Abstract
AbstractA terrain-following coordinate (σ-coordinate) in which the computational form of pressure gradient force (PGF) is two-term (the so-called classic method) has significant PGF errors near steep terrain. Using the covariant equations of the σ-coordinate to create a one-term PGF (the covariant method) can reduce the PGF errors. This study investigates the factors inducing the PGF errors of these two methods, through geometric analysis and idealized experiments. The geometric analysis first demonstrates that the terrain slope and the vertical pressure gradient can induce the PGF errors of the classic method, and then generalize the effect of the terrain slope to the effect of the slope of each vertical layer (φ). More importantly, a new factor, the direction of PGF (α), is proposed by the geometric analysis, and the effects of φ and α are quantified by tan φ·tan α. When tan φ·tan α is greater than 1/9 or smaller than −10/9, the two terms of PGF of the classic method are of the same order but opposite i...
Highlights
Since a terrain-following coordinate (σ-coordinate) (Phillips 1957) can transform the complex surface of Earth into a regular coordinate surface, the σ-coordinate becomes a common choice for atmospheric and oceanic models
Where h(x) is the terrain, H is the maximum height of the terrain, p0 = 1015.0 hPa is the surface pressure, pt = 10.0 hPa is the top pressure, λ = 8 km is the typical height of the atmosphere, Ht is the top of the domain, and Hp is a parameter to adjust the direction of pressure gradient force (PGF)
We use the results obtained by the experiments of H = 14 km as an example to analyze the variation of the PGF errors of the classic and covariant method according to the increasing TT (Figure 4)
Summary
Since a terrain-following coordinate (σ-coordinate) (Phillips 1957) can transform the complex surface of Earth into a regular coordinate surface, the σ-coordinate becomes a common choice for atmospheric and oceanic models. The third type is to design an orthogonal terrain-following coordinate to bypass the PGF errors (Li et al 2014) Both methods proposed by Li, Wang, and Wang (2012) and Li et al (2014) create a one-term computational form of PGF, while Li, Wang, and Wang (2012) proposed to use the covariant scalar equations of the σ-coordinate (the covariant method).
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