An explicit flux‐form semi‐lagrangian shallow‐water model on the sphere

Abstract

SUMMARYA global shallow-water model based on the flux-form semi-lagrangian scheme is described. The massconserving flux-form semi-Lagrangian scheme is a multidimensional semi-lagrangian extension of the higher order Godunov-type finite-volume schemes (e.g., the piece-wise parabolic method). Unlike the piece-wise parabolic methodology, neither directional splitting nor a Riemann solver is involved. A reverse engineering procedure is introduced to achieve the goal of consistent transport of the absolute vorticity and the mass, and hence, the potential vorticity. Gravity waves are treated explicitly, in a manner that is consistent with the forward-in-time flux-form semi-Lagrangian transport scheme. Due to the finite-volume nature of the flux-form semi-lagrangian scheme and the application of the monotonicity constraint, which can be regarded as a subgrid-scale flux parametrization, essentially noise-free solutions are obtained without additional diffusion. Two selected shallow-water test cases proposed by Williamson et al. (1992) and a stratospheric vortex erosion simulation are presented. Discussions on the accuracy and computational efficiency are given based on the comparisons with a Eulerian spectral model and two advective-form semi-implicit semi-Lagrangian models.

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

“…The ÿnite volume shallow water equations model of Lin and Rood [18] has been used for integrating the above SW equations. The 1-D advection schemes described thus-far have been implemented in two dimensions by using a sequential operator-split approach, details of which have been provided in Reference [15].…”

“…More details are available in Reference [18]. This represents excellent conservation of integral invariants of the shallow water equations.…”

“…Figures 3-10 provide results obtained by integration of the ÿnite volume SW equations model of Reference [18] using the di erent advection schemes for 14 and 30 days, respectively (the initial condition was speciÿed to be a Rossby-Haurwitz wave). The resolution of the model is the same as in Reference [18], 128 grid cells along the longitude and 64 along the latitude, and a time step of 600 s. The DAY-14 solution in the case of constrained van Leer and PPM schemes is similar, whereas the DAY-30 solution obtained by using the constrained van Leer is more di used than that of the PPM scheme.…”

“…The resolution of the model is the same as in Reference [18], 128 grid cells along the longitude and 64 along the latitude, and a time step of 600 s. The DAY-14 solution in the case of constrained van Leer and PPM schemes is similar, whereas the DAY-30 solution obtained by using the constrained van Leer is more di used than that of the PPM scheme. The ÿrst-order advection scheme is extremely dissipative, as evident from Figures 3 and 4.…”

“…Lin and Rood [15] have compared the ÿrst-order upwind, central di erence, PPM (modiÿed monotonic and positive deÿnite) and monotonic van Leer schemes [16]. Towards the development of a fully operational atmospheric general circulation model based on FV discretization [17], Lin and Rood [18] have implemented slope limited van Leer schemes and the PPM scheme on a shallow water equations model using a semi-Lagrangian semi-implicit time integration scheme. For a discussion and applications of other popular schemes such as MPDATA of Smolarkiewicz [19,20] and QUICK of Leonard [21][22][23], see Reference [24].…”