This research is devoted to the solution of the barotropic vorticity equation on the gnomonic cubed sphere grid by using the high-order finite difference schemes. The classical fourth-order Runge-Kutta method and the leapfrog method with a second-order first-degree polynomial regression time filter have been used for time integration. The spatial discretization is based on the explicit centered finite difference schemes of 2nd-, 4th- and 6th-order and artificial dissipation is provided with the use of high-order filters.

A well-known test case for the barotropic vorticity equation on the sphere is the Rossby-Haurwitz test case, which is relatively well-behaved and has an analytical solution. While it can be used to examine the accuracy of the schemes and to make sense of basis of model behavior in rather simple flow configurations, the absence of active cascade of enstrophy makes it less challenging for high-order methods. Indeed, the preliminary results for the error norms and diagnostics exhibited the expected convergence rates for the finite-difference methods examined. Therefore, a jet instability has been used to assess various aspects of the model. The jet instability test case for barotropic vorticity equation has been constructed based on the inviscid variant of the well-known jet instability test case for shallow water equations. The flow resulting from this test case is more complicated than the Rossby-Haurwitz wave, due to the presence of an active cascade of enstrophy, generation of very small scales and lack of an analytical solution. The unstable basic state in this test could lead to exponential growth of computational errors, making it rather susceptible to grid imprinting. For the jet instability test case, solution of the pseudospectral method with triangular truncation T3071 has been provided as a benchmark.

To assess the accuracy and conservation properties of the model as well as the effect of the artificial dissipation on the solution, several error norms and diagnostics have been used. These diagnostics are absolute enstrophy, total kinetic energy, maximum and minimum absolute vorticity, maximum speed and maximum magnitude of vorticity gradient.

In the jet instability test case, the distinction between low- and high-order methods is quite pronounced. This is because the computational errors grow exponentially within the unstable basic state. In this test case, there is significant downscale cascade of enstrophy and very small scales are generated in the flow. This means that the effect of the filter on the flow is also significant. The error diagnostics could be divided into three groups: 1) those affected only by the order of accuracy of the finite difference operators, 2) those that are affected by the application of the filter, 3) and those which are due to generation of unresolved scales in the flow. The effect of the time integration scheme on the solution was minimal. The overall features of the flow, generation of steep gradients and conservation of kinetic energy are all better simulated by the higher-order finite-difference methods. Results demonstrate the superiority of the high-order methods in terms of accuracy and performance. Though, generalizing these conclusions to multi-scale problems or ones with lower differentiability classes must be done with care.