Block-Structured Adaptive Grids on the Sphere: Advection Experiments

Abstract

Abstract A spherical 2D adaptive mesh refinement (AMR) technique is applied to the so-called Lin–Rood advection algorithm, which is built upon a conservative and oscillation-free finite-volume discretization in flux form. The AMR design is based on two modules: a block-structured data layout and a spherical AMR grid library for parallel computer architectures. The latter defines and manages the adaptive blocks in spherical geometry, provides user interfaces for interpolation routines, and supports the communication and load-balancing aspects for parallel applications. The adaptive grid simulations are guided by user-defined adaptation criteria. Both statically and dynamically adaptive setups that start from a regular block-structured latitude–longitude grid are supported. All blocks are logically rectangular, self-similar, and independent data units that are split into four in the event of refinement requests, thereby doubling the horizontal resolution. Grid coarsenings reverse this refinement principle. Refinement and coarsening levels are constrained so that there is a uniform 2:1 mesh ratio at all fine–coarse-grid interfaces. The adaptive advection model is tested using three standard advection tests with increasing complexity. These include the transport of a cosine bell around the sphere, the advection of a slotted cylinder, and a smooth deformational flow that describes the roll-up of two vortices. The latter two examples exhibit very sharp edges and gradients that challenge not only the numerical scheme but also the AMR approach. The adaptive simulations show that all features of interest are reliably detected and tracked with high-resolution grids. These are steered by either a threshold- or gradient-based adaptation criterion that depends on the characteristics of the advected tracer field. The additional resolution clearly helps preserve the shape and amplitude of the transported tracer while saving computing resources in comparison to uniform-grid model runs.