Some results

Consider a simple initially-linear internal wave (300KB avi) produced in a hydrostatic simulation

The model shows that nonlinear steepening causes the internal wave to evolve into a train of solitary waves. However, in the real world, these solitons (waves of permanent form) occur because of a balance between nonlinear steepening and wave dispersion - the latter requiring the non-hydrostatic pressure. The hydrostatic model above doesn't have include non-hydrostatic pressure, so why are we obtaining solitons?

The key to understanding what happens in modeling internal waves with an hydrostatic model requires insight into the interplay between numerical dissipation and numerical diffusion.

more to be posted later...

<To an index page for results>

To see an animated comparison of hydrostatic and non-hydrostatic non-breaking internal waves at a web page with 6 movies of 500 MB each (takes awhile to load) you can <follow this link to movies > . For some still frames of the hydrostatic model <follow this link to jpgs>. For some still frames of the non-hydrostatic model <follow this link to jpgs>.

For a comparison of breaking internal waves - again 6 movies at 500 MB each, <follow this link to movies> . For some still frames of the hydrostatic model with breaking waves <follow this link to jpgs>. For some still frames of the non-hydrostatic model with breaking waves <follow this link to jpgs>.

How the research addresses the sponsors' interests:

One of the interests of the Office of Naval Research related to internal waves is the mixing processes caused by these waves and how such mixing affects the thermocline (temperature stratification) in the ocean. Changes in the thermal structure of the ocean will change the propogation of sensor signals (e.g. sonar). Predicting the evolution of the thermal structure requires three-dimensional models of flow and transport. For large-scale applications, models use the "hydrostatic approximation" which neglects the effects of dynamic pressure. Unfortunately, the neglected dynamic pressure plays a key role in the development of internal waves, so hydrostatic models should not correctly predict the physical processes. ONR has sponsored this research to quantify the errors associated with hydrostatic modeling of internal waves and to begin to look at new ways of modeling.

A second interest of ONR related to internal waves is their propagation, evolution and breaking in coastal zones. Nonlinearl internal waves may affect subsurface AUV, ROV or swimmer operations, so there is a need for models that can accurately predict their occurence and evolution. The present work is developing methods to analyze the error associated with modeling internal waves.

Publications and presentations (fully or partially supported)

Dallimore, C.J., J. Imberger and B.R. Hodges (2004), “Modeling a Plunging Underflow,” Journal of Hydraulic Engineering, 130(11): 1068-1076.

Dallimore, C., B.R. Hodges, J. Imberger, (2003) “Coupling an underflow model to a 3D hydrodynamic model,” Journal of Hydraulic Engineering, 129(10): 1-10.

Delavan, S.K. (2003). Accumulation of Numerical Errors in Hydrostatic Models of Internal Waves. M.S. Thesis, Civil Engineering, The University of Texas at Austin, 70 pgs.

Delavan, S.K., and B.R. Hodges (2003), “Limitations of the shallow water equations for modeling stratified water bodies,” ASLO spring meeting, February 9-14, 2003, Salt Lake City, Utah.

Hodges, B.R. and S. K. Delavan (2004), “Numerical diffusion and dissipation in hydrostatic models in internal waves,” 17th ASCE Engineering Mechanics Conference, June 13-16, 2004, University of Delaware, Newark, Electronic Proceedings (CD-ROM).

Hodges, B.R. (2003), “A second-order correction for semi-implicit shallow water methods,” 16th ASCE Engineering Mechanics Conference, July 16-18 2003, University of Washington, Seattle, Electronic Proceedings (CD-ROM). <more info>

Hodges, B.R. (2003), “Accuracy order of Crank-Nicolson Discretization for Hydrostatic Free Surface Flow” (2004), Journal of Engineering Mechanics, 130(8): 904-910. <more info>

Laval, B., B.R. Hodges and J. Imberger, (2003). “Reducing Numerical Diffusion Effects with Pycnocline Filter” Journal of Hydraulic Engineering, 129(3): 215-224.

Laval, B., J. Imberger, B.R. Hodges, R. Stocker, (2003). “Modeling Circulation in Lakes: Spatial and Temporal Variations,” Limnology and Oceanography, 48(3): 983-994.

Wadzuk, B.M. (2004) Hydrostatic and Non-hydrostatic Internal Wave Models. Ph.D. Dissertation, Department of Civil Engineering, University of Texas at Austin, Dec 2004, 144 pgs.

Wadzuk, B.M., and B.R. Hodges (2004), “Isolation of hydrostatic regions within a basin,” 17th ASCE Engineering Mechanics Conference, June 13-16, 2004, University of Delaware, Newark, Electronic Proceedings (CD-ROM).

Wadzuk, B.M., and B.R. Hodges (2003), “Comparing hydrostatic and nonhydrostatic Navier-Stokes models of internal waves,” 16th ASCE Engineering Mechanics Conference, July 16-18 2003, University of Washington, Seattle, Electronic Proceedings (CD-ROM).

Wadzuk, B. (2002). Evolution of Internal Waves in Hydrostatic Models: A Study of Dynamic Pressure, M.S. Report, Department of Civil Engineering, University of Texas, Austin, May, 2002.