3-D tide model Description

The three-dimensional baroclinic tide model is modified from the Princeton Ocean Model described in Blumberg and Mellor (1987). The nonlinear primitive equation model with Boussinesq and hydrostatic approximations is driven by the barotropic tidal forcing. The vertical axis is transformed to the σ-coordinate by σ=(z-η)/(H+η), where z is positive upward with the origin placed at the mean sea level, η is the sea level fluctuation and H is the mean water depth. The governing equations are as follows.

where (U, V) are horizontal velocity components in the x and y directions, w is the velocity component normal to σ-surface, D is total water depth (D=Η+η), t is time, f is the Coriolis parameter,ρ ' is perturbation density, P’ is perturbation pressure, (ADV_U, ADV_V), (GRAD_xP’, GRAD_yP’) and (DIF_U, DIF_V are advection, pressure gradient and diffusion terms, respectively, in the horizontal momentum equations (Mellor, 2004), β (=0.940 for diurnal tides and 0.953 for semidiurnal tides) represents the loading effect due to ocean tides (Foreman et al., 1993), g is the gravitational acceleration, represent depth-averaged horizontal velocities, φ can be temperature (T) or salinity (S), and ADV_φ and DIF_φ are advection and diffusion terms in temperature and salinity equations. The nonlinear interactions between internal tides and internal waves are parameterized by linear damping termsandin (1) and (2), respectively, where r is a damping coefficient which is set to 0.2 days^-1 as suggested by Niwa and Hibiya (2001 and 2004). The adjust height of equilibrium tides (ζ) is calculated using the formulation described in Pugh (1987) as

where f_c is nodal factor, ω_c is frequency of corresponding tidal constituent, V₀ is initial phase angle of the equilibrium tides, μ is nodal angle, m=1 or 2 accounts for diurnal or semidiurnal constituents, respectively, λ is longitude, subscript c represents tidal constituent, and H_c (=a1 sin2φ or a2 cos²φ for diurnal or semidiurnal tides, φ:latitude) is amplitude of equilibrium tides multiplied by the factor 1+k-h (k and h are Love numbers due to the elastic response and the redistribution of the mass of earth). Table 1 lists a1 and a2 for diurnal and semidiurnal constituents, respectively.

Table 1. Parameters for calculating the adjusted height of equilibrium tides.

The model is bounded within 99.25°–135.25°E and 2.25°–43.25°N with (1/12)° horizontal resolution. Figure 1 shows the model domain. There are 51 uneven σ layers in the vertical with σ_k=(0, 0.002, 0.004, 0.006, 0.008, 0.01, 0.012, 0.014, 0.018, 0.022, 0.026, 0.03, 0.034, 0.037, 0.045, 0.053, 0.061, 0.069, 0.077, 0.085, 0.1, 0.116, 0.132, 0.148, 0.179, 0.211, 0.243, 0.274, 0.306, 0.337, 0.369, 0.4, 0.432, 0.464, 0.495, 0.527, 0.558, 0.59, 0.621, 0.653, 0.684, 0.716, 0.748, 0.779, 0.811, 0.842, 0.874, 0.905, 0.937, 0.968, 1), from k=1 (surface) to 51 (bottom). The bottom topography was established using the revised ETOPO2
(http://www.ngdc.noaa.gov/mgg/global/relief/ETOPO2/ETOPO2v2-2006/ETOPO2v2c)
supplement with a 1-min depth archive in the region of 105°–135°E and 2.25°–35°N provided by the National Center for Ocean Research (NCOR) of Taiwan.

The motionless ocean is subsequently driven by the tidal potential and prescribed tidal sea levels on all open-ocean boundaries through a forced radiation condition similar to that used by Blumberg and Kantha (1985). The tidal sea levels on the open boundaries are computed using harmonic constants compiled in a database (hereafter NAO.99) described in Matsumoto et al. (2000). The depth-averaged tidal current velocity normal to the open boundaries is determined by

where u_2D is calculated from a fine-tuned, two-dimensional tide model of Jan et al. [2004], C (= ) is the phase speed of a shallow water gravity wave, η_B is sea level calculated by the three-dimensional tidal model, and η_pre is prescribed tidal sea level on open boundaries. A flow relaxation scheme described by Engedahl (1995) is applied to the internal mode velocity and temperature in a 1° wide strip adjacent to open boundaries to eliminate artificial reflections. The essential model settings are summarized in Table 2.

Table 2. Model specification.

Blumberg, A. F. and L. H. Kantha (1985), Open boundary condition for circulation models. J. of Hydra. Eng., 111, 2, 237-255.

Blumberg, A. F. and G. F. Mellor (1987), A description of a three dimensional coastal ocean circulation model. In: Three-Dimensional Coastal Ocean Models, Coastal and Estuarine Stud., vol. 4, edited by N. Heaps, pp. 1-16, AGU, Washington D.C.

Engedahl, H. (1995), Use of the flow relaxation scheme in a three-dimensional baroclinic ocean model with realistic topography, Tellus, Ser. A, 47, 365-382.

Foreman, M. G. G., R. F. Henry, R. A. Walters, and V. A. Ballantyne (1993), A finite element model for tides and resonance along the north coast of British Columbia. J. Geophys. Res., 98, 2509-2532.

Matsumoto, K., T. Takanezawa, and M. Ooe (2000), Ocean tide models developed by assimilating Topex/poseidon altimeter data into hydrodynamical model: a global model and a regional model around Japan. J. Oceanogr., 56, 567-581.

Mellor, G. L. and T. Yamada (1982), Development of turbulence closure model for geophysical fluid problems. Rev. Geophys., 20, 851-875.

Mellor, G. L. (2004), Users guide for a three-dimensional, primitive equation, numerical ocean model.
http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/

Niwa, Y. and T. Hibiya (2001), Numerical study of the spatial distribution of the M2 internal tide in the Pacific Ocean. J. Geophys. Res., 106, 22441-22449.

Niwa, Y. and T. Hibiya (2004), Three-dimensional numerical simulation of M2 internal tides in the East China Sea. J. Geophys. Res., 109, C04027, doi:10.1029/2003JC001923.

Pugh, D. T. (1987), Tides, surges and mean sea-level. Wiley, Chichester, 471 pp.

Smagorinsky, J. S. (1963), General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weather Rev., 91, 99-164.

U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Geophysical Data Center (2001), 2-minute Gridded Global Relief Data (ETOPO2)
http://www.ngdc.noaa.gov/mgg/fliers/01mgg04.html