# Various Vertical Coordinate Systems Used for Numerical Weather Prediction

Akira Kasahara National Center for Atmospheric Research, Boulder, Colo. 80303

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## Abstract

For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated.

Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in that the features of prediction equations in each system are all distinct. We place special emphasis on handling the earth's orography as the lower boundary condition. As an extension of the original idea of Phillips applied to the pressure-coordinate system, we propose transformed height and isentropic systems. In those systems, both the top of the model atmosphere and the earth's surface are always coordinate surfaces. It is hoped that these new schemes, as in the case of the Phillips' sigma-system, will enable us to handle the effect of the earth's orography in the prediction models without lengthy coding logic.

## Abstract

For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated.

Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in that the features of prediction equations in each system are all distinct. We place special emphasis on handling the earth's orography as the lower boundary condition. As an extension of the original idea of Phillips applied to the pressure-coordinate system, we propose transformed height and isentropic systems. In those systems, both the top of the model atmosphere and the earth's surface are always coordinate surfaces. It is hoped that these new schemes, as in the case of the Phillips' sigma-system, will enable us to handle the effect of the earth's orography in the prediction models without lengthy coding logic.

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