Hydrostatic versus Nonhydrostatic Euler-Equation Modeling of Nonlinear Internal Waves
Publication: Journal of Engineering Mechanics
Volume 135, Issue 10
Abstract
Basin-scale internal waves are inherently nonhydrostatic; however, they are frequently resolved features in three-dimensional hydrostatic lake and coastal ocean models. Comparison of hydrostatic and nonhydrostatic formulations of the Centre for Water Research Estuary and Lake Computer Model provides insight into the similarities and differences between these representations of internal waves. Comparisons to prior laboratory experiments are used to demonstrate the expected wave evolution. The hydrostatic model cannot replicate basin-scale wave degeneration into a solitary wave train, whereas a nonhydrostatic model does represent the downscaling of energy. However, the hydrostatic model produces a nonlinear traveling borelike feature that has similarities to the mean evolution of the nonhydrostatic wave.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was partially funded by the Office of Naval Research Young Investigator Award through Grant No. UNSPECIFIEDN00014-01-1-0574. The writers also appreciate the collaboration of Professor Jörg Imberger at the Center for Water Research, University of Western Australia in the use of CWR-ELCOM.
References
Apel, J. R. (1980). “Satellite sensing of ocean surface dynamics.” Annu. Rev. Earth Planet Sci., 8, 303–342.
Appt, J., Imberger, J., and Kobus, H. (2004). “Basin-scale motion in stratified upper lake constance.” Limnol. Oceanogr., 49(4), 919–933.
Antenucci, J. P., and Imberger, J. (2003). “The seasonal evolution of wind/internal wave resonance in Lake Kinneret.” Limnol. Oceanogr., 48(5), 2055–2061.
Azevedo, A., daSilva, J. C. B., and New, A. L. (2006). “On the generation and propagation of internal solitary waves in the southern Bay of Biscay.” Deep-Sea Res., Part I, 53(6), 927–9421.
Blumberg, A. F., and Mellor, G. L. (1987). “A description of a three-dimensional coastal ocean circulation model.” Three-dimensional coastal ocean models, N. S. Heaps, ed., American Geophysical Union, Washington, D.C., 1–16.
Boegman, L., Imberger, J., Ivey, G. N., and Antenucci, J. P. (2003). “High frequency internal waves in large stratified lakes.” Limnol. Oceanogr., 48(12), 895–919.
Boegman, L., Ivey, G. N., and Imberger, J. (2005). “The energetics of large-scale internal wave degeneration in lakes.” J. Fluid Mech., 531, 159–180.
Botelho, D., Imberger, J., Dallimore, C., and Hodges, B. R. (2008). “A hydrostatic/non-hydrostatic grid-switching strategy for computing high-frequency, high wave number motions embedded in geophysical flows.” Environ. Modell. Software, 24(4), 473–488.
Casulli, V. (1999). “A semi-implicit finite difference method for non-hydrostatic, free-surface flows.” Int. J. Numer. Methods Fluids, 30, 425–440.
Casulli, V., and Cheng, R. T. (1992). “Semi-implicit finite difference methods for three-dimensional shallow water flow.” Int. J. Numer. Methods Fluids, 15, 629–648.
Casulli, V., and Stelling, G. S. (1998). “Numerical simulation of 3D quasi-hydrostatic free-surface flows.” J. Hydraul. Eng., 124, 678–686.
Daily, C., and Imberger, J. (2003). “Modelling solitons under the hydrostatic and Boussinesq approximations.” Int. J. Numer. Methods Fluids, 43(3), 231–252.
Dullin, H. R., Gottwald, G., and Holm, D. D. (2001). “An integrable shallow water equation with linear and nonlinear dispersion.” Phys. Rev. Lett., 87, 194501.
Farmer, D. M. (1978). “Observations of long nonlinear internal waves in a lake.” J. Phys. Oceanogr., 8, 63–73.
Fringer, O. B., Gerritsen, M., and Street, R. L. (2006). “An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator.” Ocean Model., 14, 139–173.
Haidvogel, D. B., Arango, H. G., Hedstrom, K., Beckmann, A., Malanotte-Rizzoli, P., and Shchepetkin, A. F. (2000). “Model evaluation experiments in the North Atlantic Basin: simulations in nonlinear terrain-following coordinates.” Dyn. Atmos. Oceans, 32, 239–281.
Hamrick, J. M., (1992). “A three-dimensional environmental fluid dynamics computer code: Theoretical and computational aspects.” Special Rep. No. 317, The College of William and Mary, Virginia Institute of Marine Science.
Hodges, B. R. (2000). “Numerical Techniques in CWR-ELCOM (code release v.1).” Rep. No. WP 1422 BH, Centre Water Research, Nedlands, Australia.
Hodges, B. R., Imberger, J., Saggio, A., and Winters, K. (2000). “Modeling basin-scale internal waves in a stratified lake.” Limnol. Oceanogr., 45(7), 1603–1620.
