Many types of atmospheric and oceanic motions possess an oscillatory structure in both space and time, such as inertia-gravity waves and Rossby waves. This paper concentrates on these waves. Two-dimensional shallow-water models are usually used to describe these kinds of waves. The advantages of the shallow-water equations are their computational and mathematical simplicity relative to more complicated three-dimensional models. The single-layer shallow water models are extensively used in the numerical study of large scale atmospheric and oceanic motions. These simple models, however, provide no information regarding vertical motions in the atmosphere or oceans. The simple multilayer shallow-water models are usually employed to resolve this issue. In some regions of the oceans and seas, two-layer shallow-water equations are sufficient to account for the dynamics of fluids (e.g., the Strait of Gibraltar in the Atlantic Ocean, the Mediterranean Sea and the Strait of Hormoz in the Persian Gulf and the Oman Sea). Two-layer models are not only simple models of rotating-stratified fluid dynamics, but they are also proper models for the simulation of many phenomena in the ocean and atmosphere.
The compact finite-difference schemes are simple and powerful ways to reach the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite-difference schemes of the same-order, compact schemes have proved to be significantly more accurate along with the benefit of using smaller stencil sizes, which can be essential in treating non-periodic boundary conditions. Applications of some families of the compact schemes, in particular the super compact finite difference method and the combined compact finite difference method, to spatial differencing in some idealized models of the atmosphere and oceans show that compact finite difference schemes can be considered as promising methods for the numerical simulation of atmosphere–ocean dynamics. Most of these studies apply compact finite difference methods to single-layer models of the atmosphere and oceans, but application to more complicated multi-layer model is lacking.
The linearized single-layer shallow-water equations have been used in many research studies as a tool for numerical accuracy assessment of different numerical schemes in a linear extent. In the present work, the extension of this idea to two-layer shallow-water equations is used. To this end, two general discrete dispersion relations, those of inertial-gravity and Rossby waves, for the linearized two-layer shallow-water equations on different numerical grids are derived. These general discrete dispersion relations can be used for the evaluation of the performance of any numerical scheme.
This paper is also focused on accuracy assessment of the sixth-order super compact (SCFDM) and sixth-order combined compact (CCFDM) finite difference schemes for spatial differencing of the linearized two-layer shallow-water equations on different numerical grids (i.e., Arakawa's A-E and Randall's Z grids). General discrete dispersion relations derived for the inertial-gravity waves and Rossby waves on different numerical grids are used to evaluate the accuracy of the sixth-order SCFDM and sixth-order CCFDM schemes for spatial differencing of the linearized two-layer shallow-water equations.
In general, for both inertia-gravity and Rossby waves, the minimum error occurs on the Z grid using either the sixth-order SCFDM or the sixth-order CCFDM method. For Randall's Z grid, it is observed that the sixth-order CCFDM method exhibits a substantial improvement in measuring the frequency of linear inertia-gravity waves of the two-layer shallow-water model on the sixth-order SCFDM method. This property is not observed for other numerical grids. For Rossby waves, the sixth-order CCFDM shows improvement on the sixth-order SCFDM method on Arakawa’s C grid. In addition, for Arakawa’s C grid it can be observed that the baroclinc and barotropic modes of the inertia gravity waves in the under-resolved case show dissimilar behavior.