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Discrete generalized hybrid vertical coordinates by a mass, energy and angular momentum conserving vertical finite-difference scheme
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    A detailed discretization of a hydrostatic primitive equation global atmospheric model on spherical and generalized hybrid vertical coordinates is described. The discretization in the horizontal using a spectral method with spherical transformation is not the major theme and is not described in this manuscript. Only the vertical discretiztion is described in detail up to the level of readiness for programming. Energy and angular momentum conservation are used as constraints to discretize the vertical integration by finite difference scheme. The entire atmosphere is divided into several layers; only pressure and vertical flux are specified at the interfaces, and other variable such as horizontal wind, temperature, specific humidity at the interfaces, and other variable such as horizontal wind, temperature, specific humidity and specific amount of tracers are specified at each layer. Conservation is a constraint that requires the pressure at each layer to be averages by the pressures at the immediate neighbor interfaces (the one above and one below a given layer). Since pressures are not combined from a pressure gradient and density in a logarithmic form, the relationship for pressure between layers and interfaces becomes simple, and with pressure equation not in logarithmic form, it provides mass conservation as an extra. Due to the generalized vertical coordinate, vertical flux is solved by applying local changes in the pressure and vertical temperature equations to the definition of the vertical coordinate, It solves vertical fluxes at all interfaces by a simple algebraic equation through matix inversion. For the sake of time splitting between dynamics and physics, the vertical flux obtained in the dynamics is without local changes from model physics, and then the vertical advection is required in the model physics. The semi-implicit time integration scheme is also given by this finite difference scheme in generalized vertical coordinates. The details of the matixes are described so as to be ready for programming. A specific definition of a generalized hybrid coordinate, including pressure and isentropic surfaces, is introduced. Due to this definition, pressure is given by surface pressure and virtual temperature. These modifications of the generalized form are neccesary to save computational resources. Instead of solving for the pressure at all interfaces, only the surface pressure equation is needed. Though the elements in the matrixes for semi-implicit computations become more complicated than those in the generalized pressure equation at all levels, the computing time is not increased because the matrixes are of the same degree; even two matrixes reduce to vector computation only because the surface pressure is solved instead of pressure at all the interfaces.
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