Comparing Hydrostatic and Nonhydrostatic Navier-stokes Models of Internal Waves

Internal waves occur at a variety of temporal and spatial scales and are mechanisms by which momentum, energy, nutrients and biota are transported in lakes, estuaries and coastal oceans. In stratified systems, accurate prediction of water quality effects requires modeling internal wave evolution and propagation. Lake, estuary, and coastal ocean models typically apply the hydrostatic approximation to the Navier-Stokes equations, which limits the accuracy of internal wave predictions. The hydrostatic approximation exploits the small aspect ratio (vertical:horizontal length scale) of natural systems, which makes the vertical acceleration and dynamic pressure negligible relative to the horizontal acceleration and hydrostatic pressure (Marshall et al, 1997). The hydrostatic approximation also eliminates the need to solve a three-dimensional Poisson problem for dynamic pressure, thereby dramatically decreasing computational requirements. The hydrostatic approximation is adequate for large-scale ocean processes, but breaks down for scales less than ten kilometers (Kantha and Clayson, 2000). At the mesoscale (10-100 km in horizontal extent, depths of order 1000 m, and horizontal velocities of 0.1 – 1 m s), the aspect ratio may no longer be considered small, therefore dynamic pressure and vertical acceleration are not negligible and the hydrostatic approximation will not effectively model system dynamics. The exclusion of dynamic pressure in a model may distort the balance between internal wave steepening and dispersion such that the small error of neglecting dynamic pressure accumulates into a large error in the long-term prediction of wave propagation. Thus, inclusion of vertical acceleration and nonhydrostatic pressure is necessary to properly model the evolution of internal waves (Long, 1972). As a simple, monochromatic wave in a nonlinear system evolves, the system’s nonlinearities cause the internal wave to slowly steepen. If there is no force balancing the steepening, the wave will propagate unabated, shorten, and eventually overtop itself causing a mixing event to occur. The full Reynolds-averaged Navier-Stokes (RANS) momentum equation contains both vertical acceleration (the advective term) and dynamic pressure, Eq. (1):