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Derivation of Slope Flow Equations Using Two Different Coordinate Representations

R. A. PielkeDepartment of Atmospheric Science, Colorado State University, Fort Collins, CO 80523

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M. SegalDepartment of Atmospheric Science, Colorado State University, Fort Collins, CO 80523

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R. T. McNiderK.E. Johnson Environmental and Energy Center, University of Alabama, Huntsville, AL 35899

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Y. MahrerCooperative Institute for Research in the Atmosphere, Colorado State Univesity, Fort Collins, CO 80523

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Abstract

This paper examines two coordinate representations for slope flow models, one a rotation of the coordinate axes, the other a generalized vertical coordinate transformation. An analytic solution is developed in both representations for a uniform slope to examine the differences due to slightly different forms of a generalised hydrostatic equation. For the first transformation, velocity acclerations in the direction of the generalized vertical coordinate are ignored, while for the second transformation, velocity accelerations perpendicular to the terrain are neglected. Surprisingly, only the period of flow oscillation and not the mean strength of the slope flow was changed in using the first coordinate representation instead of the second. Only for slopes greater than 45° does the difference in periods between the two transformations 30%. Differences which may occur for nonuniform slopes, however, still need to be examined.

Abstract

This paper examines two coordinate representations for slope flow models, one a rotation of the coordinate axes, the other a generalized vertical coordinate transformation. An analytic solution is developed in both representations for a uniform slope to examine the differences due to slightly different forms of a generalised hydrostatic equation. For the first transformation, velocity acclerations in the direction of the generalized vertical coordinate are ignored, while for the second transformation, velocity accelerations perpendicular to the terrain are neglected. Surprisingly, only the period of flow oscillation and not the mean strength of the slope flow was changed in using the first coordinate representation instead of the second. Only for slopes greater than 45° does the difference in periods between the two transformations 30%. Differences which may occur for nonuniform slopes, however, still need to be examined.

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