Discussing sources of the β-term in a vorticity equation in rotating coordinates

Abstract

In the study of atmospheric dynamics, the vorticity equation in a rotating coordinate system plays a crucial role. However, a paradox arises when one considers the term related to spatial variations in Coriolis parameters known as the “β-term”. The β-term should not appear in the vorticity equation because the three-dimensional (3D) planetary vorticity is a constant vector. However, it is always in the vorticity equation. In this article, the source of the β-term in different rotating coordinates are investigated. The results show that in the spherical coordinate system, the β-term comes from the directions changing of one of the unit vectors (j→) with the spatial position and originates from the tilting term. By contrast, in the height coordinate system, the β-term cannot be derived from the tilting term as the individual changes of the coordinate frames with time are omitted, Instead it is proven to be related to the advection term. Although the both coordinate systems are rotating coordinate systems, the sources of their β-terms differ due to the simplification levels of the coordinate systems. Although the 3D planetary vorticity is a constant vector in the spherical coordinate system, the conversions between its components are allowable and spatial derivatives of its components can be observed, eliminating the paradox of the β-term. However, in the height coordinate system, the 3D planetary vorticity vector is not a constant vector in order to maintain the conservation of the absolute angular momentum and mechanical energy. To account for the influence of the earth’s curvature on atmospheric motion, the β-term of the Coriolis parameters varying with the latitude appears. So, the origin of the β-term paradox proposed in the height coordinate system comes from a misunderstanding of the physical constraints of the height coordinate system.