Abstract
Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.
Highlights
The basic terrain-following (BTF) coordinate has been widely employed in numerical weather prediction (NWP) models due to its simplicity and ability to accurately represent the terrain [1,2,3,4,5,6,7].advection errors can occur in a BTF coordinate system when the velocity field is not aligned with terrain-following (TF) vertical layers (VLs)
Such dynamically adaptive meshes have numerous advantages over other widely used meshes and existing adaptive mesh refinement (AMR) methods; for example, a dynamically adaptive mesh can accurately represent the boundary conditions over complex bathymetries
Advection errors pose a significant computational problem in atmospheric modelling over steep terrains. Such errors are induced primarily by the misalignment of the flow direction with respect to the VLs, distortions of the grid and the complexity of the underlying terrain. To reduce these errors and to avoid wasting computational resources, this study introduces adaptive meshes that adapt to variations in the tracer concentration as the flow evolves and become sparser at locations where the gradients of variable solutions are relatively low
Summary
The basic terrain-following (BTF) coordinate has been widely employed in numerical weather prediction (NWP) models due to its simplicity and ability to accurately represent the terrain [1,2,3,4,5,6,7]. Atmosphere 2018, 9, 444 in NWP models are becoming progressively steeper This resolution improvement is making it increasingly difficult to accurately represent advection processes over steep terrains. This difficulty may even cause numerical instabilities [8] To reduce this source of advection errors, model developers have proposed the use of either special vertical coordinates or computational grid/mesh approaches to address this problem. Shaw and Weller [5] confirmed that the advection accuracy depends on the alignment of the velocity field with the VLs, indicating that a TF coordinate should be able to effectively simulate TF advection processes, while the cut-cell method is preferred for horizontal advection problems.
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