Jian-Guo Li

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.

The shallow‐water equations (SWEs) are discretized on a spherical multiple‐cell (SMC) grid and validated with classical tests, including steady zonal flow over flat or hill floor, the Rossby–Haurwitz wave and the unstable zonal jet. The numerical schemes follow the conventional latitude–longitude (lat‐lon) grid ones at low latitudes but switch to a fixed‐reference direction system to define vector variables at high latitudes for reduction of polar curvature errors. Semi‐implicit schemes are used for both the Coriolis terms and the potential energy gradients. A C‐grid mass‐conserving advection–diffusion scheme is used for the thickness variable and mixed A‐ and D‐grid schemes are applied on the momentum equation. Tests demonstrate that the two reference directions work fine on the SMC grid and the SWEs model is stable as long as numerical noises are suppressed with enough smoothing. This implies that other reduced grids could be used for dynamical models if the vector polar problem is properly solved. The unstructured SMC grid also supports multi‐resolutions in mesh refinement style and a refined area is included to show grid refinement effects. For steady smooth flows, the refinement is almost unnoticeable while in unstable jet case, it may stir up instability ripples in the jet flow unless strong average is applied.

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.

The Advanced Synthetic Aperture Radar (ASAR) on board the Envisat satellite is an important resource for observation of global ocean surface wave spectra. However, assessment of this valuable dataset is not straightforward as a result of a lack of other independent ocean wave spectral observations. The radar altimeter (RA-2) on board the same satellite measures ocean wave height at the same time as the ASAR but at a location about 200 km distant. A small number of moored buoys produce one-dimensional (1D) ocean wave spectra but few ASAR spectra fall on the buoy positions in a given period. Indirect comparison of the Envisat ASAR 2D wave spectra with the RA-2 wave heights and 1D spectra of three selected buoys from July 2004 to February 2006 is facilitated by a wave model, which provides coherent spatial and temporal links between these observations. In addition to the conventional significant wave height (SWH), four spectral subrange wave heights (SRWHs) are used to illustrate the spectral characteristics of these observations. A comparison of three Envisat ASAR 2D spectra with the closest model and buoy spectra is also attempted to illustrate the qualities of these different observations and to demonstrate the restrictions to their direct comparison. Results indicate that these three independent observations are in good agreement in terms of SWH, though the Envisat ASAR shows the largest variance. Comparison of SRWHs indicates that the ASAR spectra agree well with buoy and model in moderately long waves, but the ASAR instrument does not resolve high-frequency waves, especially along the satellite track.