. Introducing a parameterized nonhydrostatic pressure profile or increasing the number of layers is akin to retaining high-order terms in the Boussinesq-type equations with comparable rational function approximations of linear and nonlinear wave properties. Although both approaches converge to the exact solution at high order, the spatial derivatives in nonhydrostatic models always remain at first order. The simple numerical framework along with grid refinement techniques enable application of a single model over a wide range of spatial scales for practical application.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors received support from the National Tsunami Hazard Mitigation Program (Grant NA15NWS4670025) via the Hawaii Emergency Management Agency. SOEST Contribution No. 10243.
References
Agnon, Y., Madsen, P. A., and Schäffer, H. A. (1999). “A new approach to high-order Boussinesq models.” J. Fluid Mech., 399, 319–333.
Ai, C., Jin, S., and Lv, B. (2011). “A new fully non-hydrostatic 3D free surface model for water wave motion.” Int. J. Numer. Methods Fluids, 66(11), 1354–1370.
Aricò, C., and Re, C. L. (2016). “A non-hydrostatic pressure distribution solver for the nonlinear shallow water equations over irregular topography.” Adv. Water Resour., 98, 47–69.
Bai, Y., and Cheung, K. F. (2012). “Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation.” Int. J. Numer. Methods Fluids, 69(2), 411–429.
Bai, Y., and Cheung, K. F. (2013a). “Depth-integrated free-surface flow with parameterized non-hydrostatic pressure.” Int. J. Numer. Methods Fluids, 71(4), 403–421.
Bai, Y., and Cheung, K. F. (2013b). “Dispersion and nonlinearity of multi-layer non-hydrostatic free-surface flow.” J. Fluid Mech., 726, 226–260.
Bai, Y., and Cheung, K. F. (2015). “Dispersion and kinematics of multi-layer non-hydrostatic models.” Ocean Modell., 92, 11–27.
Bai, Y., and Cheung, K. F. (2016). “Linear and nonlinear properties of reduced two-layer models for non-hydrostatic free-surface flow.” Ocean Modell., 107, 64–81.
Bai, Y., and Cheung, K. F. (2018). “Linear shoaling of free-surface waves in multilayer non-hydrostatic models.” Ocean Modell., 121, 90–104.
Bai, Y., Lay, T., Cheung, K. F., and Ye, L. (2017). “Two regions of seafloor deformation generated the tsunami for the 13 November 2016, Kaikoura, New Zealand Earthquake.” Geophys. Res. Lett., 44(13), 6597–6606.
Casulli, V. (1995). “Recent developments in semi-implicit numerical methods for free surface hydrodynamics.” Adv. Hydro-Sci. Eng., 2, 2174–2181.
Casulli, V., and Stelling, G. (1998). “Numerical simulation of 3D quasi-hydrostatic, free surface flows.” J. Hydraul. Eng., 678–686.
Chazel, B. F., Benoit, M, Ern, A., and Piperno, S. (2009). “A double layer Boussinesq-type model for highly nonlinear and dispersive waves.” Proc. R. Soc. A, 465, 2319–2346.
Cheung, K. F., Bai, Y., and Yamazaki, Y. (2013). “Surges around the Hawaiian Islands from the 2011 Tohoku tsunami.” J. Geophys. Res. Oceans, 118(10), 5703–5719.
Cui, H., Pietrzak, J. D., and Stelling, G. S. (2014). “Optimal dispersion with minimized Poisson equations for non-hydrostatic free surface flows.” Ocean Modell., 81, 1–12.
Donahue, A., Zhang, Y., Kennedy, A., Westerink, J., and Panda, N. (2015). “A Boussinesq-scaled pressure-Poisson water wave model.” Ocean Modell., 86, 36–57.
Fang, K., Liu, Z., and Zou, Z. (2015). “Efficient computation of coastal waves using a depth-integrated, non-hydrostatic model.” Coastal. Eng., 97, 21–36.
Gobbi, M. F., Kirby, J. T., and Wei, G. (2000). “A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4.” J. Fluid Mech., 405, 181–210.
Jeschke, A., Pedersen, G. K., Vater, S., and Behrens, J. (2017). “Depth-averaged non-hydrostatic extension for shallow water equations with quadratic vertical pressure profile: Equivalence to Boussinesq-type equations.” Int. J. Numer. Methods Fluids, 84(10), 569–583.
Kazolea, M., Delis, A. I., Nikolos, I. K., and Synolakis, C. E. (2012). “An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations.” Coastal. Eng., 69, 42–66.
Kirby, J. T. (2016). “Boussinesq models and their application to coastal processes across a wide range of scales.” J. Waterway, Port, Coastal, Ocean Eng., 03116005.
Li, N., Roeber, V., Yamazaki, Y., Heitmann, T. W., Bai, Y., and Cheung, K. F. (2014). “Integration of coastal inundation modeling from storm tides to individual waves.” Ocean Modell., 83, 26–42.
Liu, Z., and Fang, K. (2016). “A new two-layer Boussinesq model for coastal waves from deep to shallow water: Derivation and analysis.” Wave Motion, 67, 1–14.
Lu, X., and Xie, S. (2016). “Depth-averaged non-hydrostatic numerical modeling of nearshore wave propagations based on the FORCE scheme.” Coastal Eng., 114, 208–219.
Lynett, P. J., et al. (2017). “Inter-model analysis of tsunami-induced coastal current.” Ocean Modell., 114, 14–32.
Lynett, P. J., and Liu, P. L. (2004). “A two-layer approach to wave modelling.” Proc. R. Soc. A, 460(2049), 2637–2669.
