A higher order tracer transport scheme for icosahedral hexagonal grids

Weather and climate models require both efficient and accurate numerical methods to simulate tracer (e.g., moisture, salinity) advection. The distribution of the tracer used in transport equations is approximated by a Taylor expansion. A scheme is developed that builds upon a simpler, second-order convergent method. This original method describes tracer distributions with a first-order Taylor expansion, while the extension uses a second-order expansion to describe the distribution. The original method is conservative and defines a simple departure region, but violates monotonicity preservation. This scheme, due to its simplicity, is not very accurate with more complex tracer and velocity flow configurations. The extension of the method requires three specific modifications: Green's Theorem is used to calculate these next order terms and minimize the computational stencil, Gauss Quadrature is employed to calculate the tracer advected in a departure region, and the cell-averaged value is re-normalized to correct for the addition of these higher order terms. Two tests are run on a planar, perfect hexagonal grid: a solid body rotational case and a time-dependent deformational-flow case. The extension of the tracer distribution function shows marked improvements over the original method, and this extended scheme is third-order convergent for the solid-body rotation case. The improvements, however, are not as obvious when in the deformational-flow test. Nonetheless, the results indicate that the scheme warrants further testing. The successful application of a flux limiter shows that the method can be prepared further for possible implementation into weather and climate models.