Abstract
Abstract. We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to overcome two well-known problems in the classic σ coordinate, namely, pressure gradient force (PGF) errors and advection errors. First, in the design of basis vectors, we rotate the basis vectors of the z coordinate in a particular way in order to reduce the PGF errors and add a special rotation parameter b to each rotation angel in order to reduce the advection errors. Second, the corresponding definition of each OS coordinate is solved through its basis vectors. Third, the scalar equations of the OS coordinate are solved by expanding the vector equation using the basis vectors. Since the computational form of PGF has only one term in each momentum equation of the OS coordinate, the PGF errors will be significantly reduced, according to Li et al. (2012). When a proper b is chosen, the σ levels over a steep terrain can be significantly smoothed, therefore alleviating the advection errors in the OS coordinate. This is demonstrated by a series of 2-D linear advection experiments under a unified framework.