2013 , Volume 18, № 1, p.45-64

Various geometrical approximations for the rotating shallow water equations are considered. The first approximation is a transformation from the equations on the ellipsoid to the equations on the sphere. The second approximation is to proceed from the equations on the sphere to the equations on the tangent plane or the surface. The approximate equations for all geometric approximations are obtained. The main requirement for constructing these approximations is a conservation of the Hamiltonian structure. This goal was achieved in two ways. First, because the metric tensor of the surface determines the Hamiltonian structure of the equations on it, so the first way was to appropriately choose the approximate equations, part of the complete system, which is associated with the corresponding expansion of the metric tensor. Second, it was observed that the Poisson bracket for the covariant components of the velocity is almost independent of the Lame coefficients, so the Hamiltonian contains the main dependence on the Lame coefficients and all approximations can be reduced to the expansion of this Hamiltonian.