Accuracy assessment of sixth order combined compact and super compact methods on C-D and LE grids: Representation of linear inertia-gravity and Rossby waves

The oceanic and atmospheric models have been developed on different numerical grids. The Arakawa's C grid is well-known because of the advantages of the C-grid discretization at high resolutions. The C grid, however, is well suited for reproducing high frequency inertia-gravity waves in resolved cases, but there are difficulties in dealing with the Coriolis terms and low-frequency processes. In particular, the C-grid approach is unfavorable in the under-resolved cases with grid-scale noise. Several fixes have been proposed for the C-grid problem. One such method is the C-D grid approach which improves spectral properties of the inertia-gravity waves at low resolutions. The C-D grid approach employs a combination of the C and D grids such that all terms are the same as in a conventional C-grid discretization except for the Coriolis terms where the D-grid velocities are used so that they require no interpolation. Another grid is the LE grid that comprises the same structure of Arakawa’s E grid with a different grid space. Most of these studies apply the traditional second-order finite difference method to spatial differencing on the C-D grid, but the application to higher accurate finite difference methods is lacking. Finite difference methods are commonly used to simulate the dynamical behavior of geophysical fluids. Numerical simulations of the complicated flows such as vortices, turbulent currents and instabilities need high accuracy methods as well as high resolutions. The compact finite difference methods are powerful ways to reach the objectives of high accuracy and low computational cost. The super compact and combined compact finite difference methods can be considered as promising methods for large scale computations in atmosphere–ocean dynamics with high accuracy. In this study, we derived the general discrete dispersion relations of inertia-gravity and Rossby waves on the C-D and LE grids. The linearized single-layer and two-layershallow-water models were used to describe these kinds of waves which play an important role in the setup of the ocean circulation. These relations were used to assess the performances of the sixth-order super compact finite difference (SCD6) and sixthorder combined compact finite difference (CCD6) schemes on the C-D and LE grids. The results on these grids were compared to Randall’s Z grid and Arakawa’s C and D grids. The general discrete dispersion relations of inertia-gravity waves on the C-D grid were similar to the LE grid at both single layer and two-layer models, but they were different for Rossby waves. The results of the present work revealed that the CCD6 scheme exhibits a substantial improvement over the SCD6 scheme for the frequency and group velocity of inertia– gravity waves on the C-D and LE grids. In the same manner, for the frequency of Rossby waves, the performance of the CCD6 scheme is better than that of SCD6 scheme, but for the group velocity of Rossby waves, the SCD6 scheme is slightly more accurate than CCD6 scheme. In general, the C-D grid is, however, composed of Arakawa’s C and D grids which are susceptible to grid scale noise but its behavior is favorable for both inertia-gravity and Rossby waves. In addition, for inertia-gravity waves, it could be observed that the accuracy of the SCD6 scheme on the C-D grid is similar to the Z grid and even the CCD6 scheme exhibits higher accuracy on the C-D grid.