Hodges, B. R., Laval, B., and Wadzuk, B. M. (2006). “Numerical error assessment and a temporal horizon for internal waves in a hydrostatic model.” Ocean Model., 13(1), 44–64.
Horn, D. A., Imberger, J., and Ivey, G. N. (2001). “The degeneration of large-scale interfacial gravity waves in lakes.” J. Fluid Mech., 434, 181–207.
Horn, D. A., Imberger, J., Ivey, G. N., and Redekopp, L. G. (2002). “A weakly nonlinear model of long internal waves in closed basins.” J. Fluid Mech., 467, 269–287.
Hutter, K., Bauer, G., Wang, Y., and Güting, P. (1998). “Forced motion response in enclosed lakes.” Coastal and estuarine studies, 54, American Geophysical Union, Washington, DC, 137–166.
Kao, T. W., Pan, F., and Renouard, D. (1985). “Internal solitary waves on the pycnocline: Generation, propagation, and shoaling and breaking over a slope.” J. Fluid Mech., 159, 19–53.
Kennedy, A. B., Chen, Q., Kirby, J. T., and Dalrymple, R. A. (2000). “Boussinesq modeling of wave transformation, breaking and run-up. I: 1D.” J. Waterway, Port, Coastal, Ocean Eng., 126, 39–47.
Killworth, P. D., Stainforth, D., Webb, D. J., and Patterson, S. M. (1991). “Development of a free-surface Bryan–Cox–Semtner ocean model.” J. Phys. Oceanogr., 21(9), 1333–1348.
Kim, J., and Moin, P. (1985). “Application of a fractional-step method to incompressible Navier-Stokes equations.” J. Comput. Phys., 59, 308–323.
Laval, B., Hodges, B. R., and Imberger, J. (2003a). “Reducing numerical diffusion effects with pycnocline filter.” J. Hydraul. Eng., 129, 215–224.
Laval, B., Imberger, J., Hodges, B. R., and Stocker, R. (2003b). “Modeling circulation in lakes: Spatial and temporal variations.” Limnol. Oceanogr., 48(3), 983–994.
Laval, B. E., Imberger, J., and Findikakis, A. N. (2005). “Dynamics of a large tropical lake: Lake Maracaibo.” Aquat. Sci., 67(3), 337–349.
Leonard, B. P. (1991). “The ultimate conservative difference scheme applied to unsteady one-dimensional advection.” Comput. Methods Appl. Mech. Eng., 88, 17–74.
Long, R. R. (1972). “The steepening of long, internal waves.” Tellus, 24, 88–99.
Lorenz, E. N. (1955). “Available potential energy and the maintenance of the general circulation.” Tellus, 72, 157–167.
Marshall, J., Hill, C., Perlman, L., and Adcroft, A. (1997). “Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling.” J. Geophys. Res., 102, 5733–5752.
Michallet, H., and Ivey, G. N. (1999). “Experiments on mixing due to internal solitary waves breaking on uniform slopes.” J. Geophys. Res., 104(C6), 13467–13477.
Roache, P. J. (2002). “Code verification by the method of manufactured solutions.” J. Fluids Eng., 124(1), 4–10.
Romero, J. R., and Imberger, J. (2003). “Effect of a flood underflow on reservoir water quality: Data and three-dimensional modeling.” Archiv Hydrobiol., 157(1), 1–25.
Roy, C. J. (2003). “Grid convergence error analysis for mixed-order numerical schemes.” Amer. Inst. Aeronautics Astronautics, 41(4), 595–604.
Rueda, F. J., Sanmiguel-Rojas, E., and Hodges, B. R. (2007). “Baroclinic stability for a family of two-level, semi-implicit numerical methods for the 3D shallow water equations.” Int. J. Numer. Methods Fluids, 3, 237–268.
Shchepetkin, A. F., and McWilliams, J. C. (2005). “The regional oceanic modeling system (ROMS): A split-explicit, free-surface, topography-following-coordinate oceanic model.” Ocean Model., 9(4), 347–404.
Segur, H., and Hammack, J. L. (1982). “Solitary wave models of long internal waves.” J. Fluid Mech., 118, 285–304.
Venayagamoorthy, S. K., and Fringer, O. B. (2005). “Nonhydrostatic and nonlinear contributions to the energy flux budget in nonlinear internal waves.” Geophys. Res. Lett., 32, L15603.
Wadzuk, B. M., and Hodges, B. R. (2004). “Hydrostatic and non-hydrostatic internal wave models.” CRWR Online Rep. No. 04-9, Univ. of Texas at Austin, Austin, Tex., ⟨http://www.crwr.utexas.edu/online.shtml⟩ (November 8, 2008).
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.
Welch, P. D. (1967). “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms.” IEEE Trans. Audio Electroacoust., AU-15, 70–73.
Information & Authors
Information
Published In
Copyright
© 2009 ASCE.
History
Received: Mar 21, 2006
Accepted: May 7, 2009
Published online: Sep 15, 2009
Published in print: Oct 2009
Notes
Note. Associate Editor: Brett F. Sanders
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.