Optimized Vertical Layers for the Hybrid Terrain-Following Coordinate Minimizing Numerical Errors in a 2D Rising Bubble Experiment near Steep Terrain

The basic terrain-following (BTF) coordinate simplifies the lower boundary conditions of a numerical model but leads to numerical error and instability on steep terrain. Hybrid terrain-following (HTF) coordinates with smooth slopes of vertical layers (slopeVL) generally overcome this difficulty. Therefore, the HTF coordinate becomes very desirable for atmospheric and oceanic numerical models. However, improper vertical layering in HTF coordinates may also increase the incidence of error. Except for the slopeVL of an HTF coordinate, this study further optimizes the HTF coordinate focusing on the thickness of vertical layers (thickVL). Four HTF coordinates (HTF1–HTF4) with similar slopeVL but different vertical transition methods of thickVL are designed, and the relationship between thickVL and numerical errors in each coordinate is compared in the classic idealized thermal convection two-dimensional (2D) rising bubble experiment over steep terrain. The errors of potential temperature \theta and vertical velocity w are reduced most, by approximately 70% and 40%, respectively, in the HTF1 coordinate, with a monotonic increase in thickVL according to the increasing height; however, the errors of \theta increased in all the other HTF coordinates, with nonmonotonic thickVLs. Furthermore, analyses of the errors of vertical pressure gradient force (VPGF) show that due to the interpolation errors of thickVL, the inflection points in the vertical transition of thickVL induce the initial VPGF errors; therefore, the HTF1 coordinate with a monotonic increase in thickVL has the smallest errors among all the coordinates. More importantly, the temporal evolution of VPGF errors manifests top-type VPGF errors that propagate upward gradually during the time integration. Only the HTF1 and HTF4 coordinates with a monotonic increase in thickVL near the top of the terrain can suppress this propagation. This optimized HTF coordinate (i.e., HTF1) can be a reference for designing a vertical thickVL in a numerical model.