An efficient multi-resolution grid for global models and coupled systems

The latitude-longitude (lat-lon) grid is the most widely used global coordinate system for various purposes but its singularity at the Pole and the vector polar problems associated with its converging meridians hinder its applications. Well from the very start of numerical modelling history, quite a few grids have been attempted to tackle these problems and reduced grid is the simplest one among other grids. However, the reduced grid is almost abandoned by modern numerical modellers due to its unsatisfactory results for dynamical models in the polar region. Spherical multiple-cell (SMC) grid is similar to the reduced grid apparently but uses the unstructured technique for efficiency. It merges longitudinal cells at high latitudes like the reduced grid to overcome the CFL restriction and introduces a polar cell to remove the polar singularity. It also supports quad-tree-like mesh refinement to form a multi-resolution grid. To tackle the vector polar problem associated with the increased curvature at high latitudes, the SMC grid uses a new fixed reference direction to define vector components near the poles for improved polar performance. Global transportation is quite efficient on the SMC grid with optional second or third order transportation scheme. Present applications of the SMC grid, particularly in ocean surface wave models, are presented and possible future usage in global models and coupled systems are proposed.

Global Transport on a Spherical Multiple-Cell Grid

Second- and third-order upstream nonoscillatory (UNO) advection schemes are applied on a spherical multiple-cell (SMC) grid for global transport. Similar to the reduced grid, the SMC grid relaxes the Courant–Friedrichs–Lewy (CFL) restriction of the Eulerian advection time step on the conventional latitude–longitude grid by zonally merging cells toward the poles. Round polar cells are introduced to remove the polar singularity of the spherical coordinate system. The unstructured feature of the SMC grid allows unused cells to be removed out of memory and transport calculations. Solid-body rotation and deformation flow tests are used for comparison with other transport schemes. Application on the global ocean surface is used to demonstrate the flexibility of the SMC grid by removing all land points and making possible the extension of global ocean surface wave models to cover the Arctic in response to the retreating sea ice in recent summers. Numerical results suggest that UNO schemes on the SMC grid are suitable for global transport.

Averaging of Earth-Crossing Orbits

The orbits of planet-crossing asteroids (and comets) can undergo close approaches and collisions with some major planet. This introduces a singularity in the N-body Hamiltonian, and the averaging of the equations of motion, traditionally used to compute secular perturbations, is undefined. We have shown (Gronchi and Milani, 1998) that it is possible to define in a rigorous way some generalised averaged equations of motion, in such a way that the generalised solutions are unique and piecewise smooth, with corners on the node crossing lines.The model is the averaged equations of motion first introduced by Kozai (1962): the perturbing planets are assumed to move in circular, coplanar orbits, and the equations of motion are averaged over the anomalies of the asteroid and of the planets. In the non-crossing case the averaging is integrable; in the planet-crossing case there is a polar singularity of order two in the equations of motion, and averaging is not possible. To define a generalized solution, we decrease the order of the polar singularity by the method of extraction of the singularities by Kantorovich. The singularity of the perturbing function is approximated by a modified inverse distance, the one between the straight lines tangent to the two orbits at the nodal points. In this approximation the averaged perturbing function has an analytical expression, allowing explicit computation with elliptic integrals and elementary functions.