On the Importance of the Nontraditional Coriolis Terms in Large-Scale Motions Associated with Cumulus Convective Forcing in the Tropics: Forced Response Problem

In meteorological dynamics, when the shallow atmosphere approximation is assumed in the momentum equations, the “traditional approximation” must also be used, which excludes some metric terms and two cosine Coriolis terms (hereafter, nontraditional Coriolis terms or NCTs). However, some studies suggest that the elimination of the NCTs may not be adequate in the tropics, especially in motions interacting with cumulus convection (e.g., White and Bromley 1995). We therefore quantitatively investigate the effect of the NCTs on large-scale motions in the tropics. Numerical calculations are conducted using the Quasi-Hydrostatic Equations (QHEs) with local positive-only diabatic forcing mimicking cumulus convection. The forcing is assumed to move eastward with an intraseasonal period. We hereafter use a word, contribution, as the difference between results with the NCTs and without them, which denotes just the effect of the NCTs.

Results show that contributions have the following four features. (i) Contributions to vertical vorticity (horizontal winds) and perturbations of pressure, potential temperature and density are large; on the contrary, vertical motion and horizontal divergence are little affected. (ii) Equivalent barotropic structure is shown up in contributions to vertical vorticity and pressure perturbation. (iii) Contributions to vorticity and pressure perturbation become large when the meridional gradient of the forcing is large. (iv) All variables are largely affected at the west-side of the forcing.

The above features are physically and comprehensively understood by considering the tilting of meridional component of the planetary vorticity vector due to the meridional gradient of vertical motion, i.e., diabatic heating. If the meridional gradient is large, this tilting also becomes strong. Because the tilting is directly connected to vertical vorticity, and the tilted vorticity adjusts pressure, then contributions to them are large. The equivalent barotropic structure comes from that of vertical motion, which has vertically standing one. Then, potential temperature and density perturbations are “quasi-hydrostatically” balanced with the others so that contributions to them are also large. In contrast, because of the strong relation between vertical motion and diabatic heating, the NCTs little affect vertical motion and horizontal divergence, which closely related to vertical motion. In addition, large contributions at the west-side of the forcing are explained as a Rossby response. The above results indicate that in order to reduce dynamical error caused by the exclusion of the NCTs, we should use the nonhydrostatic deep (i.e., not assuming the shallow atmosphere approximation) or quasi-hydrostatic equations as a dynamical core in global numerical models.