An orthogonal terrain-following coordinate and its preliminary tests using 2-D idealized advection experiments

Y Li,D Wang,J Li,B Wang,L Dong

doi:10.5194/gmd-7-1767-2014

Abstract

Abstract. We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to reduce the advection errors in the classic σ coordinate. First, we rotate the basis vectors of the z 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 b 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 b, 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.

Highlights

The complex surface of the Earth is the lower boundary of a numerical atmospheric model, which has become more and more important for operational forecast and scientific research
There are mainly two kinds of methods to deal with the terrain in a model: using a proper vertical coordinate, such as the terrain-following coordinate proposed by Phillips (1957), or using the cut-cell method that has been used in computational fluid dynamics and recently been adapted by many researchers for simulating atmospheric and oceanic flows over irregular geometry (Adcroft et al, 1997; Yamazaki and Satomura, 2010; Lock et al, 2012; Adcroft, 2013; Good et al, 2014; Steppeler et al, 2013)
We aim to reduce the well-known advection errors of the classic σ coordinate (CS coordinate) through designing a 3-D orthogonal curvilinear terrain-following coordinate (OS coordinate) in a unique way

Summary

Introduction

The complex surface of the Earth is the lower boundary of a numerical atmospheric model, which has become more and more important for operational forecast and scientific research. A smooth terrain-following 2 The orthogonal curvilinear terrain-following (STF) coordinate was proposed, which can smooth the σ levcoordinate els much more than the SLEVE and which has been implemented in the MPAS model (Klemp, 2011, 2012) Note that all these methods have been successful at alleviating the advection errors in the σ coordinate via smoothing the σ coordinate levels above a steep terrain; they did not tackle the non-orthogonal basis vectors of the σ coordinate. Through the two rotation angles θ and λ shown, we can solve the expressions of these basis vectors, which are orthogonal and terrain following (see Supplement A for the detail).

Idealized experiments used to compare advection errors

Parameters in 2-D linear advection experiments

Conclusion and discussion