Preferred equilibrium solutions of the barotropic vorticity equation

The long-term behaviour of the barotropic vorticity equation (BVE), which is beneficial for understanding long-term climate variations, has been investigated by applying an explicit quadratic conserved finite difference integration scheme. The results suggest that an initial pattern predominated at the beginning of time evolves to an equilibrium state after a long transition period in which the conserved prerequisites, such as the integral kinetic energy, square vorticity, and vorticity, are broken by accumulated errors. This could cause the BVE to be indeterministic. Similar phenomena could reappear in integration cases starting at the same initial pattern with different small perturbations, initial patterns, time and grid intervals, and difference schemes. The eventual equilibrium states for the above cases are different from each other but spatially similar. They all possess a single-wave structure in the horizontal plane but with random phases. The results further imply that despite the indeterministic characteristics due to the nonlinearity, the long-term behaviour of the BVE is deterministic in a spatial structure, shedding light on long-term climate change studies.