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The deep-atmosphere Euler equation set has been derived into a form similar to the shallow-atmosphere Euler equation set without any approximation, which makes it more convenient to evolve any existing shallow-atmosphere dynamics model to deep-atmosphere dynamics.The deep atmospheric dynamics system is a fully non-approximated Euler equation set, which includes all-dimensional Coriolis force, vertically expanded cells (r=a+z), and geocentric gravitational force with height. The deep-atmosphere Euler equation in spherical height coordinates was transferred into spherical generalized vertical coordinates with pseudo horizontal wind in NCEP Office Note 477 (Juang 2014). The deep-atmosphere continuity equation in generalized vertical coordinates can be easily converted into a shallow-atmosphere-alike form by introducing scaled horizontal winds and scaled height with a hydrostatic relationship in generalized coordinates to represent density. The relationship provides a coordinate pressure change with respect to scaled height for a given density. From this relationship we can use geopotential height to define scaled vertical motion, thus all the three-dimensional winds in deep-atmosphere dynamics can be replaced by scaled winds. Since all winds are using a scaled form, the momentum equations are converted into scaled momentum equations with scaled terms plus some add-on terms.

The deep-atmospheric nonhydrostatic global dynamics are introduced with detailed discretization on spherical and generalized vertical coordinates. Based on the NCEP global spectral model, the horizontal discretization (which is not described in this manuscript) uses the spectral method with spherical spectral transformation; the vertical discretization described in this paper is illustrated in detail up to the level of readiness for programming. The primitive equations contain three-dimensional momentum, enthalpy as a thermodynamic variable, density, and tracers in height coordinates which are used to convert to generalized vertical coordinates with virtual horizontal winds for spherical coordinates. The equations are examined to show their characteristics of multiple conservations, which are mass conservation, angular momentum conservation, entropy conservation, and total energy conservation. The concept of mean pressure at any given level by projecting unit air weight on mean earth radius surface is utilized to have a mass coordinate, which results in a similar formulation of the density equation in a hydrostatic system. The mean pressure at a given model level, obtained from the weight concept, is called a coordinate pressure, which has the property of a monotonic decrease with height suitable for the coordinate system. The angular momentum conservation leads to a discretization for the relationship among coordinate pressure, height, and temperature, which is similar to the hydrostatic relationship in a hydrostatic system, also deduces a relationship for heights between model levels and model layers. The total energy conservation is obtained from three dimensional momentum equations, geopotential height, and the thermodynamic equation. To do total energy conservation, we have a discretization for the total derivative of pressure, which is discretized from the momentum equation and used for the thermodynamic equation, to ensure total energy conservation. The potential enthalpy conservation is also applied to the vertical advection for enthalpy in Eulerian system, which requires multiplying enthalpy to vertical advection of logarithmic enthalpy. Since sigma-pressure vertical coordinates are used in the NCEP GFS, we give a specific discretization in sigma-pressure hybrid vertical coordinates. The two-time-level semi-implicit semi-Lagrangian scheme is used as example for time integration discretization. The linearization of all prognostic equations is required for the semiimplicit time scheme. The matrices used in the semi-implicit time scheme for linear terms are listed in appendices along with cold start initial fields from the hydrostatic system and detailed derivations for the continuity equation from the height coordinate to generalized hybrid vertical coordinates.