An Accurate Semi-Lagrangian Scheme with a Unified Interpolation Function Constructed from Vorticity and Velocity Components(<Special Issue>CIP)

For atmospheric and oceanic modeling, the equations of motion are numerically solved in either momentum form or vorticity form. Since vorticity is a conservative quantity in the Lagrangian sense, it has been considered that the vorticity form discretization scheme is more appropriate for the simulation of atmospheric and oceanic flows. However, it requires a Poisson solver to obtain the streamfunction from the vorticity: the use of a Poisson solver is thought to be a drawback for high-resolution atmospheric and oceanic modeling. In contrast, a Poisson solver is not required if the momentum form discretization scheme is applied to compressible flows. In this study, we propose a new advection scheme which possesses the advantages of both schemes: conservation of vorticity and no need for a Poisson solver. Both velocity components and vorticity are temporally integrated using the semi-Lagrangian method by constructing a unified interpolation function for velocity components and vorticity. We apply this scheme to a two-dimensional shear instability problem, and have found that this scheme gives a result competitive with the vorticity form scheme and is more accurate than the momentum form scheme.