Abstract
Z-grid finite volume models conserve all-scalar quantities as well as energy and potential enstrophy and yield better dispersion relations for shallow water equations than other finite volume models, such as C-grid and C-D grid models; however, they are more expensive to implement. During each time integration, a Z-grid model must solve Poisson equations to convert its vorticity and divergence to a stream function and velocity potential, respectively. To optimally utilize these conversions, we propose a model in which the stability and possibly accuracy on the sphere are improved by introducing more stencils, such that a generalized Z-grid model can utilize longer time-integration steps and reduce computing time. Further, we analyzed the proposed model’s dispersion relation and compared it to that of the original Z-grid model for a linearly rotating shallow water equation, an important property for numerical models solving primitive equations. The analysis results suggest a means of balancing stability and dispersion. Our numerical results also show that the proposed Z-grid model for a shallow water equation is more stable and efficient than the original Z-grid model, increasing the time steps by more than 1.4 times.
Highlights
The societal demand for methods that facilitate future accurate weather prediction is pushing resolvable scales of global weather forecasts down to the kilometer range
They choose different global grid structures and forms of the finite element methods. These finite element methods can be classified into several types of models according to the grid arrangements of the model state variables: A-grid, C-grid, and C-D grid [1,2,3,4,5] as well as Z-grid (Heikes and Randall [6])
We investigated the possibility of modifying the Z-grid scheme to improve its stability such that a longer time step could be used in model integration
Summary
The societal demand for methods that facilitate future accurate weather prediction is pushing resolvable scales of global weather forecasts down to the kilometer range. To improve the efficiency and possible accuracy on spheres, we propose the generalization of the Voronoi Z-grid model by using state variables at the centroidal centers and the same grid stencil for calculating both the normal and tangential derivatives. The motivation of this generalization in this study was to seek a more stable scheme so that it allowed us to apply longer time steps than those applied in the Voronoi Z-grid.
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