GMD Geoscientific Model Development GMD Geosci. Model Dev. 1991-9603 Copernicus Publications Göttingen, Germany 10.5194/gmd-7-1767-2014 An orthogonal terrain-following coordinate and its preliminary tests using 2-D idealized advection experiments Li Y. 1 Wang B. 1 2 Wang D. 3 Li J. 1 Dong L. 1 State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China Ministry of Education Key Laboratory for Earth System Modeling, and Center for Earth System Science, Tsinghua University, Beijing, China State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China 25 08 2014 7 4 1767 1778 Copyright: © 2014 Y. Li et al. 2014 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://gmd.copernicus.org/articles/7/1767/2014/gmd-7-1767-2014.html The full text article is available as a PDF file from https://gmd.copernicus.org/articles/7/1767/2014/gmd-7-1767-2014.pdf

We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal &sigma; coordinate, or the OS coordinate) to reduce the advection errors in the classic &sigma; coordinate. First, we rotate the basis vectors of the <i>z</i> coordinate in a specific way in order to obtain the orthogonal, terrain-following basis vectors of the OS coordinate, and then add a rotation parameter <i>b</i> to each rotation angle to create the smoother vertical levels of the OS coordinate with increasing height. Second, we solve the corresponding definition of each OS coordinate through its basis vectors; and then solve the 3-D coordinate surfaces of the OS coordinate numerically, therefore the computational grids created by the OS coordinate are not exactly orthogonal and its orthogonality is dependent on the accuracy of a numerical method. Third, through choosing a proper <i>b</i>, we can significantly smooth the vertical levels of the OS coordinate over a steep terrain, and, more importantly, we can create the orthogonal, terrain-following computational grids in the vertical through the orthogonal basis vectors of the OS coordinate, which can reduce the advection errors better than the corresponding hybrid σ coordinate. However, the convergence of the grid lines in the OS coordinate over orography restricts the time step and increases the numerical errors. We demonstrate the advantages and the drawbacks of the OS coordinate relative to the hybrid σ coordinate using two sets of 2-D linear advection experiments.

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