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    The Derivation of a Terrain-Following Coordinate System for Use in a Hydrostatic Model

    Source: Journal of the Atmospheric Sciences:;1981:;Volume( 038 ):;issue: 008::page 1707
    Author:
    Pielke, Roger A.
    ,
    Martin, Charles L.
    DOI: 10.1175/1520-0469(1981)038<1707:TDOATF>2.0.CO;2
    Publisher: American Meteorological Society
    Abstract: This article uses tensor transformation procedures in order to derive a terrain-following coordinate system that is frequently used in a number of regional and mesoscale hydrostatic models. Tensor transformation procedures are used so as to ensure physical invariance of the primitive equations between the Cartesian and terrain-following systems. Among the major conclusions are as follows: Applying the chain rule to the hydrostatic equation, before transforming from a Cartesian to a terrain-following coordinate system, yields a different set of equations than if the hydrostatic assumption is applied after the tensor transformation is made. The hydrostatic equations in the two terrain-following representations are the same only when the slope of the terrain in the model is much less than 45°. Variations of the metric tensor across a grid volume appear in the set of conservation equations as a result of averaging the equations over a grid volume. Such deviations have always been ignored in existing non-hydrostatic and hydrostatic meteorological models. Care must be taken to assure that parameterizations which are a function of distance above the ground be defined in terms of the original Cartesian system, and not the new generalized vertical coordinate σ. The profile exchange coefficient K(z), for example, cannot be defined simply by replacing z by σ.
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      The Derivation of a Terrain-Following Coordinate System for Use in a Hydrostatic Model

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    http://yetl.yabesh.ir/yetl1/handle/yetl/4154153
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    contributor authorPielke, Roger A.
    contributor authorMartin, Charles L.
    date accessioned2017-06-09T14:22:25Z
    date available2017-06-09T14:22:25Z
    date copyright1981/08/01
    date issued1981
    identifier issn0022-4928
    identifier otherams-18177.pdf
    identifier urihttp://onlinelibrary.yabesh.ir/handle/yetl/4154153
    description abstractThis article uses tensor transformation procedures in order to derive a terrain-following coordinate system that is frequently used in a number of regional and mesoscale hydrostatic models. Tensor transformation procedures are used so as to ensure physical invariance of the primitive equations between the Cartesian and terrain-following systems. Among the major conclusions are as follows: Applying the chain rule to the hydrostatic equation, before transforming from a Cartesian to a terrain-following coordinate system, yields a different set of equations than if the hydrostatic assumption is applied after the tensor transformation is made. The hydrostatic equations in the two terrain-following representations are the same only when the slope of the terrain in the model is much less than 45°. Variations of the metric tensor across a grid volume appear in the set of conservation equations as a result of averaging the equations over a grid volume. Such deviations have always been ignored in existing non-hydrostatic and hydrostatic meteorological models. Care must be taken to assure that parameterizations which are a function of distance above the ground be defined in terms of the original Cartesian system, and not the new generalized vertical coordinate σ. The profile exchange coefficient K(z), for example, cannot be defined simply by replacing z by σ.
    publisherAmerican Meteorological Society
    titleThe Derivation of a Terrain-Following Coordinate System for Use in a Hydrostatic Model
    typeJournal Paper
    journal volume38
    journal issue8
    journal titleJournal of the Atmospheric Sciences
    identifier doi10.1175/1520-0469(1981)038<1707:TDOATF>2.0.CO;2
    journal fristpage1707
    journal lastpage1713
    treeJournal of the Atmospheric Sciences:;1981:;Volume( 038 ):;issue: 008
    contenttypeFulltext
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