A mathematical model of the bathymetry-generated external gravitational field: The currently available global geopotential models and the global elevation and bathymetry data allow modelling the topography-corrected and bathymetry stripped reference gravity field to a very high spectral resolution (up to degree 2160 of spherical harmonics) using methods for a spherical harmonic analysis and synthesis of the gravity field. When modelling the topography-corrected and crust-density-contrast stripped reference gravity field, additional stripping corrections are applied due to the ice, sediment and other major known density contrasts within the Earth's crust. The currently available data of global crustal density structures have, however, a very low resolution and accuracy. The compilation of the global crust density contrast stripped gravity field is thus limited to a low spectral resolution, typically up to degree 180 of spherical harmonics. In this study we derive the expressions used in forward modelling of the bathymetry-generated gravitational field quantities and the corresponding bathymetric stripping corrections to gravity field quantities by means of the spherical bathymetric (ocean bottom depth) functions. The expressions for the potential and its radial derivative are formulated for the adopted constant (average) ocean saltwater density contrast and for the spherical approximation of the geoid surface. These newly derived expressions are utilized in numerical examples to compute the gravitational potential and attraction generated by the ocean density contrast. The computation is realized globally on a 1 x 1 arc-deg geographical grid at the Earth's surface.