A New Terrain-Following Vertical Coordinate Formulation for Atmospheric Prediction Models

Abstract

Most numerical weather prediction models rely on a terrain-following coordinate framework. The computational mesh is thus characterized by inhomogeneities with scales determined by the underlying topography. Such inhomogeneities may affect the truncation error of numerical schemes. In this study, a new class of terrainfollowing coordinate systems for use in atmospheric prediction models is proposed. Unlike conventional systems, the new smooth level vertical (SLEVE) coordinate yields smooth coordinates at mid-and upper levels. The basic concept of the new coordinate is to employ a scale-dependent vertical decay of underlying terrain features. The decay rate is selected such that small-scale topographic variations decay much faster with height than their large-scale counterparts. This generalization implies a nonlocal coordinate transformation. The new coordinate is tested and compared against standard sigma and hybrid coordinate systems using an idealized advection test. It is demonstrated that the presence of coordinate transformations induces substantial truncation errors. These are critical for grid inhomogeneities with wavelengths smaller than approximately eight grid increments, and may overpower the regular-grid truncation error of the underlying finite-difference approximation. These results are confirmed by a theoretical analysis of the truncation error. In addition, the new coordinate is tested in idealized and real-case numerical experiments using a nonhydrostatic model. The simulations using the new coordinate yield a substantial reduction of small-scale noise in dynamical and thermodynamical model fields.

Fine‐scale modeling of the boundary layer wind field over steep topography

“…Smoothing of the coordinate surfaces above a certain height helps to reduce errors associated with the deformation of the computational mesh. This property of ARPS is attractive as the truncation error due to grid transformations has the same leading order as that associated with the solution of the dynamical equations [Schär et al, 2002].…”

Quantitative precipitation forecasting in the Alps: The advances achieved by the Mesoscale Alpine Programme

“…Besides the above-mentioned deficiency in the cloud microphysical scheme (Smith et al, 2003), the freeatmosphere temperature and wind fields were found to be heavily disturbed by grid-scale numerical noise over the Alps despite heavy smoothing of the model topography. Schär et al (2002) suggested that numerical errors related to the terrain-following coordinate system (Gal-Chen and Somerville, 1975) were responsible for these undesirable noisy structures. These numerical errors were found to be roughly proportional to the steepness of the coordinate surfaces and to severely reduce the numerical accuracy of horizontal advection.…”

“…These numerical errors were found to be roughly proportional to the steepness of the coordinate surfaces and to severely reduce the numerical accuracy of horizontal advection. To remedy these deficiencies, Schär et al (2002) have developed a generalized coordinate transformation that allows for a rapid decay with height of small-scale topographic structures in the coordinate surfaces. Their so-called SLEVE (Smooth-LEvel VErtical) coordinate has been demonstrated to greatly improve the accuracy of advection over steep topography and to reduce the noisy behaviour of the MC2.…”

Advection errors in an orthogonal terrain‐following coordinate: idealized 2‐D experiments using steep terrains

“…Note that the maximum h(x) equals h 0 , only if n is twice of an odd number. The three kinds of terrains proposed by Schär et al (2002) are shown in Figure 1(a)-(c), the one used in Li et al (2014) is shown in Figure 1(c), and three kinds of new terrains designed in this paper are illustrated in Figure 1(e)-(g). Specifically, the new terrains have the same width but double the number of the peaks, respectively, corresponding to the ones proposed by Schär et al (2002), and then lead to the increase of the slope as well as the number of the maximum slope that is mostly relevant to the high skewness of the computational grids (Figure 1(d) and (h)).…”