Ma, G., Chou, Y.-J., and Shi, F. (2014a). “A wave-resolving model for nearshore suspended sediment transport.” Ocean Modell., 77, 33–49.
Ma, G., Shi, F., Hsiao, S., and Kirby, J. T. (2014b). “Non-hydrostatic modeling of wave interactions with porous structures.” Coastal Eng., 91, 84–98.
Ma, G., Shi, F., and Kirby, J. T. (2012). “Shock-capturing non-hydrostatic model for fully dispersive surface wave processes.” Ocean Modell., 43–44, 22–35.
Madsen, P. A., Bingham, H. B., and Schäffer, H. A. (2003). “Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis.” Proc. R. Soc. A, 459(2033), 1075–1104.
Madsen, P. A., Fuhrman, D. R., and Wang, B. (2006). “A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry.” Coastal Eng., 53(5–6), 487–504.
Madsen, P. A., Murray, R., and Sørensen, O. R. (1991). “A new form of Boussinesq equations with improved linear dispersion characteristics.” Coastal Eng., 15(4), 371–388.
Madsen, P. A., and Sørensen, O. R. (1992). “A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry.” Coastal Eng., 18, 183–204.
Nwogu, O. (1993). “Alternative form of Boussinesq equations for nearshore wave propagation.” J. Waterway, Port, Coastal, Ocean Eng., 618–638.
Peregrine, D. H. (1967). “Long waves on a beach.” J. Fluid Mech., 27(4), 815–827.
Rijnsdorp, D. P., Smit, P. B., Zijlema, M., and Reniers, A. (2017). “Efficient non-hydrostatic modelling of 3D wave induced currents using a subgrid approach.” Ocean Modell., 116, 118–133.
Rijnsdorp, D. P., and Zijlema, M. (2016). “Simulating waves and their interactions with a restrained ship using a non-hydrostatic wave-flow model.” Coastal Eng., 114, 119–136.
Roeber, V., and Cheung, K. F. (2012). “Boussinesq-type model for energetic breaking waves in fringing reef environments.” Coastal Eng., 70, 1–20.
Shi, F., Kirby, J. T., Harris, J. C., Geiman, J. D., and Grilli, S. T. (2012). “A high-order adaptive time stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation.” Ocean Modell., 43–44, 36–51.
Shi, J., et al. (2015). “Pressure decimation and interpolation (PDI) method for a baroclinic non-hydrostatic model.” Ocean Modell., 96, 265–279.
Stansby, P. K., and Zhou, J. G. (1998). “Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems.” Int. J. Numer. Methods Fluids, 28(3), 541–563.
Stelling, G. S., and Duinmeijer, S. P. A. (2003). “A staggered conservative scheme for every Froude number in rapidly varied shallow water flows.” Int. J. Numer. Methods Fluids, 43(12), 1329–1354.
Stelling, G. S., and Zijlema, M. (2003). “An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation.” Int. J. Numer. Methods Fluids, 43(1), 1–23.
Tonelli, M., and Petti, M. (2012). “Shock-capturing Boussinesq model for irregular wave propagation.” Coastal Eng., 61, 8–19.
Walters, R. A. (2005). “A semi-implicit finite element model for non-hydrostatic (dispersive) surface waves.” Int. J. Numer. Methods Fluids, 49(7), 721–737.
Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R. (1995). “A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves.” J. Fluid Mech., 294, 71–92.
Wei, Z., and Jia, Y. (2013). “A depth-integrated non-hydrostatic finite element model for wave propagation.” Int. J. Numer. Methods Fluids, 73(11), 976–1000.
Yamazaki, Y., Cheung, K. F., and Kowalik, Z. (2011a). “Depth-integrated, non-hydrostatic model with grid-nesting for tsunami generation, propagation and run-up.” Int. J. Numer. Methods Fluids, 67(12), 2081–2107.
Yamazaki, Y., Kowalik, Z., and Cheung, K. F. (2009). “Depth-integrated, non-hydrostatic model for wave breaking and run-up.” Int. J. Numer. Methods Fluids, 63, 1448–1470.
Yamazaki, Y., Lay, T., Cheung, K. F., Yue, H., and Kanamori, H. (2011b). “Modeling near-field tsunami observations to improve finite-fault slip models for the 11 March 2011 Tohoku Earthquake.” Geophys. Res. Lett., 38(7).
Young, C.-C., Wu, C. H., Liu, W.-C., and Kuo, J.-T. (2009). “A higher-order non-hydrostatic σ model for simulating non-linear refraction-diffraction of water waves.” Coastal Eng., 56(9), 919–930.
Zhang, J., Liang, D., and Liu, H. (2017). “An efficient 3D non-hydrostatic model for simulating near-shore breaking waves.” Ocean Eng., 140, 19–28.
Zhu, L., Chen, Q., and Wan, X. (2014). “Optimization of non-hydrostatic Euler model for water waves.” Coastal Eng., 91, 191–199.
Zijlema, M., and Stelling, G. S. (2005). “Further experiences with computing non-hydrostatic free-surface flows involving water waves.” Int. J. Numer. Methods Fluids, 48(2), 169–197.
Zijlema, M., and Stelling, G. S. (2008). “Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure.” Coastal Eng., 55(10), 780–790.
Zijlema, M., Stelling, G., and Smit, P. (2011). “SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters.” Coastal Eng., 58(10), 992–1012.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 144 • Issue 2 • March 2018
Copyright
© 2018 American Society of Civil Engineers.
History
Received: May 26, 2017
Accepted: Sep 27, 2017
Published online: Jan 9, 2018
Published in print: Mar 1, 2018
Discussion open until: Jun 9, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.