Backward Integration of Nonlinear Shallow Water Model: Part 2: Vortex Merger

Using the same initial condition in vortex merger, the backward integrations of height and vorticity from Sun’s shallow water model are the images of forward integrations. The system becomes very stable after long forward or backward integration (integration time > 120). If we integrate forward in time from initial t = t0 to t = tf, then, use the result as the initial condition to integrate backward from tf to t0. The difference between backward integration and the forward initial system at t = t0 increases as integration time increases. The difference is small if tf is less than 40 and backward integration can retrieve the original initial pattern well. If tf > 60, the difference becomes noticeable and the backward integration cannot retrieve the original pattern because of the accumulation of small difference generated at each time-step between forward and backward numerical calculations. The reverse time can be extended using implicit schemes or more complicated time-step schemes.