A detailed derivation of improved ocean tidal equations in continuous (COTEs) and discrete (DOTEs) forms is presented. These equations feature the Boussinesq linear eddy dissipation law with a novel eddy viscosity that depends on the lateral mesh area, i.e., on mesh size and ocean depth. Analogously, the linear law of bottom friction is used with a new bottom friction coefficient depending on the bottom mesh area. The primary astronomical tide#x2010;generating potential is modified by secondary effects due to the oceanic and terrestrial tides. The fully linearized equations are defined in a single#x2010;layer ocean basin of realistic bathymetry varying from 50 m to 7,000 m. The DOTEs are set up on a 1oby 1ospherically graded grid system, using central finite differences in connection with Richardson's staggered computation scheme. Mixed single#x2010;step finite differences in time are introduced, which enhance decay, dispersion, and stability properties of the DOTEs and facilitate#x2014;in Part II of this paper#x2014;a unique hydrodynamical interpolation of empirical tide data. The purely hydrodynamical modeling is completed by imposing boundary conditions consisting of no#x2010;flow across and free#x2010;slip along the mathematical ocean shorelines. Shortcomings of the constructed preliminary M2ocean tide charts are briefly discussed. Needed improvements of the model are left to Part II.
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